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0 1 0 0]def/BuildChar{exch begin m 0 3 -1 roll put l setfont m stringwidth setcharwidth .05 setlinewidth 0 setlinecap FontInfo/UnderlinePosition get 1000 div dup -.07 exch moveto .07 exch m stringwidth 3 -1 roll add 3 1 roll add exch lineto stroke 0 0 moveto m show end}def end c o definefont}ifelse}if end}a /EPSBegin{save userdict/BeachHead get begin/la b count/ma b userdict/showpage{}put end 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath}a /EPSEnd{userdict/BeachHead get begin count ma sub dup 0 gt{{pop}repeat}{pop}ifelse la end restore}a /cimage{userdict/BeachHead get begin{{readstring}}{{readhexstring}}ifelse/u b/colorimage where {pop 4 index dup string/v b dup string/w b dup string/x b dup string/y b string/z b {currentfile v u pop}{currentfile w u pop}{currentfile x u pop}{currentfile y u pop currentfile z u pop pop}5 -1 roll{true 4 A}{true 4/colorimage load exec}ifelse}{4 index dup string/z b 4 mul string/B b{currentfile B u pop pop currentfile z u pop} exch{transimage}{/image load exec}ifelse}ifelse end}a/C{D{gsave E F 3 index idtransform translate G 1 4 index 4 index{H}/image load exec grestore/I 0 def/G 0 def/D false def}if}a /transimage{userdict/BeachHead get begin 2 index 8 ne{/image load exec}{4 index cvi string /H b/J 0 string def/K 0 def/D false def/I 0 def/G 0 def 0 1 5 index 1 sub{/F b 0 1 6 index 1 sub {K J length ge{1 index dup type/stringtype ne{exec}if/J b/K 0 def}if J K get/K K 1 add def dup 255 eq{pop pop C}{H I 3 -1 roll put/I I 1 add def/G G 1 add def D not{/E b/G 1 def /D true def}{pop}ifelse}ifelse}for C}for 5{pop}repeat}ifelse end}a/L{D{gsave E F 8 index idtransform translate I 1 8 9 index{M}{N}{O}{P}true 4/colorimage load exec grestore/I 0 def /D false def}if}a/A{9 index cvi dup string/M b dup string/N b dup string/O b string/P b /Q 0 string def/K 0 def/D false def/I 0 def/G 0 def 0 1 10 index 1 sub{/F b 0 1 11 index 1 sub {K Q length ge{6 index exec/R b 5 index exec/S b 4 index exec/T b 3 index exec/Q b/K 0 def }if R K get S K get T K get Q K get/K K 1 add def dup 0 eq 2 index 0 eq and 3 index 0 eq and 4 index 0 eq and{5{pop}repeat L}{M I 6 -1 roll put N I 5 -1 roll put O I 4 -1 roll put P I 3 -1 roll put/I I 1 add def D not{/E b/D true def}{pop}ifelse}ifelse}for L}for 10{pop}repeat}a /bps 8 string def/bpm[8 0 0 8 0 0]def/bpp{bps}def/overlaybackpat{userdict/BeachHead get begin gsave setrgbcolor bps copy pop dup 0 get 8 div floor cvi 8 mul 1 index 2 get 8 div floor cvi 8 mul 2 index 1 get 8 div floor cvi 8 mul 8 4 index 3 get 8 div floor cvi 8 mul{2 index 8 3 index{1 index gsave translate 8 8 scale 8 8 false bpm/bpp load imagemask grestore}for pop}for pop pop pop grestore end}a /U{userdict/BeachHead get begin/V b/W b countdictstack save V 2 add 2 roll count V sub /X b /W load end{exec}stopped userdict/BeachHead get begin/Y b count X sub{pop}repeat Y 3 1 roll restore countdictstack exch sub{end}repeat end}a/Z( )def/aa{moveto{ba setfont Z end gsave 0 setgray stringwidth grestore userdict/BeachHead get begin rmoveto/ca load null ne{/da da 1 add def da ea length le{fa ea da get ca}if}{ax ay rmoveto fa ga eq{cx cy rmoveto}if}ifelse}stopped currentdict userdict/BeachHead get ne{userdict/BeachHead get begin}if}a /filltextpath{userdict/BeachHead get begin/ea b dup type dup/integertype eq exch/realtype eq or {/ay b/ax b/ga b/cy b/cx b/ca null def}{/ca b}ifelse/ha b/ia b ia{currentfont/_isSBShadowFont get {gsave 0 setgray/ca load null ne{/ca load ea kshow}{cx cy ga ax ay ea awidthshow}ifelse grestore}if}if gsave currentfont ia{begin r FontMatrix makefont l FontMatrix makefont end }{null exch}ifelse/ja b/ka b/ba currentfont def _doTexturePat{systemdict/makepattern known} {false}ifelse{matrix currentmatrix _strtxtmatrix null ne{_strtxtmatrix setmatrix}if 1 -1 scale txTrnsX txTrnsY translate settexturepat setmatrix/da 0 def ea{/fa b Z 0 fa put ja setfont currentpoint Z show aa{exit}if}forall}{10 setlinewidth/da 0 def currentpoint newpath 0 dup dup dup moveto lineto closepath moveto ea{/fa b Z 0 fa put currentpoint ja setfont count 1 add dup 1 roll Z true{charpath}stopped count count -1 roll sub{pop}repeat currentpoint{ha}0 U pop newpath 0 dup dup dup moveto lineto closepath moveto aa{exit}if}forall}ifelse grestore ka null ne{gsave 0 setgray/da 0 def ea{/fa b Z 0 fa put ka setfont currentpoint Z show aa{exit}if}forall grestore}if /_doTexturePat false def/_strtxtmatrix null def end}a/na[256{0}repeat]def mark 161 176 173 185 176 165 177 177 178 163 179 179 181 109 182 182 183 229 184 213 185 112 186 242 189 87 195 214 197 187 198 68 214 184 215 224 240 240 counttomark 2 div cvi{na 3 1 roll put}repeat /krnshow{dup type dup/integertype ne exch/realtype ne and{12}if/Symbol findfont exch scalefont/oa exch def/ua currentfont def/pa 0 def exch{dup na exch get dup 0 eq isWinPS or{pop Z 0 3 -1 roll put currentpoint Z userdict begin show end moveto}{oa setfont Z 0 3 -1 roll put currentpoint Z show moveto ua setfont pop}ifelse dup pa get/pa pa 1 add def 1 index pa get/pa pa 1 add def rmoveto }forall pop}a/setcmykcolor where{pop/bhsetcmykcolor/setcmykcolor load def}{/bhsetcmykcolor {4 1 roll 3{3 index add neg 1 add dup 0 lt{pop 0}if 3 1 roll}repeat setrgbcolor pop}a}ifelse end BeachHead begin/isWinPS false def end 3 setflat gsave gsave newpath [1 0 0 -1 57.2495 297.408] concat -56.9992 -30.0991 moveto 57.0016 -30.0991 lineto 57.0016 35.4002 lineto -56.9992 35.4002 lineto -56.9992 -30.0991 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore grestore gsave newpath [1 0 0 -1 57.2495 297.408] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 101.752 310.008] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (y) [4.49 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 57.2495 297.408] concat grestore gsave newpath [1 0 0 -1 57.2495 297.408] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 66.7513 276.008] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (n) [5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 57.2495 297.408] concat grestore grestore gsave gsave newpath [1 0 0 -1 303 297.408] concat -56.9992 -30.0991 moveto 57.0016 -30.0991 lineto 57.0016 35.4002 lineto -56.9992 35.4002 lineto -56.9992 -30.0991 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore grestore gsave newpath [1 0 0 -1 303 297.408] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 347.503 310.008] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (y) [4.49 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 303 297.408] concat grestore gsave newpath [1 0 0 -1 303 297.408] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 312.502 276.008] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (n) [5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 303 297.408] concat grestore grestore gsave gsave newpath [1 0 0 -1 181.67 297.408] concat -56.9992 -30.0991 moveto 57.0016 -30.0991 lineto 57.0016 35.4002 lineto -56.9992 35.4002 lineto -56.9992 -30.0991 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore grestore gsave newpath [1 0 0 -1 181.67 297.408] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 226.172 310.008] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (y) [4.49 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 181.67 297.408] concat grestore gsave newpath [1 0 0 -1 181.67 297.408] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 191.171 276.008] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (n) [5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 181.67 297.408] concat grestore grestore gsave gsave newpath [1 0 0 -1 57.2495 171.761] concat -56.9992 -30.0991 moveto 57.0016 -30.0991 lineto 57.0016 35.4002 lineto -56.9992 35.4002 lineto -56.9992 -30.0991 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore grestore gsave newpath [1 0 0 -1 57.2495 171.761] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 101.752 184.361] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (y) [4.49 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 57.2495 171.761] concat grestore gsave newpath [1 0 0 -1 57.2495 171.761] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 66.7513 150.361] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (n) [5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 57.2495 171.761] concat grestore grestore gsave gsave newpath [1 0 0 -1 56.9992 97.1451] concat -56.9992 -30.0991 moveto 57.0016 -30.0991 lineto 57.0016 35.4002 lineto -56.9992 35.4002 lineto -56.9992 -30.0991 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore grestore gsave newpath [1 0 0 -1 56.9992 97.1451] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 101.501 109.745] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (y) [4.49 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 56.9992 97.1451] concat grestore gsave newpath [1 0 0 -1 56.9992 97.1451] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 66.501 75.745] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (n) [5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 56.9992 97.1451] concat grestore grestore gsave gsave newpath [1 0 0 -1 70.7318 158.618] concat -55.9992 -27.4997 moveto 29.0369 -27.4997 lineto 29.0369 6.5152 lineto -55.9992 6.5152 lineto -55.9992 -27.4997 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 setlinewidth 0 setlinecap 0 setlinejoin stroke grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 23.0645 178.905] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (Is the transpose ) [2.49 0 4.49 0 3.15 0 2.49 0 5.00 0 5.00 0 3.15 0 2.49 0 2.99 0 5.00 0 5.00 0 4.49 0 5.00 0 5.00 0 4.49 0 5.00 0 3.15 0 ] 9.00 krnshow end grestore gsave userdict begin BeachHead end begin [1 0 0 -1 23.0645 178.905] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 17.5675 moveto [1 0 0 -1 0 0] concat (available?) [5.00 0 4.49 0 5.00 0 1.99 0 1.99 0 5.00 0 5.00 0 1.99 0 5.00 0 5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave newpath [1 0 0 -1 70.7318 83.8582] concat -55.9992 -27.4997 moveto 29.0369 -27.4997 lineto 29.0369 6.5152 lineto -55.9992 6.5152 lineto -55.9992 -27.4997 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 setlinewidth 0 setlinecap 0 setlinejoin stroke grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 31.6318 104.289] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (Is storage at) [2.49 0 4.49 0 3.15 0 4.49 0 2.49 0 5.00 0 2.99 0 5.00 0 5.00 0 5.00 0 3.15 0 5.00 0 2.49 0 ] 9.00 krnshow end grestore gsave userdict begin BeachHead end begin [1 0 0 -1 31.6318 104.289] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 17.5675 moveto [1 0 0 -1 0 0] concat (a premium?) [5.00 0 3.15 0 5.00 0 2.99 0 5.00 0 7.49 0 1.99 0 5.00 0 7.49 0 5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave newpath [1 0 0 -1 316.483 215.971] concat -55.9992 -27.4997 moveto 29.0369 -27.4997 lineto 29.0369 6.5152 lineto -55.9992 6.5152 lineto -55.9992 -27.4997 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 setlinewidth 0 setlinecap 0 setlinejoin stroke grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 288.359 230.857] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (Try CG) [5.17 0 2.99 0 4.49 0 3.15 0 6.49 0 7.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave newpath [1 0 0 -1 70.7318 284.265] concat -55.9992 -27.4997 moveto 29.0369 -27.4997 lineto 29.0369 6.5152 lineto -55.9992 6.5152 lineto -55.9992 -27.4997 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 setlinewidth 0 setlinecap 0 setlinejoin stroke grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 32.1387 304.552] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (Is the matrix) [2.49 0 4.49 0 3.15 0 2.49 0 5.00 0 5.00 0 3.15 0 7.49 0 5.00 0 2.49 0 2.99 0 1.99 0 4.49 0 ] 9.00 krnshow end grestore gsave userdict begin BeachHead end begin [1 0 0 -1 32.1387 304.552] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 17.5675 moveto [1 0 0 -1 0 0] concat (symmetric?) [4.49 0 4.49 0 7.49 0 7.49 0 5.00 0 2.49 0 2.99 0 1.99 0 4.49 0 5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave newpath [1 0 0 -1 195.152 284.265] concat -55.9992 -27.4997 moveto 29.0369 -27.4997 lineto 29.0369 6.5152 lineto -55.9992 6.5152 lineto -55.9992 -27.4997 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 setlinewidth 0 setlinecap 0 setlinejoin stroke grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 156.559 304.552] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (Is the matrix) [2.49 0 4.49 0 3.15 0 2.49 0 5.00 0 5.00 0 3.15 0 7.49 0 5.00 0 2.49 0 2.99 0 1.99 0 4.49 0 ] 9.00 krnshow end grestore gsave userdict begin BeachHead end begin [1 0 0 -1 156.559 304.552] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 17.5675 moveto [1 0 0 -1 0 0] concat (definite?) [5.00 0 5.00 0 2.49 0 1.99 0 5.00 0 1.99 0 2.49 0 5.00 0 5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave newpath [1 0 0 -1 316.483 284.265] concat -55.9992 -27.4997 moveto 29.0369 -27.4997 lineto 29.0369 6.5152 lineto -55.9992 6.5152 lineto -55.9992 -27.4997 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 setlinewidth 0 setlinecap 0 setlinejoin stroke grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 261.972 304.552] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (Are the outer) [6.00 0 2.99 0 5.00 0 3.15 0 2.49 0 5.00 0 5.00 0 3.15 0 5.00 0 5.00 0 2.49 0 5.00 0 2.99 0 ] 9.00 krnshow end grestore gsave userdict begin BeachHead end begin [1 0 0 -1 261.972 304.552] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 17.5675 moveto [1 0 0 -1 0 0] concat (eigenvalues known?) [5.00 0 1.99 0 5.00 0 5.00 0 5.00 0 4.49 0 5.00 0 1.99 0 5.00 0 5.00 0 4.49 0 3.15 0 4.49 0 5.00 0 5.00 0 6.49 0 5.00 0 5.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave newpath [1 0 0 -1 429.483 284.265] concat -55.9992 -27.4997 moveto 29.0369 -27.4997 lineto 29.0369 6.5152 lineto -55.9992 6.5152 lineto -55.9992 -27.4997 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 setlinewidth 0 setlinecap 0 setlinejoin stroke grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 385.627 304.552] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 6.76758 moveto [1 0 0 -1 0 0] concat (Try Chebyshev) [5.17 0 2.99 0 4.49 0 3.15 0 6.49 0 5.00 0 5.00 0 5.00 0 4.49 0 4.49 0 5.00 0 5.00 0 4.49 0 ] 9.00 krnshow end grestore gsave userdict begin BeachHead end begin [1 0 0 -1 385.627 304.552] concat /Helvetica findbeachheadfont 9.00 scalefont setfont userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 17.5675 moveto [1 0 0 -1 0 0] concat (or CG) [5.00 0 2.99 0 3.15 0 6.49 0 7.00 0 ] 9.00 krnshow end grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave newpath [1 0 0 -1 70.7318 6.7652] concat -55.9992 -27.4997 moveto 29.0369 -27.4997 lineto 29.0369 6.5152 lineto -55.9992 6.5152 lineto -55.9992 -27.4997 lineto closepath userdict begin BeachHead end begin 0 0 0 0 bhsetcmykcolor end gsave eofill grestore userdict begin BeachHead end begin 0 0 0 1 bhsetcmykcolor end 0 setlinewidth 0 setlinecap 0 setlinejoin stroke grestore grestore gsave newpath [1 0 0 -1 251.197 96.2078] concat grestore gsave gsave userdict begin BeachHead end begin [1 0 0 -1 24.9006 27.052] concat /Helvetica findbeachheadfont 9.00 scalefont setfont 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b(INTR)o(ODUCTION)450 391 y FC(the)11 b(linear)f(system)g(only)g(via)g(the)h(matrix-v)o(ector)e (pro)q(duct)i Fy(y)j FC(=)d Fy(A)r Fu(\001)r Fy(x)g FC(\(and)f(p)q(erhaps)i Fy(z)i FC(=)d Fy(A)1905 376 y Fx(T)1934 391 y Fu(\001)r Fy(x)p FC(\).)450 441 y(Th)o(us)h(the)h(user)g(need)g(only)e(supply)h(a)g (subroutine)g(for)g(computing)e Fy(y)k FC(\(and)e(p)q(erhaps)h Fy(z)r FC(\))g(giv)o(en)e Fy(x)p FC(,)450 491 y(whic)o(h)j(p)q(ermits)f(full) g(exploitation)f(of)i(the)g(sparsit)o(y)g(or)g(other)g(sp)q(ecial)h (structure)h(of)d Fy(A)p FC(.)512 541 y(W)m(e)f(b)q(eliev)o(e)h(that)g(after) g(reading)f(this)h(b)q(o)q(ok,)f(applications)f(dev)o(elop)q(ers)j(will)d(b)q (e)j(able)e(to)g(use)450 591 y(templates)i(to)g(get)g(their)h(program)e (running)h(on)g(a)g(parallel)f(mac)o(hine)g(quic)o(kly)m(.)18 b(Nonsp)q(ecialists)450 640 y(will)d(kno)o(w)h(ho)o(w)g(to)h(c)o(ho)q(ose)g (and)f(implemen)o(t)e(an)i(approac)o(h)h(to)f(solv)o(e)g(a)h(particular)f (problem.)450 690 y(Sp)q(ecialists)f(will)e(b)q(e)i(able)g(to)f(assem)o(ble)g (and)h(mo)q(dify)d(their)j(co)q(des|without)g(ha)o(ving)e(to)i(mak)o(e)450 740 y(the)i(h)o(uge)g(in)o(v)o(estmen)o(t)e(that)i(has,)g(up)f(to)h(no)o(w,)f (b)q(een)i(required)f(to)f(tune)i(large-scale)e(applica-)450 790 y(tions)g(for)h(eac)o(h)g(particular)f(mac)o(hine.)25 b(Finally)m(,)15 b(w)o(e)i(hop)q(e)g(that)g(all)e(users)j(will)e(gain)f(a)i(b)q(etter)450 840 y(understanding)f(of)e(the)i(algorithms)c(emplo)o(y)o(ed.)20 b(While)14 b(education)i(has)f(not)g(b)q(een)h(one)f(of)g(the)450 889 y(traditional)10 b(goals)h(of)g(mathematical)e(soft)o(w)o(are,)i(w)o(e)h (b)q(eliev)o(e)g(that)g(our)g(approac)o(h)f(will)g(go)g(a)g(long)450 939 y(w)o(a)o(y)i(in)h(pro)o(viding)e(suc)o(h)j(a)e(v)n(aluable)g(service.) 450 1074 y Fp(1.1)70 b(Wh)n(y)23 b(Use)g(T)-6 b(emplates?)450 1165 y FC(T)m(emplates)9 b(o\013er)i(three)h(signi\014can)o(t)e(adv)n(an)o (tages.)17 b(First,)11 b(templates)f(are)h(general)f(and)h(reusable.)450 1215 y(Th)o(us,)h(they)g(can)h(simplify)c(p)q(orts)j(to)g(div)o(erse)h(mac)o (hines.)j(This)c(feature)g(is)g(imp)q(ortan)o(t)e(giv)o(en)i(the)450 1264 y(div)o(ersit)o(y)i(of)f(parallel)g(arc)o(hitectures.)512 1314 y(Second,)h(templates)e(exploit)h(the)h(exp)q(ertise)h(of)d(t)o(w)o(o)h (distinct)g(groups.)18 b(The)c(exp)q(ert)g(n)o(umer-)450 1364 y(ical)i(analyst)g(creates)j(a)e(template)f(re\015ecting)i(in-depth)f(kno)o (wledge)f(of)h(a)f(sp)q(eci\014c)j(n)o(umerical)450 1414 y(tec)o(hnique.)g (The)14 b(computational)d(scien)o(tist)j(then)g(pro)o(vides)g(\\v)n (alue-added")e(capabilit)o(y)g(to)i(the)450 1464 y(general)e(template)e (description,)i(customizing)e(it)h(for)g(sp)q(eci\014c)h(con)o(texts)h(or)e (applications)f(needs.)512 1513 y(And)16 b(third,)g(templates)f(are)i(not)e (language)g(sp)q(eci\014c.)25 b(Rather,)16 b(they)h(are)f(displa)o(y)o(ed)f (in)g(an)450 1563 y(Algol-lik)o(e)i(structure,)23 b(whic)o(h)d(is)g(readily)f (translatable)g(in)o(to)g(the)i(target)f(language)f(suc)o(h)h(as)450 1613 y FA(FORTRAN)14 b FC(\(with)h(the)h(use)g(of)f(the)h(Basic)g(Linear)g (Algebra)f(Subprograms,)f(or)i FA(BLAS)p FC(,)e(whenev)o(er)450 1663 y(p)q(ossible\))19 b(and)f FA(C)p FC(.)g(By)h(using)f(these)i(familia)o (r)c(st)o(yles,)k(w)o(e)f(b)q(eliev)o(e)g(that)f(the)h(users)h(will)d(ha)o(v) o(e)450 1713 y(trust)f(in)f(the)h(algorithms.)21 b(W)m(e)14 b(also)h(hop)q(e)h(that)g(users)g(will)e(gain)h(a)g(b)q(etter)i (understanding)f(of)450 1763 y(n)o(umerical)c(tec)o(hniques)k(and)d(parallel) g(programming)o(.)512 1812 y(F)m(or)h(eac)o(h)g(template,)f(w)o(e)h(pro)o (vide)f(some)g(or)h(all)f(of)g(the)i(follo)o(wing:)512 1888 y Fu(\017)21 b FC(a)13 b(mathematical)e(description)j(of)g(the)g(\015o)o(w)g (of)f(the)h(iteration;)512 1965 y Fu(\017)21 b FC(discussion)14 b(of)f(con)o(v)o(ergence)j(and)e(stopping)f(criteria;)512 2042 y Fu(\017)21 b FC(suggestions)11 b(for)e(applying)g(a)h(metho)q(d)g(to)g(sp)q (ecial)g(matrix)e(t)o(yp)q(es)k(\()p Fq(e.g.)p FC(,)e(banded)g(systems\);)512 2119 y Fu(\017)21 b FC(advice)15 b(for)f(tuning)g(\(for)h(example,)e(whic)o (h)h(preconditioners)i(are)f(applicable)f(and)g(whic)o(h)554 2168 y(are)g(not\);)512 2245 y Fu(\017)21 b FC(tips)14 b(on)f(parallel)g (implemen)o(tations;)d(and)512 2322 y Fu(\017)21 b FC(hin)o(ts)14 b(as)g(to)f(when)i(to)e(use)i(a)f(metho)q(d,)e(and)i(wh)o(y)m(.)512 2398 y(F)m(or)g(eac)o(h)g(of)f(the)i(templates,)e(the)h(follo)o(wing)e(can)i (b)q(e)g(obtained)g(via)f(electronic)i(mail.)512 2474 y Fu(\017)21 b FC(a)13 b FA(MATLAB)g FC(implemen)o(tatio)o(n)e(based)k(on)f(dense)h (matrices;)512 2550 y Fu(\017)21 b FC(a)13 b FA(FORTRAN-77)f FC(program)g(with)i(calls)f(to)h FA(BLAS)1312 2535 y Fv(1)1330 2550 y FC(.)512 2626 y(See)h(App)q(endix)f(A)g(for)g(details.)p 450 2655 620 2 v 496 2681 a Fo(1)514 2693 y Fn(F)m(or)f(a)g(discussion)e(of)i Fm(BLAS)f Fn(as)h(building)e(blo)q(c)o(ks,)h(see)h([66)o(,)g(67)o(,)h(68)o(,) f(141)o(])h(and)e Fm(LAPACK)f Fn(routines)h([3)o(].)21 b(Also,)450 2733 y(see)11 b(App)q(endix)f(B.)p eop %%Page: 3 15 14 bop 150 275 a Fr(1.2.)31 b(WHA)m(T)13 b(METHODS)h(ARE)g(CO)o(VERED?)723 b FC(3)150 391 y Fp(1.2)70 b(What)23 b(Metho)r(ds)g(Are)g(Co)n(v)n(ered?)150 482 y FC(Man)o(y)16 b(iterativ)o(e)g(metho)q(ds)g(ha)o(v)o(e)g(b)q(een)h(dev) o(elop)q(ed)g(and)f(it)f(is)h(imp)q(ossible)f(to)h(co)o(v)o(er)h(them)e(all.) 150 532 y(W)m(e)e(c)o(hose)h(the)g(metho)q(ds)e(b)q(elo)o(w)h(either)h(b)q (ecause)h(they)f(illustrate)f(the)h(historical)e(dev)o(elopmen)o(t)150 582 y(of)g(iterativ)o(e)h(metho)q(ds,)f(or)h(b)q(ecause)i(they)e(represen)o (t)i(the)f(curren)o(t)g(state-of-the-art)g(for)e(solving)150 632 y(large)i(sparse)h(linear)e(systems.)18 b(The)d(metho)q(ds)e(w)o(e)h (discuss)h(are:)201 715 y(1.)20 b(Jacobi)201 798 y(2.)g(Gauss-Seidel)201 881 y(3.)g(Successiv)o(e)c(Ov)o(er-Relaxation)c(\(SOR\))201 964 y(4.)20 b(Symmetric)12 b(Successiv)o(e)j(Ov)o(er-Relaxation)e(\(SSOR\)) 201 1047 y(5.)20 b(Conjugate)13 b(Gradien)o(t)h(\(CG\))201 1130 y(6.)20 b(Minimal)11 b(Residual)i(\(MINRES\))h(and)g(Symmetric)e(LQ)i (\(SYMMLQ\))201 1213 y(7.)20 b(Conjugate)13 b(Gradien)o(ts)h(on)g(the)g (Normal)e(Equations)i(\(CGNE)g(and)f(CGNR\))201 1296 y(8.)20 b(Generalized)14 b(Minimal)d(Residual)i(\(GMRES\))201 1379 y(9.)20 b(Biconjugate)14 b(Gradien)o(t)f(\(BiCG\))180 1462 y(10.)20 b(Quasi-Minimal)11 b(Residual)i(\(QMR\))180 1545 y(11.)20 b(Conjugate)13 b(Gradien)o(t)h(Squared)g(\(CGS\))180 1628 y(12.)20 b(Biconjugate)14 b(Gradien)o(t)f(Stabilized)h(\(Bi-CGST)m(AB\))180 1711 y(13.)20 b(Cheb)o(yshev)15 b(Iteration)150 1794 y(F)m(or)c(eac)o(h)h (metho)q(d)f(w)o(e)g(presen)o(t)j(a)d(general)g(description,)i(including)d(a) h(discussion)h(of)f(the)h(history)150 1844 y(of)g(the)i(metho)q(d)e(and)h(n)o (umerous)f(references)k(to)d(the)h(literature.)k(W)m(e)13 b(also)f(giv)o(e)g (the)i(mathemat-)150 1894 y(ical)f(conditions)h(for)f(selecting)i(a)e(giv)o (en)h(metho)q(d.)212 1943 y(W)m(e)f(do)h(not)f(in)o(tend)h(to)f(write)h(a)f (\\co)q(okb)q(o)q(ok",)g(and)g(ha)o(v)o(e)g(delib)q(erately)h(a)o(v)o(oided)f (the)h(w)o(ords)150 1993 y(\\n)o(umerical)k(recip)q(es",)j(b)q(ecause)g (these)g(phrases)f(imply)d(that)j(our)f(algorithms)e(can)i(b)q(e)h(used)150 2043 y(blindly)11 b(without)h(kno)o(wledge)g(of)f(the)i(system)f(of)g (equations.)17 b(The)c(state)g(of)e(the)i(art)f(in)g(iterativ)o(e)150 2093 y(metho)q(ds)19 b(do)q(es)h(not)f(p)q(ermit)f(this:)28 b(some)19 b(kno)o(wledge)f(ab)q(out)h(the)h(linear)f(system)g(is)g(needed)150 2143 y(to)d(guaran)o(tee)g(con)o(v)o(ergence)i(of)d(these)j(algorithms,)c (and)h(generally)h(the)h(more)e(that)h(is)g(kno)o(wn)150 2192 y(the)d(more)f(the)h(algorithm)d(can)i(b)q(e)i(tuned.)k(Th)o(us,)12 b(w)o(e)h(ha)o(v)o(e)g(c)o(hosen)g(to)f(presen)o(t)j(an)d(algorithmic)150 2242 y(outline,)f(with)g(guidelines)g(for)f(c)o(ho)q(osing)h(a)g(metho)q(d)g (and)g(implemen)o(ting)d(it)i(on)h(particular)g(kinds)150 2292 y(of)g(high-p)q(erformance)g(mac)o(hines.)16 b(W)m(e)c(also)f(discuss)i(the)f (use)h(of)e(preconditioners)i(and)e(relev)n(an)o(t)150 2342 y(data)j(storage)g(issues.)p eop %%Page: 4 16 15 bop 450 275 a FC(4)902 b Fr(CHAPTER)14 b(1.)31 b(INTR)o(ODUCTION)p eop %%Page: 5 17 16 bop 150 706 a Fz(Chapter)34 b(2)150 918 y FB(Iterativ)m(e)41 b(Metho)s(ds)150 1137 y FC(The)15 b(term)f(\\iterativ)o(e)h(metho)q(d")e (refers)j(to)f(a)f(wide)h(range)g(of)f(tec)o(hniques)i(that)e(use)i (successiv)o(e)150 1187 y(appro)o(ximations)c(to)i(obtain)g(more)f(accurate)j (solutions)e(to)g(a)g(linear)g(system)g(at)h(eac)o(h)g(step.)20 b(In)150 1237 y(this)13 b(b)q(o)q(ok)h(w)o(e)f(will)f(co)o(v)o(er)i(t)o(w)o (o)f(t)o(yp)q(es)h(of)f(iterativ)o(e)g(metho)q(ds.)k(Stationary)c(metho)q(ds) g(are)h(older,)150 1287 y(simpler)k(to)g(understand)j(and)d(implemen)o(t,)f (but)i(usually)f(not)h(as)g(e\013ectiv)o(e.)35 b(Nonstationary)150 1337 y(metho)q(ds)12 b(are)h(a)f(relativ)o(ely)f(recen)o(t)j(dev)o(elopmen)o 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y(Gauss-Seidel)14 b(metho)q(d:)254 2059 y Fy(x)278 2038 y Fv(\()p Fx(k)q Fv(\))278 2071 y Fx(i)335 2059 y FC(=)e(\()p Fy(b)413 2065 y Fx(i)436 2059 y Fu(\000)478 2020 y Ft(X)481 2108 y Fx(j)r(i)793 2059 y Fy(a)815 2065 y Fx(i;j)854 2059 y Fy(x)878 2038 y Fv(\()p Fx(k)q Fw(\000)p Fv(1\))878 2071 y Fx(j)967 2059 y FC(\))p Fy(=a)1026 2065 y Fx(i;i)1061 2059 y Fy(:)541 b FC(\(2.5\))212 2184 y(Tw)o(o)16 b(imp)q(ortan)o(t)e(facts)i (ab)q(out)f(the)i(Gauss-Seidel)e(metho)q(d)g(should)h(b)q(e)g(noted.)24 b(First,)17 b(the)150 2234 y(computations)e(in)h(\(2.5\))g(app)q(ear)h(to)f (b)q(e)h(serial.)25 b(Since)17 b(eac)o(h)g(comp)q(onen)o(t)f(of)g(the)h(new)g (iterate)150 2284 y(dep)q(ends)i(up)q(on)e(all)e(previously)i(computed)g (comp)q(onen)o(ts,)g(the)g(up)q(dates)h(cannot)g(b)q(e)f(done)h(si-)150 2334 y(m)o(ultaneously)10 b(as)j(in)e(the)i(Jacobi)f(metho)q(d.)17 b(Second,)c(the)g(new)f(iterate)h Fy(x)1321 2319 y Fv(\()p Fx(k)q Fv(\))1379 2334 y FC(dep)q(ends)h(up)q(on)f(the)150 2384 y(order)k(in)g(whic)o(h)f(the)i(equations)e(are)h(examined.)26 b(The)17 b(Gauss-Seidel)g(metho)q(d)e(is)i(sometimes)150 2434 y(called)e(the)h Fq(metho)n(d)g(of)g(suc)n(c)n(essive)g(displac)n(ements)j FC(to)c(indicate)g(the)h(dep)q(endence)i(of)c(the)i(iter-)150 2483 y(ates)h(on)f(the)g(ordering.)25 b(If)15 b(this)h(ordering)h(is)e(c)o (hanged,)i(the)g Fq(c)n(omp)n(onents)j FC(of)15 b(the)i(new)g(iterate)150 2533 y(\(and)d(not)g(just)g(their)g(order\))h(will)d(also)h(c)o(hange.)212 2583 y(These)f(t)o(w)o(o)f(p)q(oin)o(ts)f(are)i(imp)q(ortan)o(t)d(b)q(ecause) j(if)e Fy(A)h FC(is)g(sparse,)h(the)f(dep)q(endency)i(of)d(eac)o(h)i(com-)150 2633 y(p)q(onen)o(t)17 b(of)f(the)i(new)f(iterate)g(on)g(previous)g(comp)q (onen)o(ts)f(is)h(not)g(absolute.)26 b(The)18 b(presence)h(of)150 2683 y(zeros)13 b(in)e(the)i(matrix)d(ma)o(y)g(remo)o(v)o(e)h(the)h (in\015uence)h(of)e(some)g(of)g(the)i(previous)f(comp)q(onen)o(ts.)17 b(Us-)150 2733 y(ing)10 b(a)g(judicious)f(ordering)i(of)f(the)g(equations,)h 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y(o\013)e(\(parallelism)d(v)o(ersus)j(con)o(v)o(ergence)i(rate\))e(and)f (some)g(standard)h(reorderings,)g(the)h(reader)f(is)450 1703 y(referred)i(to)d(Chapter)i(3)f(and)f Fu(x)q FC(4.4.)512 1755 y(In)d(matrix)f(terms,)h(the)g(de\014nition)g(of)g(the)g(Gauss-Seidel)h (metho)q(d)e(in)g(\(2.5\))h(can)g(b)q(e)h(expressed)450 1805 y(as)554 1897 y Fy(x)578 1880 y Fv(\()p Fx(k)q Fv(\))635 1897 y FC(=)h(\()p Fy(D)f Fu(\000)f Fy(L)p FC(\))826 1880 y Fw(\000)p Fv(1)871 1897 y FC(\()p Fy(U)5 b(x)944 1880 y Fv(\()p Fx(k)q Fw(\000)p Fv(1\))1041 1897 y FC(+)10 b Fy(b)p FC(\))p Fy(:)785 b FC(\(2.6\))450 1986 y(As)24 b(b)q(efore,)i Fy(D)q FC(,)g Fu(\000)p Fy(L)e FC(and)f Fu(\000)p Fy(U)29 b FC(represen)o(t)d(the)e (diagonal,)f(lo)o(w)o(er-triangular,)h(and)f(upp)q(er-)450 2036 y(triangular)13 b(parts)h(of)g Fy(A)p FC(,)f(resp)q(ectiv)o(ely)m(.)512 2088 y(The)i(pseudo)q(co)q(de)g(for)f(the)g(Gauss-Seidel)g(algorithm)d(is)j (giv)o(en)f(in)h(Figure)g(2.2.)450 2216 y Fl(2.2.3)55 b(The)19 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1576 y Fk(1)p 1188 1581 V 1188 1598 a(2)1207 1588 y Fv(\))1126 1623 y Fx(i)1222 1612 y FC(\))599 1662 y Fs(end)599 1711 y FC(c)o(hec)o(k)15 b(con)o(v)o(ergence;)g(con)o(tin)o(ue)f(if)f(necessary)511 1761 y Fs(end)p 1523 1830 2 1438 v 325 1832 1200 2 v 643 1909 a FC(Figure)h(2.4:)j(The)e(SSOR)e(Metho)q(d)150 2117 y(South)o(w)o(ell's)j (original)f(relaxation)i(metho)q(d.)26 b(In)17 b(c)o(haotic)g(metho)q(ds,)g (the)h(order)g(of)e(relaxation)150 2167 y(is)c(unconstrained,)i(thereb)o(y)g (eliminating)9 b(costly)k(sync)o(hronization)g(of)f(the)h(pro)q(cessors,)h (though)150 2217 y(the)g(e\013ect)i(on)e(con)o(v)o(ergence)h(is)f(di\016cult) f(to)h(predict.)212 2300 y(The)e(notion)f(of)g(accelerating)i(the)f(con)o(v)o (ergence)h(of)f(an)f(iterativ)o(e)h(metho)q(d)f(b)o(y)g(extrap)q(olation)150 2350 y(predates)j(the)e(dev)o(elopmen)o(t)g(of)f(SOR.)h(Indeed,)h(South)o(w)o (ell)e(used)i(o)o(v)o(errelaxation)f(to)g(accelerate)150 2400 y(the)19 b(con)o(v)o(ergence)h(of)e(his)g(original)f(relaxation)g(metho)q(d.) 31 b(More)18 b(recen)o(tly)m(,)i(the)f Fq(ad)g(ho)n(c)h(SOR)150 2450 y FC(metho)q(d,)d(in)f(whic)o(h)h(a)g(di\013eren)o(t)h(relaxation)e (factor)h Fy(!)i FC(is)e(used)h(for)f(up)q(dating)f(eac)o(h)i(v)n(ariable,) 150 2500 y(has)c(giv)o(en)f(impressiv)o(e)h(results)h(for)e(some)g(classes)i (of)f(problems)f(\(see)i(Ehrlic)o(h)f([80)o(]\).)212 2583 y(The)24 b(three)h(main)c(references)26 b(for)d(the)h(theory)g(of)f(stationary)g (iterativ)o(e)g(metho)q(ds)g(are)150 2633 y(V)m(arga)17 b([206)o(],)g(Y)m (oung)g([212)o(])g(and)g(Hageman)f(and)i(Y)m(oung)e([119)o(].)29 b(The)18 b(reader)h(is)e(referred)i(to)150 2683 y(these)g(b)q(o)q(oks)f (\(and)f(the)h(references)j(therein\))d(for)f(further)i(details)e(concerning) i(the)f(metho)q(ds)150 2733 y(describ)q(ed)e(in)d(this)h(section.)p eop %%Page: 14 26 25 bop 450 275 a FC(14)755 b Fr(CHAPTER)15 b(2.)31 b(ITERA)m(TIVE)14 b(METHODS)450 391 y Fp(2.3)70 b(Nonstationary)24 b(Iterativ)n(e)d(Metho)r(ds) 450 482 y FC(Nonstationary)16 b(metho)q(ds)f(di\013er)i(from)d(stationary)i (metho)q(ds)g(in)f(that)h(the)h(computations)e(in-)450 532 y(v)o(olv)o(e)10 b(information)d(that)k(c)o(hanges)g(at)g(eac)o(h)g (iteration.)16 b(T)o(ypically)m(,)9 b(constan)o(ts)i(are)g(computed)f(b)o(y) 450 582 y(taking)j(inner)h(pro)q(ducts)g(of)f(residuals,)h(or)f(other)i(v)o (ectors)f(arising)f(from)f(the)i(iterativ)o(e)g(metho)q(d.)450 697 y Fl(2.3.1)55 b(Conjugate)19 b(Gradien)n(t)g(Metho)r(d)f(\(CG\))450 774 y FC(The)12 b(Conjugate)f(Gradien)o(t)h(metho)q(d)e(is)i(an)f(e\013ectiv) o(e)i(metho)q(d)e(for)g(symmetric)f(p)q(ositiv)o(e)h(de\014nite)450 823 y(systems.)36 b(It)20 b(is)f(the)i(oldest)f(and)f(b)q(est)i(kno)o(wn)e (of)h(the)g(nonstationary)f(metho)q(ds)h(discussed)450 873 y(here.)e(The)12 b(metho)q(d)f(pro)q(ceeds)i(b)o(y)e(generating)h(v)o(ector)g (sequences)i(of)c(iterates)j(\()p Fq(i.e.)p FC(,)e(successsiv)o(e)450 923 y(appro)o(ximations)k(to)i(the)h(solution\),)f(residuals)h(corresp)q (onding)g(to)f(the)h(iterates,)h(and)e(searc)o(h)450 973 y(directions)g(used) g(in)e(up)q(dating)h(the)h(iterates)g(and)f(residuals.)25 b(Although)15 b(the)i(length)f(of)g(these)450 1023 y(sequences)c(can)e(b)q(ecome)f(large,)g (only)g(a)g(small)e(n)o(um)o(b)q(er)i(of)f(v)o(ectors)j(needs)g(to)e(b)q(e)h (k)o(ept)g(in)f(memory)m(.)450 1072 y(In)h(ev)o(ery)h(iteration)f(of)g(the)h (metho)q(d,)f(t)o(w)o(o)g(inner)g(pro)q(ducts)i(are)f(p)q(erformed)f(in)f (order)j(to)e(compute)450 1122 y(up)q(date)19 b(scalars)f(that)g(are)g (de\014ned)h(to)f(mak)o(e)e(the)j(sequences)h(satisfy)e(certain)g (orthogonalit)o(y)450 1172 y(conditions.)f(On)12 b(a)f(symmetric)f(p)q (ositiv)o(e)i(de\014nite)g(linear)f(system)h(these)h(conditions)e(imply)e (that)450 1222 y(the)14 b(distance)h(to)f(the)g(true)h(solution)e(is)h(minim) o(ized)e(in)h(some)g(norm.)512 1272 y(The)19 b(iterates)g Fy(x)778 1257 y Fv(\()p Fx(i)p Fv(\))836 1272 y FC(are)g(up)q(dated)g(in)f(eac)o(h)h (iteration)e(b)o(y)i(a)f(m)o(ultiple)e(\()p Fy(\013)1712 1278 y Fx(i)1725 1272 y FC(\))i(of)g(the)h(searc)o(h)450 1322 y(direction)14 b(v)o(ector)h Fy(p)769 1306 y Fv(\()p Fx(i)p Fv(\))808 1322 y FC(:)554 1399 y Fy(x)578 1382 y Fv(\()p Fx(i)p Fv(\))629 1399 y FC(=)d Fy(x)697 1382 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))788 1399 y FC(+)d Fy(\013)856 1405 y Fx(i)870 1399 y Fy(p)891 1382 y Fv(\()p Fx(i)p Fv(\))930 1399 y Fy(:)450 1477 y FC(Corresp)q(ondingly)14 b(the)g(residuals)h Fy(r)1025 1462 y Fv(\()p Fx(i)p Fv(\))1076 1477 y FC(=)d Fy(b)d Fu(\000)g Fy(Ax)1243 1462 y Fv(\()p Fx(i)p Fv(\))1297 1477 y FC(are)14 b(up)q(dated)g(as)554 1555 y Fy(r)574 1538 y Fv(\()p Fx(i)p Fv(\))625 1555 y FC(=)e Fy(r)689 1538 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))780 1555 y Fu(\000)e Fy(\013q)869 1538 y Fv(\()p Fx(i)p Fv(\))991 1555 y FC(where)84 b Fy(q)1200 1538 y Fv(\()p Fx(i)p Fv(\))1252 1555 y FC(=)12 b Fy(Ap)1348 1538 y Fv(\()p Fx(i)p Fv(\))1387 1555 y Fy(:)494 b FC(\(2.10\))450 1640 y(The)21 b(c)o(hoice)h Fy(\013)h FC(=)g Fy(\013)805 1646 y Fx(i)842 1640 y FC(=)g Fy(r)917 1625 y Fv(\()p Fx(i)p Fv(\))955 1612 y Fj(T)980 1640 y Fy(r)1000 1625 y Fv(\()p Fx(i)p Fv(\))1040 1640 y Fy(=p)1082 1625 y Fv(\()p Fx(i)p Fv(\))1120 1612 y Fj(T)1144 1640 y Fy(Ap)1196 1625 y Fv(\()p Fx(i)p Fv(\))1257 1640 y FC(minim)o(i)o(zes) d Fy(r)1478 1625 y Fv(\()p Fx(i)p Fv(\))1516 1612 y Fj(T)1540 1640 y Fy(A)1571 1625 y Fw(\000)p Fv(1)1616 1640 y Fy(r)1636 1625 y Fv(\()p Fx(i)p Fv(\))1696 1640 y FC(o)o(v)o(er)h(all)f(p)q(ossible)450 1689 y(c)o(hoices)15 b(for)e Fy(\013)h FC(in)f(equation)h(\(2.10\).)512 1739 y(The)h(searc)o(h)g(directions)f(are)g(up)q(dated)h(using)f(the)g (residuals)554 1817 y Fy(p)575 1800 y Fv(\()p Fx(i)p Fv(\))626 1817 y FC(=)e Fy(r)690 1800 y Fv(\()p Fx(i)p Fv(\))739 1817 y FC(+)d Fy(\014)803 1823 y Fx(i)p Fw(\000)p Fv(1)860 1817 y Fy(p)881 1800 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))963 1817 y Fy(;)918 b FC(\(2.11\))450 1902 y(where)15 b(the)g(c)o(hoice)f Fy(\014)788 1908 y Fx(i)815 1902 y FC(=)e Fy(r)879 1886 y Fv(\()p Fx(i)p Fv(\))917 1874 y Fj(T)941 1902 y Fy(r)961 1886 y Fv(\()p Fx(i)p Fv(\))1001 1902 y Fy(=r)1042 1886 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))1123 1874 y Fj(T)1147 1902 y Fy(r)1167 1886 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))1263 1902 y FC(ensures)k(that)e Fy(p)1520 1886 y Fv(\()p Fx(i)p Fv(\))1574 1902 y FC(and)g Fy(Ap)1707 1886 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))1803 1902 y FC({)g(or)g(equiv-)450 1951 y(alen)o(tly)m(,)g Fy(r)616 1936 y Fv(\()p Fx(i)p Fv(\))671 1951 y FC(and)h Fy(r)773 1936 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))871 1951 y FC({)g(are)h(orthogonal.)21 b(In)15 b(fact,)h(one)f(can)h(sho)o(w)f (that)h(this)f(c)o(hoice)h(of)f Fy(\014)1985 1957 y Fx(i)450 2001 y FC(mak)o(es)e Fy(p)596 1986 y Fv(\()p Fx(i)p Fv(\))649 2001 y FC(and)h Fy(r)750 1986 y Fv(\()p Fx(i)p Fv(\))803 2001 y FC(orthogonal)f(to)h Fq(al)r(l)k FC(previous)c Fy(Ap)1342 1986 y Fv(\()p Fx(j)r Fv(\))1399 2001 y FC(and)g Fy(r)1500 1986 y Fv(\()p Fx(j)r Fv(\))1557 2001 y FC(resp)q(ectiv)o(ely)m(.)512 2051 y(The)19 b(pseudo)q(co)q(de)h(for)e(the)h(Preconditioned)g(Conjugate)f (Gradien)o(t)g(Metho)q(d)i(is)e(giv)o(en)g(in)450 2101 y(Figure)d(2.5.)20 b(It)15 b(uses)h(a)e(preconditioner)i Fy(M)5 b FC(;)14 b(for)h Fy(M)j FC(=)13 b Fy(I)19 b FC(one)c(obtains)f(the)h(unpreconditioned)450 2151 y(v)o(ersion)20 b(of)e(the)i(Conjugate)f(Gradien)o(t)g(Algorithm.)32 b(In)20 b(that)f(case)h(the)g(algorithm)d(ma)o(y)h(b)q(e)450 2200 y(further)13 b(simpli\014ed)e(b)o(y)i(skipping)e(the)j(\\solv)o(e")d (line,)h(and)h(replacing)f Fy(z)1572 2185 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))1667 2200 y FC(b)o(y)g Fy(r)1743 2185 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))1838 2200 y FC(\(and)h Fy(z)1955 2185 y Fv(\(0\))450 2250 y FC(b)o(y)h Fy(r)528 2235 y Fv(\(0\))572 2250 y FC(\).)450 2357 y Fs(Theory)450 2434 y FC(The)g(unpreconditioned)g (conjugate)f(gradien)o(t)g(metho)q(d)g(constructs)i(the)f Fy(i)p FC(th)f(iterate)h Fy(x)1852 2419 y Fv(\()p Fx(i)p 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y(minim)o(izatio)o(n)h(prop)q(ert)o(y)m(,)k(although)e(o)o(v)o(er)h (this)f(di\013eren)o(t)i(subspace.)26 b(It)16 b(requires)h(in)e(addition)450 2733 y(that)f(the)g(preconditioner)h Fy(M)k FC(is)13 b(symmetric)g(and)g(p)q (ositiv)o(e)h(de\014nite.)p eop %%Page: 15 27 26 bop 150 275 a Fr(2.3.)31 b(NONST)m(A)m(TIONAR)m(Y)13 b(ITERA)m(TIVE)h (METHODS)564 b FC(15)p 250 393 1350 2 v 250 1326 2 933 v 452 484 a(Compute)12 b Fy(r)651 469 y Fv(\(0\))707 484 y FC(=)g Fy(b)d Fu(\000)h Fy(Ax)875 469 y Fv(\(0\))933 484 y FC(for)k(some)f(initial)f (guess)i Fy(x)1353 469 y Fv(\(0\))452 534 y Fs(for)28 b Fy(i)12 b FC(=)g(1)p Fy(;)7 b FC(2)p Fy(;)g(:)g(:)g(:)539 585 y Fs(solv)o(e)13 b Fy(M)5 b(z)721 570 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))815 585 y FC(=)12 b Fy(r)879 570 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))539 642 y Fy(\032)560 648 y Fx(i)p Fw(\000)p Fv(1)629 642 y FC(=)g Fy(r)693 627 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))774 615 y Fj(T)798 642 y Fy(z)819 627 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))539 692 y Fs(if)h Fy(i)f FC(=)g(1)581 744 y Fy(p)602 729 y Fv(\(1\))658 744 y FC(=)g Fy(z)723 729 y Fv(\(0\))539 794 y Fs(else)581 843 y Fy(\014)604 849 y Fx(i)p Fw(\000)p Fv(1)672 843 y FC(=)g Fy(\032)737 849 y Fx(i)p Fw(\000)p Fv(1)794 843 y Fy(=\032)836 849 y Fx(i)p Fw(\000)p Fv(2)581 895 y Fy(p)602 880 y Fv(\()p Fx(i)p Fv(\))653 895 y FC(=)g Fy(z)718 880 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))810 895 y FC(+)d Fy(\014)874 901 y Fx(i)p Fw(\000)p Fv(1)931 895 y Fy(p)952 880 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))539 945 y Fs(endif)539 997 y Fy(q)559 982 y Fv(\()p Fx(i)p Fv(\))611 997 y FC(=)j Fy(Ap)707 982 y Fv(\()p Fx(i)p Fv(\))539 1054 y Fy(\013)566 1060 y Fx(i)591 1054 y FC(=)g Fy(\032)656 1060 y Fx(i)p Fw(\000)p Fv(1)713 1054 y Fy(=p)755 1038 y Fv(\()p Fx(i)p Fv(\))793 1026 y Fj(T)817 1054 y Fy(q)837 1038 y Fv(\()p Fx(i)p Fv(\))539 1105 y Fy(x)563 1090 y Fv(\()p Fx(i)p Fv(\))614 1105 y FC(=)g Fy(x)682 1090 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))773 1105 y FC(+)e Fy(\013)842 1111 y Fx(i)855 1105 y Fy(p)876 1090 y Fv(\()p Fx(i)p Fv(\))539 1157 y Fy(r)559 1142 y Fv(\()p Fx(i)p Fv(\))611 1157 y FC(=)h Fy(r)674 1142 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))766 1157 y Fu(\000)e Fy(\013)834 1163 y Fx(i)848 1157 y Fy(q)868 1142 y Fv(\()p Fx(i)p Fv(\))539 1207 y FC(c)o(hec)o(k)15 b(con)o(v)o(ergence;)g(con)o(tin)o(ue)f(if)f (necessary)452 1257 y Fs(end)p 1598 1326 V 250 1328 1350 2 v 374 1405 a FC(Figure)h(2.5:)j(The)e(Preconditioned)f(Conjugate)g(Gradien)o (t)f(Metho)q(d)212 1546 y(The)i(ab)q(o)o(v)o(e)e(minimi)o(zation)e(of)i(the)i (error)g(is)f(equiv)n(alen)o(t)f(to)h(the)g(residuals)h Fy(r)1441 1531 y Fv(\()p Fx(i)p Fv(\))1492 1546 y FC(=)d Fy(b)d Fu(\000)h Fy(Ax)1660 1531 y Fv(\()p Fx(i)p Fv(\))150 1600 y FC(b)q(eing)k Fy(M)307 1585 y Fw(\000)p Fv(1)366 1600 y FC(orthogonal)f(\(that)h(is,)g Fy(r)754 1585 y Fv(\()p Fx(i)p Fv(\))792 1572 y Fj(T)816 1600 y Fy(M)861 1585 y Fw(\000)p Fv(1)906 1600 y Fy(r)926 1585 y Fv(\()p Fx(j)r Fv(\))981 1600 y FC(=)e(0)i(if)f Fy(i)g Fu(6)p FC(=)f Fy(j)r FC(\).)19 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y(but)e(useful)g(b)q(ounds)h (can)f(often)g(b)q(e)h(obtained.)27 b(F)m(or)17 b(the)h(Conjugate)e(Gradien)o (t)h(metho)q(d,)g(the)150 2533 y(error)j(can)g(b)q(e)g(b)q(ounded)g(in)f (terms)g(of)g(the)h(sp)q(ectral)g(condition)f(n)o(um)o(b)q(er)f Fy(\024)1410 2539 y Fv(2)1448 2533 y FC(of)h(the)h(matrix)150 2583 y Fy(M)195 2568 y Fw(\000)p Fv(1)239 2583 y Fy(A)p FC(.)36 b(\(Recall)19 b(that)h(if)f Fy(\025)629 2589 y Fv(max)713 2583 y FC(and)g Fy(\025)823 2589 y Fv(min)900 2583 y FC(are)i(the)f(largest)g(and) g(smallest)e(eigen)o(v)n(alues)i(of)150 2633 y(a)d(symmetric)e(p)q(ositiv)o (e)h(de\014nite)i(matrix)d Fy(B)r FC(,)i(then)h(the)f(sp)q(ectral)h (condition)f(n)o(um)o(b)q(er)f(of)g Fy(B)j FC(is)150 2683 y Fy(\024)174 2689 y Fv(2)193 2683 y FC(\()p Fy(B)r FC(\))12 b(=)g Fy(\025)338 2689 y Fv(max)401 2683 y FC(\()p Fy(B)r FC(\))p Fy(=\025)511 2689 y Fv(min)569 2683 y FC(\()p Fy(B)r FC(\)\).)18 b(If)12 b(^)-24 b Fy(x)9 b FC(is)h(the)g(exact)h(solution)d(of)i(the)g (linear)f(system)h Fy(Ax)h FC(=)h Fy(b)p FC(,)e(with)150 2733 y(symmetric)15 b(p)q(ositiv)o(e)i(de\014nite)h(matrix)d Fy(A)p FC(,)j(then)f(for)g(CG)g(with)f(symmetric)g(p)q(ositiv)o(e)g(de\014nite)p eop %%Page: 16 28 27 bop 450 275 a FC(16)755 b Fr(CHAPTER)15 b(2.)31 b(ITERA)m(TIVE)14 b(METHODS)450 391 y FC(preconditioner)h Fy(M)5 b FC(,)13 b(it)g(can)h(b)q(e)h (sho)o(wn)f(that)554 463 y Fu(k)p Fy(x)599 446 y Fv(\()p Fx(i)p Fv(\))647 463 y Fu(\000)e FC(^)-23 b Fy(x)o Fu(k)733 469 y Fx(A)772 463 y Fu(\024)12 b FC(2)p Fy(\013)864 446 y Fx(i)877 463 y Fu(k)p Fy(x)922 446 y Fv(\(0\))975 463 y Fu(\000)g FC(^)-23 b Fy(x)o Fu(k)1061 469 y Fx(A)1893 463 y FC(\(2.12\))450 534 y(where)24 b Fy(\013)j FC(=)h(\()709 508 y Fu(p)p 743 508 43 2 v 743 534 a Fy(\024)767 540 y Fv(2)801 534 y Fu(\000)16 b FC(1\))p Fy(=)p FC(\()923 508 y Fu(p)p 957 508 V 957 534 a Fy(\024)981 540 y Fv(2)1015 534 y FC(+)g(1\))23 b(\(see)i(Golub)d(and)h(V)m (an)g(Loan)f([108)o(,)j Fu(x)q FC(10.2.8],)d(and)450 584 y(Kaniel)d([131)o (]\),)g(and)g Fu(k)p Fy(y)q Fu(k)869 569 y Fv(2)869 596 y Fx(A)917 584 y Fu(\021)h FC(\()p Fy(y)q(;)7 b(Ay)q FC(\).)36 b(F)m(rom)17 b(this)i(relation)g(w)o(e)g(see)h(that)g(the)f(n)o(um)o(b)q(er)g(of)450 634 y(iterations)14 b(to)g(reac)o(h)g(a)g(relativ)o(e)g(reduction)g(of)f Fy(\017)h FC(in)g(the)g(error)h(is)f(prop)q(ortional)f(to)1793 607 y Fu(p)p 1828 607 V 27 x Fy(\024)1852 640 y Fv(2)1870 634 y FC(.)512 684 y(In)k(some)f(cases,)i(practical)f(application)e(of)h(the)i (ab)q(o)o(v)o(e)e(error)i(b)q(ound)f(is)f(straigh)o(tforw)o(ard.)450 734 y(F)m(or)g(example,)f(elliptic)g(second)j(order)f(partial)e(di\013eren)o (tial)h(equations)g(t)o(ypically)f(giv)o(e)h(rise)g(to)450 783 y(co)q(e\016cien)o(t)11 b(matrices)f Fy(A)g FC(with)g Fy(\024)959 789 y Fv(2)977 783 y FC(\()p Fy(A)p FC(\))i(=)g Fy(O)q FC(\()p Fy(h)1169 768 y Fw(\000)p Fv(2)1214 783 y FC(\))e(\(where)i Fy(h)e FC(is)g(the)g(discretization)h(mesh)f(width\),)450 833 y(indep)q(enden)o(t)15 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g(that)g(is,)g(con)o(v)o(ergence)h(at)f(a)f(rate)h(that)g(increases)i(p)q(er) e(iteration.)26 b(This)17 b(phe-)450 1232 y(nomenon)e(is)h(explained)f(b)o(y) h(the)h(fact)f(that)g(CG)f(tends)i(to)f(eliminate)e(comp)q(onen)o(ts)i(of)f (the)i(er-)450 1281 y(ror)g(in)f(the)h(direction)g(of)f(eigen)o(v)o(ectors)h (asso)q(ciated)h(with)e(extremal)f(eigen)o(v)n(alues)i(\014rst.)27 b(After)450 1331 y(these)18 b(ha)o(v)o(e)e(b)q(een)h(eliminated,)e(the)i (metho)q(d)f(pro)q(ceeds)i(as)e(if)g(these)i(eigen)o(v)n(alues)e(did)g(not)g (ex-)450 1381 y(ist)f(in)g(the)h(giv)o(en)e(system,)h Fq(i.e.)p FC(,)g(the)g(con)o(v)o(ergence)i(rate)f(dep)q(ends)h(on)e(a)g(reduced)i (system)e(with)450 1431 y(a)k(\(m)o(uc)o(h\))f(smaller)f(condition)h(n)o(um)o (b)q(er)g(\(for)h(an)g(analysis)f(of)g(this,)i(see)g(V)m(an)e(der)i(Sluis)e (and)450 1481 y(V)m(an)e(der)h(V)m(orst)g([194)o(]\).)25 b(The)17 b(e\013ectiv)o(eness)j(of)c(the)h(preconditioner)g(in)f(reducing)h(the)h (condi-)450 1531 y(tion)d(n)o(um)o(b)q(er)f(and)h(in)g(separating)h(extremal) e(eigen)o(v)n(alues)h(can)h(b)q(e)g(deduced)g(b)o(y)f(studying)h(the)450 1580 y(appro)o(ximated)c(eigen)o(v)n(alues)i(of)f(the)h(related)h(Lanczos)g (pro)q(cess.)450 1686 y Fs(Implemen)o(tati)o(on)450 1762 y FC(The)21 b(Conjugate)f(Gradien)o(t)h(metho)q(d)f(in)o(v)o(olv)o(es)f(one)i (matrix-v)o(ector)f(pro)q(duct,)i(three)g(v)o(ector)450 1812 y(up)q(dates,)d(and)f(t)o(w)o(o)f(inner)h(pro)q(ducts)h(p)q(er)f(iteration.) 29 b(Some)17 b(sligh)o(t)g(computational)e(v)n(arian)o(ts)450 1862 y(exist)i(that)g(ha)o(v)o(e)g(the)h(same)e(structure)j(\(see)f(Reid)f ([174)o(]\).)27 b(V)m(arian)o(ts)16 b(that)h(cluster)h(the)g(inner)450 1912 y(pro)q(ducts,)d(a)e(fa)o(v)o(orable)g(prop)q(ert)o(y)i(on)e(parallel)g (mac)o(hines,)g(are)h(discussed)h(in)f Fu(x)p FC(4.4.)512 1962 y(F)m(or)g(a)f(discussion)h(of)f(the)i(Conjugate)e(Gradien)o(t)h(metho)q(d)e (on)i(v)o(ector)g(and)g(shared)h(memory)450 2011 y(computers,)e(see)h (Dongarra,)e Fq(et)h(al.)g FC([68)o(,)f(162].)17 b(F)m(or)12 b(discussions)i(of)e(the)i(metho)q(d)e(for)g(more)g(gen-)450 2061 y(eral)f(parallel)f(arc)o(hitectures)k(see)e(Demmel,)d(Heath)j(and)f(V)m (an)f(der)i(V)m(orst)g([64)o(])e(and)i(Ortega)f([162)o(],)450 2111 y(and)j(the)g(references)j(therein.)450 2217 y Fs(F)l(urther)d (references)450 2293 y FC(A)c(more)g(formal)e(presen)o(tation)j(of)f(CG,)f (as)i(w)o(ell)e(as)i(man)o(y)d(theoretical)j(prop)q(erties,)h(can)f(b)q(e)g (found)450 2343 y(in)j(the)h(textb)q(o)q(ok)f(b)o(y)h(Hac)o(kbusc)o(h)g([117) o(].)j(Shorter)e(presen)o(tations)f(are)g(giv)o(en)f(in)g(Axelsson)g(and)450 2393 y(Bark)o(er)j([14)o(])e(and)h(Golub)f(and)g(V)m(an)h(Loan)f([108)o(].)23 b(An)16 b(o)o(v)o(erview)g(of)f(pap)q(ers)i(published)f(in)g(the)450 2443 y(\014rst)f(25)e(y)o(ears)h(of)g(existence)h(of)f(the)g(metho)q(d)f(is)h (giv)o(en)f(in)h(Golub)f(and)g(O'Leary)h([107)o(].)450 2556 y Fl(2.3.2)55 b(MINRES)18 b(and)i(SYMMLQ)450 2633 y FC(The)h(Conjugate)f (Gradien)o(t)h(metho)q(d)e(can)i(b)q(e)h(view)o(ed)e(as)h(a)f(sp)q(ecial)h(v) n(arian)o(t)f(of)g(the)h(Lanc-)450 2683 y(zos)f(metho)q(d)f(\(see)i Fu(x)p FC(5.1\))e(for)h(p)q(ositiv)o(e)f(de\014nite)h(symmetric)e(systems.)36 b(The)20 b(MINRES)f(and)450 2733 y(SYMMLQ)14 b(metho)q(ds)f(are)i(v)n(arian)o (ts)e(that)g(can)h(b)q(e)h(applied)e(to)g(symmetric)f(inde\014nite)j (systems.)p eop %%Page: 17 29 28 bop 150 275 a Fr(2.3.)31 b(NONST)m(A)m(TIONAR)m(Y)13 b(ITERA)m(TIVE)h (METHODS)564 b FC(17)212 391 y(The)13 b(v)o(ector)f(sequences)i(in)e(the)g (Conjugate)g(Gradien)o(t)f(metho)q(d)g(corresp)q(ond)i(to)f(a)g(factoriza-) 150 441 y(tion)i(of)f(a)h(tridiagonal)e(matrix)g(similar)g(to)i(the)h(co)q (e\016cien)o(t)g(matrix.)h(Therefore,)f(a)f(breakdo)o(wn)150 491 y(of)i(the)i(algorithm)c(can)j(o)q(ccur)h(corresp)q(onding)f(to)g(a)g (zero)g(piv)o(ot)f(if)g(the)i(matrix)d(is)h(inde\014nite.)150 541 y(F)m(urthermore,)d(for)f(inde\014nite)h(matrices)g(the)g(minimi)o 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Fx(i)681 2548 y Fy(y)h Fu(\000)d(k)p Fy(r)795 2531 y Fv(\(0\))839 2548 y Fu(k)860 2554 y Fv(2)879 2548 y Fy(e)898 2531 y Fv(\(1\))943 2548 y Fu(k)964 2554 y Fv(2)150 2632 y FC(and)i(this)f(\014nal)h(expression)h(can)f(simply)d(b)q(e)k(seen)g (as)f(a)f(minim)n(um)d(norm)i(least)i(squares)h(problem.)212 2683 y(The)j(elemen)o(t)g(in)f(the)h(\()p Fy(i)10 b FC(+)g(1)p Fy(;)d(i)p FC(\))15 b(p)q(osition)f(of)956 2672 y(\026)948 2683 y Fy(T)972 2689 y Fx(i)1001 2683 y FC(can)h(b)q(e)g(annihilated)e(b)o(y) i(a)f(simple)g(Giv)o(ens)150 2733 y(rotation)i(and)h(the)g(resulting)g(upp)q (er)g(bidiagonal)e(system)h(\(the)i(other)f(sub)q(diagonal)e(elemen)o(ts)p eop %%Page: 18 30 29 bop 450 275 a FC(18)755 b Fr(CHAPTER)15 b(2.)31 b(ITERA)m(TIVE)14 b(METHODS)450 391 y FC(ha)o(ving)e(b)q(een)i(remo)o(v)o(ed)e(in)h(previous)g (iteration)g(steps\))h(can)f(simply)e(b)q(e)j(solv)o(ed,)e(whic)o(h)h(leads)g (to)450 441 y(the)h(MINRES)g(metho)q(d)f(\(see)j(P)o(aige)d(and)h(Saunders)h ([164)o(]\).)512 491 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FC(is)g(used)g(in)f(the)i(construction)f(of)f(the)i(iteration)e(co)q (e\016cien)o(ts)i(through)f(an)f(inner)h(pro)q(d-)450 2237 y(uct)j(lik)o(e)e(\()p Fy(p;)7 b(A)695 2222 y Fx(T)721 2237 y Fy(Ap)p FC(\))19 b(leads)h(to)f(the)h(suggestion)f(that)h(suc)o(h)g(an)f (inner)g(pro)q(duct)h(b)q(e)g(replaced)450 2287 y(b)o(y)14 b(\()p Fy(Ap;)7 b(Ap)p FC(\).)450 2406 y Fl(2.3.4)55 b(Generalized)17 b(Minimal)g(Residual)h(\(GMRES\))450 2483 y FC(The)13 b(Generalized)h (Minimal)c(Residual)i(metho)q(d)g(is)h(an)f(extension)i(of)e(MINRES)h(\(whic) o(h)g(is)g(only)450 2533 y(applicable)d(to)g(symmetric)f(systems\))i(to)f (unsymmetric)f(systems.)17 b(Lik)o(e)10 b(MINRES,)g(it)h(generates)450 2583 y(a)g(sequence)j(of)c(orthogonal)h(v)o(ectors,)h(but)g(in)f(the)h (absence)h(of)e(symmetry)e(this)j(can)f(no)g(longer)h(b)q(e)450 2633 y(done)f(with)f(short)h(recurrences;)j(instead,)d(all)e(previously)h (computed)g(v)o(ectors)i(in)d(the)i(orthogonal)450 2683 y(sequence)16 b(ha)o(v)o(e)e(to)g(b)q(e)g(retained.)19 b(F)m(or)13 b(this)h(reason,)g (\\restarted")i(v)o(ersions)e(of)f(the)i(metho)q(d)e(are)450 2733 y(used.)p eop %%Page: 19 31 30 bop 150 275 a Fr(2.3.)31 b(NONST)m(A)m(TIONAR)m(Y)13 b(ITERA)m(TIVE)h (METHODS)564 b FC(19)212 391 y(In)14 b(the)g(Conjugate)g(Gradien)o(t)f(metho) q(d,)f(the)j(residuals)f(form)e(an)h(orthogonal)g(basis)g(for)h(the)150 441 y(space)h(span)p Fu(f)p Fy(r)385 426 y Fv(\(0\))429 441 y Fy(;)7 b(Ar)499 426 y Fv(\(0\))543 441 y Fy(;)g(A)593 426 y Fv(2)612 441 y Fy(r)632 426 y Fv(\(0\))676 441 y Fy(;)g(:)g(:)g(:)n Fu(g)p FC(.)18 b(In)c(GMRES,)e(this)i(basis)g(is)g(formed)f(explicitly:)281 529 y Fy(w)312 514 y Fv(\()p Fx(i)p Fv(\))364 529 y FC(=)e Fy(Av)459 514 y Fv(\()p Fx(i)p Fv(\))281 579 y Fs(for)i Fy(k)g FC(=)f(1)p Fy(;)7 b(:)g(:)g(:)t(;)g(i)353 631 y(w)384 616 y Fv(\()p Fx(i)p Fv(\))436 631 y FC(=)k Fy(w)510 616 y Fv(\()p Fx(i)p Fv(\))559 631 y Fu(\000)f FC(\()p Fy(w)648 616 y Fv(\()p Fx(i)p Fv(\))687 631 y Fy(;)d(v)727 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Fu(\031)12 b FC(0,)g(o)q(ccurs)i(when)f(the)g Fy(LU)5 b FC(-decomp)q(osition)10 b(fails)450 2575 y(\(see)20 b Fu(x)p FC(2.3.5\),)d(and)i(can)f(b)q(e)h(repaired)g(b)o(y)f(using)g(another)h (decomp)q(osition.)30 b(This)18 b(is)g(done)h(in)450 2624 y(QMR)14 b(\(see)h Fu(x)p FC(2.3.6\).)512 2683 y(Sometimes,)d(breakdo)o(wn)i(or)f (near-breakdo)o(wn)i(situations)e(can)h(b)q(e)h(satisfactorily)e(a)o(v)o (oided)450 2733 y(b)o(y)20 b(a)g(restart)i(at)e(the)h(iteration)f(step)h (immediately)c(b)q(efore)k(the)g(\(near-\))g(breakdo)o(wn)g(step.)p eop %%Page: 23 35 34 bop 150 275 a Fr(2.3.)31 b(NONST)m(A)m(TIONAR)m(Y)13 b(ITERA)m(TIVE)h (METHODS)564 b FC(23)150 391 y(Another)10 b(p)q(ossibilit)o(y)e(is)i(to)f (switc)o(h)h(to)f(a)g(more)f(robust)i(\(but)g(p)q(ossibly)f(more)g(exp)q (ensiv)o(e\))i(metho)q(d,)150 441 y(lik)o(e)i(GMRES.)150 568 y Fs(Implemen)o(tati)o(on)150 651 y FC(BiCG)24 b(requires)i(computing)d(a)i (matrix-v)o(ector)e(pro)q(duct)j Fy(Ap)1196 636 y Fv(\()p Fx(k)q Fv(\))1267 651 y 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b(the)f(generation)g(of)g(the)150 1458 y(basis)j(v)o(ectors)g(is)g(relativ)o (ely)f(c)o(heap)h(and)f(the)h(memory)d(requiremen)o(ts)j(are)g(mo)q(dest.)29 b(Sev)o(eral)150 1508 y(v)n(arian)o(ts)17 b(of)f(BiCG)h(ha)o(v)o(e)g(b)q(een) i(prop)q(osed)f(that)f(increase)i(the)f(e\013ectiv)o(eness)i(of)d(this)g (class)h(of)150 1558 y(metho)q(ds)g(in)h(certain)h(circumstances.)34 b(These)20 b(v)n(arian)o(ts)e(\(CGS)h(and)g(Bi-CGST)m(AB\))f(will)g(b)q(e)150 1608 y(discussed)e(in)d(coming)f(subsections.)150 1742 y Fl(2.3.6)55 b(Quasi-Minimal)17 b(Residual)g(\(QMR\))150 1826 y FC(The)c(BiConjugate)g (Gradien)o(t)f(metho)q(d)h(often)g(displa)o(ys)f(rather)i(irregular)f(con)o (v)o(ergence)h(b)q(eha)o(v-)150 1875 y(ior.)19 b(Moreo)o(v)o(er,)c(the)g (implicit)d Fy(LU)19 b FC(decomp)q(osition)13 b(of)h(the)h(reduced)h (tridiagonal)d(system)h(ma)o(y)150 1925 y(not)i(exist,)i(resulting)e(in)g (breakdo)o(wn)h(of)f(the)h(algorithm.)23 b(A)17 b(related)g(algorithm,)d(the) j(Quasi-)150 1975 y(Minimal)11 b(Residual)j(metho)q(d)g(of)f(F)m(reund)i(and) g(Nac)o(h)o(tigal)e([100)o(],)g([101)o(])h(attempts)g(to)g(o)o(v)o(ercome)150 2025 y(these)19 b(problems.)27 b(The)18 b(main)d(idea)i(b)q(ehind)h(this)g (algorithm)c(is)j(to)h(solv)o(e)f(the)h(reduced)h(tridi-)150 2075 y(agonal)14 b(system)i(in)f(a)h(least)g(squares)g(sense,)i(similar)13 b(to)j(the)g(approac)o(h)g(follo)o(w)o(ed)e(in)h(GMRES.)150 2124 y(Since)h(the)f(constructed)j(basis)d(for)g(the)h(Krylo)o(v)e(subspace)j (is)e(bi-orthogonal,)e(rather)j(than)f(or-)150 2174 y(thogonal)g(as)h(in)g (GMRES,)g(the)g(obtained)g(solution)g(is)g(view)o(ed)h(as)f(a)g(quasi-minim)o (a)o(l)d(residual)150 2224 y(solution,)g(whic)o(h)h(explains)f(the)i(name.)j (Additionally)m(,)11 b(QMR)j(uses)h(lo)q(ok-ahead)e(tec)o(hniques)i(to)150 2274 y(a)o(v)o(oid)10 b(breakdo)o(wns)i(in)f(the)h(underlying)f(Lanczos)i (pro)q(cess,)g(whic)o(h)e(mak)o(es)g(it)g(more)f(robust)i(than)150 2324 y(BiCG.)150 2450 y Fs(Con)o(v)o(ergence)150 2533 y FC(The)h(con)o(v)o (ergence)h(b)q(eha)o(vior)e(of)g(QMR)g(is)h(t)o(ypically)e(m)o(uc)o(h)g(smo)q (other)h(than)g(for)g(BiCG.)g(F)m(reund)150 2583 y(and)h(Nac)o(h)o(tigal)e ([100)o(])h(presen)o(t)j(quite)e(general)g(error)g(b)q(ounds)h(whic)o(h)e (sho)o(w)h(that)g(QMR)g(ma)o(y)e(b)q(e)150 2633 y(exp)q(ected)k(to)f(con)o(v) o(erge)g(ab)q(out)f(as)g(fast)g(as)h(GMRES.)e(F)m(rom)g(a)h(relation)f(b)q (et)o(w)o(een)j(the)f(residuals)150 2683 y(in)e(BiCG)g(and)h(QMR)f(\(F)m (reund)i(and)e(Nac)o(h)o(tigal)g([100)n(,)h(relation)f(\(5.10\)]\))f(one)i (ma)o(y)e(deduce)j(that)150 2733 y(at)i(phases)h(in)e(the)i(iteration)e(pro)q (cess)j(where)f(BiCG)f(mak)o(es)f(signi\014can)o(t)g(progress,)i(QMR)f(has)p eop %%Page: 24 36 35 bop 450 275 a FC(24)755 b Fr(CHAPTER)15 b(2.)31 b(ITERA)m(TIVE)14 b(METHODS)p 543 434 1350 2 v 543 2533 2 2100 v 622 524 a FC(Compute)f Fy(r)822 509 y Fv(\(0\))878 524 y FC(=)f Fy(b)d Fu(\000)h Fy(Ax)1046 509 y Fv(\(0\))1104 524 y FC(for)j(some)g(initial)f(guess)j Fy(x)1524 509 y Fv(\(0\))624 576 y FC(~)-23 b Fy(v)643 561 y Fv(\(1\))700 576 y FC(=)12 b Fy(r)764 561 y Fv(\(0\))808 576 y FC(;)i Fs(solv)o(e)e Fy(M)989 582 y Fv(1)1008 576 y Fy(y)h FC(=)h(~)-23 b Fy(v)1106 561 y Fv(\(1\))1151 576 y FC(;)14 b Fy(\032)1198 582 y Fv(1)1228 576 y FC(=)e Fu(k)p Fy(y)q Fu(k)1335 582 y Fv(2)622 628 y FC(Cho)q(ose)23 b(~)-29 b Fy(w)798 613 y Fv(\(1\))856 628 y FC(,)13 b(for)h(example)21 b(~)-30 b Fy(w)1138 613 y Fv(\(1\))1194 628 y FC(=)12 b Fy(r)1258 613 y Fv(\(0\))622 680 y Fs(solv)o(e)h Fy(M)778 686 y Fv(2)797 680 y Fy(z)g FC(=)21 b(~)-30 b Fy(w)904 665 y Fv(\(1\))949 680 y FC(;)13 b Fy(\030)992 686 y Fv(1)1022 680 y FC(=)f Fu(k)p Fy(z)r Fu(k)1129 686 y Fv(2)622 730 y Fy(\015)643 736 y Fv(0)674 730 y FC(=)g(1;)7 b Fy(\021)779 736 y Fv(0)808 730 y FC(=)12 b Fu(\000)p FC(1)622 780 y Fs(for)29 b Fy(i)12 b FC(=)g(1)p Fy(;)7 b FC(2)p Fy(;)g(:)g(:)g(:)710 829 y Fs(if)13 b Fy(\032)773 835 y Fx(i)799 829 y FC(=)e(0)j(or)g Fy(\030)946 835 y Fx(i)971 829 y FC(=)e(0)i Fs(metho)q(d)g(fails)710 881 y Fy(v)731 866 y Fv(\()p Fx(i)p Fv(\))783 881 y FC(=)f(~)-22 b Fy(v)848 866 y Fv(\()p Fx(i)p Fv(\))888 881 y Fy(=\032)930 887 y Fx(i)944 881 y FC(;)14 b Fy(y)f FC(=)f Fy(y)q(=\032)1110 887 y Fx(i)710 933 y Fy(w)741 918 y Fv(\()p Fx(i)p Fv(\))792 933 y FC(=)21 b(~)-30 b Fy(w)867 918 y Fv(\()p Fx(i)p Fv(\))907 933 y Fy(=\030)946 939 y Fx(i)959 933 y FC(;)14 b Fy(z)f FC(=)f Fy(z)r(=\030)1121 939 y Fx(i)710 983 y Fy(\016)728 989 y Fx(i)754 983 y FC(=)g Fy(z)819 968 y Fx(T)845 983 y Fy(y)q FC(;)i Fs(if)f Fy(\016)952 989 y Fx(i)978 983 y FC(=)f(0)h Fs(metho)q(d)i(fails)710 1033 y(solv)o(e)29 b Fy(M)882 1039 y Fv(2)903 1033 y FC(~)-23 b Fy(y)13 b FC(=)f Fy(y)710 1082 y Fs(solv)o(e)29 b Fy(M)887 1067 y Fx(T)882 1093 y Fv(1)915 1082 y FC(~)-23 b Fy(z)13 b FC(=)f Fy(z)710 1132 y Fs(if)29 b Fy(i)12 b FC(=)f(1)768 1184 y Fy(p)789 1169 y Fv(\(1\))845 1184 y FC(=)j(~)-23 b Fy(y)q FC(;)14 b Fy(q)956 1169 y Fv(\(1\))1012 1184 y FC(=)g(~)-23 b Fy(z)768 1234 y Fs(else)768 1286 y Fy(p)789 1271 y Fv(\()p Fx(i)p Fv(\))840 1286 y FC(=)15 b(~)-24 b Fy(y)11 b Fu(\000)e FC(\()p Fy(\030)990 1292 y Fx(i)1004 1286 y Fy(\016)1022 1292 y Fx(i)1037 1286 y Fy(=\017)1075 1292 y Fx(i)p Fw(\000)p Fv(1)1131 1286 y FC(\))p Fy(p)1168 1271 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))768 1338 y Fy(q)788 1322 y Fv(\()p Fx(i)p Fv(\))839 1338 y FC(=)14 b(~)-23 b Fy(z)11 b Fu(\000)f FC(\()p Fy(\032)992 1344 y Fx(i)1006 1338 y Fy(\016)1024 1344 y Fx(i)1038 1338 y Fy(=\017)1076 1344 y Fx(i)p Fw(\000)p Fv(1)1132 1338 y FC(\))p Fy(q)1168 1322 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))710 1387 y Fs(endif)714 1439 y FC(~)-25 b Fy(p)12 b FC(=)f Fy(Ap)838 1424 y Fv(\()p Fx(i)p Fv(\))710 1496 y Fy(\017)727 1502 y Fx(i)752 1496 y FC(=)h Fy(q)816 1481 y Fv(\()p Fx(i)p Fv(\))854 1468 y Fj(T)882 1496 y FC(~)-24 b Fy(p)p FC(;)13 b Fs(if)g Fy(\017)984 1502 y Fx(i)1009 1496 y FC(=)f(0)h Fs(metho)q(d)i(fails)710 1546 y Fy(\014)733 1552 y Fx(i)759 1546 y FC(=)d Fy(\017)820 1552 y Fx(i)833 1546 y Fy(=\016)872 1552 y Fx(i)887 1546 y 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Fx(i)755 1907 y FC(=)g Fy(\032)820 1913 y Fx(i)p Fv(+1)876 1907 y Fy(=)p FC(\()p Fy(\015)934 1913 y Fx(i)p Fw(\000)p Fv(1)991 1907 y Fu(j)p Fy(\014)1026 1913 y Fx(i)1040 1907 y Fu(j)p FC(\);)h Fy(\015)1114 1913 y Fx(i)1140 1907 y FC(=)e(1)p Fy(=)1225 1871 y Ft(p)p 1266 1871 111 2 v 1266 1907 a FC(1)e(+)h Fy(\022)1358 1892 y Fv(2)1357 1918 y Fx(i)1377 1907 y FC(;)k Fs(if)e Fy(\015)1465 1913 y Fx(i)1491 1907 y FC(=)g(0)h Fs(metho)q(d)i(fails)710 1956 y Fy(\021)731 1962 y Fx(i)756 1956 y FC(=)d Fu(\000)p Fy(\021)853 1962 y Fx(i)p Fw(\000)p Fv(1)909 1956 y Fy(\032)930 1962 y Fx(i)944 1956 y Fy(\015)967 1941 y Fv(2)965 1967 y Fx(i)987 1956 y Fy(=)p FC(\()p Fy(\014)1047 1962 y Fx(i)1061 1956 y Fy(\015)1084 1941 y Fv(2)1082 1967 y Fx(i)p Fw(\000)p Fv(1)1139 1956 y FC(\))710 2006 y Fs(if)29 b Fy(i)12 b FC(=)f(1)768 2058 y Fy(d)790 2043 y Fv(\(1\))845 2058 y FC(=)h Fy(\021)910 2064 y Fv(1)929 2058 y Fy(p)950 2043 y Fv(\(1\))994 2058 y FC(;)h Fy(s)1038 2043 y Fv(\(1\))1095 2058 y FC(=)f Fy(\021)1160 2064 y Fv(1)1181 2058 y FC(~)-24 b Fy(p)710 2108 y Fs(else)768 2160 y Fy(d)790 2145 y Fv(\()p Fx(i)p Fv(\))841 2160 y FC(=)11 b Fy(\021)905 2166 y Fx(i)919 2160 y Fy(p)940 2145 y Fv(\()p Fx(i)p Fv(\))989 2160 y FC(+)e(\()p Fy(\022)1065 2166 y Fx(i)p Fw(\000)p Fv(1)1122 2160 y Fy(\015)1143 2166 y Fx(i)1158 2160 y FC(\))1174 2145 y Fv(2)1192 2160 y Fy(d)1214 2145 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))768 2211 y Fy(s)787 2196 y Fv(\()p Fx(i)p Fv(\))839 2211 y FC(=)i Fy(\021)903 2217 y Fx(i)920 2211 y FC(~)-24 b Fy(p)9 b FC(+)g(\()p Fy(\022)1023 2217 y Fx(i)p Fw(\000)p Fv(1)1080 2211 y Fy(\015)1101 2217 y Fx(i)1116 2211 y FC(\))1132 2196 y Fv(2)1150 2211 y Fy(s)1169 2196 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))710 2261 y Fs(endif)710 2313 y Fy(x)734 2298 y Fv(\()p Fx(i)p Fv(\))785 2313 y FC(=)j Fy(x)853 2298 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))944 2313 y FC(+)e Fy(d)1008 2298 y Fv(\()p Fx(i)p Fv(\))710 2365 y Fy(r)730 2350 y Fv(\()p Fx(i)p Fv(\))781 2365 y FC(=)i Fy(r)845 2350 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))937 2365 y Fu(\000)d Fy(s)997 2350 y Fv(\()p Fx(i)p Fv(\))710 2415 y FC(c)o(hec)o(k)15 b(con)o(v)o(ergence;)g(con)o(tin)o (ue)f(if)f(necessary)622 2464 y Fs(end)p 1891 2533 2 2100 v 543 2535 1350 2 v 450 2642 a FC(Figure)g(2.8:)k(The)c(Preconditioned)h(Quasi) f(Minimal)c(Residual)k(Metho)q(d)g(without)g(Lo)q(ok-ahead)p eop %%Page: 25 37 36 bop 150 275 a Fr(2.3.)31 b(NONST)m(A)m(TIONAR)m(Y)13 b(ITERA)m(TIVE)h (METHODS)564 b FC(25)150 391 y(arriv)o(ed)11 b(at)h(ab)q(out)f(the)h(same)f (appro)o(ximation)d(for)14 b(^)-24 b Fy(x)p FC(.)17 b(On)12 b(the)g(other)g(hand,)f(when)h(BiCG)f(mak)o(es)150 441 y(no)j(progress)h(at)f (all,)e(QMR)h(ma)o(y)f(still)h(sho)o(w)h(slo)o(w)f(con)o(v)o(ergence.)212 493 y(The)i(lo)q(ok-ahead)d(steps)j(in)f(the)g(QMR)g(metho)q(d)f(prev)o(en)o (t)i(breakdo)o(wn)f(in)f(all)f(cases)k(but)e(the)150 543 y(so-called)g (\\incurable)f(breakdo)o(wn".)150 662 y Fs(Implemen)o(tati)o(on)150 742 y FC(The)18 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(terpreted)g(as)g(the)f(pro)q(duct)i(of)d(BiCG)h(and)g(re-)150 2683 y(p)q(eatedly)e(applied)e(GMRES\(1\).)24 b(A)o(t)17 b(least)f(lo)q (cally)m(,)e(a)i(residual)g(v)o(ector)h(is)f(minimi)o(zed,)e(whic)o(h)150 2733 y(leads)k(to)h(a)f(considerably)g(smo)q(other)g(con)o(v)o(ergence)i(b)q (eha)o(vior.)31 b(On)19 b(the)g(other)g(hand,)g(if)f(the)p eop %%Page: 28 40 39 bop 450 275 a FC(28)755 b Fr(CHAPTER)15 b(2.)31 b(ITERA)m(TIVE)14 b(METHODS)450 391 y FC(lo)q(cal)h(GMRES\(1\))g(step)i(stagnates,)g(then)f (the)h(Krylo)o(v)e(subspace)j(is)d(not)h(expanded,)h(and)e(Bi-)450 441 y(CGST)m(AB)f(will)g(break)h(do)o(wn.)20 b(This)15 b(is)g(a)f(breakdo)o (wn)h(situation)f(that)h(can)g(o)q(ccur)h(in)e(addition)450 491 y(to)i(the)g(other)g(breakdo)o(wn)g(p)q(ossiblities)g(in)f(the)h (underlying)g(BiCG)f(algorithm.)21 b(This)16 b(t)o(yp)q(e)g(of)450 541 y(breakdo)o(wn)d(ma)o(y)e(b)q(e)j(a)o(v)o(oided)e(b)o(y)h(com)o(bining)e (BiCG)h(with)h(other)g(metho)q(ds,)g Fq(i.e.)p FC(,)f(b)o(y)h(selecting)450 591 y(other)19 b(v)n(alues)f(for)g Fy(!)785 597 y Fx(i)817 591 y FC(\(see)h(the)g(algorithm\).)29 b(One)19 b(suc)o(h)g(alternativ)o(e)f (is)g(Bi-CGST)m(AB2)g(\(see)450 640 y(Gutknec)o(h)o(t)h([114)n(]\);)h(more)d (general)h(approac)o(hes)h(are)g(suggested)g(b)o(y)f(Sleijp)q(en)g(and)g(F)m (okk)o(ema)450 690 y(in)13 b([186)o(].)450 806 y Fs(Implemen)o(tati)o(on)450 886 y FC(Bi-CGST)m(AB)k(requires)h(t)o(w)o(o)f(matrix-v)o(ector)g(pro)q (ducts)h(and)g(four)f(inner)g(pro)q(ducts,)j Fq(i.e.)p FC(,)d(t)o(w)o(o)450 936 y(inner)d(pro)q(ducts)h(more)e(than)h(BiCG)f(and)h(CGS.)512 987 y(The)i(pseudo)q(co)q(de)h(for)e(the)h(Preconditioned)g(BiConjugate)f (Gradien)o(t)g(Stabilized)g(Metho)q(d)450 1037 y(with)f(preconditioner)g Fy(M)19 b FC(is)14 b(giv)o(en)f(in)g(Figure)h(2.10.)450 1161 y Fl(2.3.9)55 b(Cheb)n(yshev)19 b(Iteration)450 1241 y FC(Cheb)o(yshev)13 b(Iteration)f(is)f(another)h(metho)q(d)f(for)h(solving)e(nonsymmetric)g (problems)h(\(see)i(Golub)450 1291 y(and)g(V)m(an)g(Loan)h([108)n(,)g Fu(x)p FC(10.1.5])d(and)j(V)m(arga)e([206)o(,)h(Chapter)i(5]\).)i(Cheb)o (yshev)e(Iteration)e(a)o(v)o(oids)450 1340 y(the)j(computation)e(of)h(inner)h (pro)q(ducts)g(as)g(is)f(necessary)j(for)d(the)h(other)g(nonstationary)f (meth-)450 1390 y(o)q(ds.)j(F)m(or)12 b(some)f(distributed)i(memory)d(arc)o (hitectures)15 b(these)e(inner)g(pro)q(ducts)h(are)e(a)g(b)q(ottlenec)o(k)450 1440 y(with)j(resp)q(ect)i(to)e(e\016ciency)m(.)21 b(The)15 b(price)h(one)f(pa)o(ys)g(for)g(a)o(v)o(oiding)d(inner)k(pro)q(ducts)g(is)f (that)g(the)450 1490 y(metho)q(d)i(requires)j(enough)e(kno)o(wledge)g(ab)q (out)g(the)h(sp)q(ectrum)g(of)f(the)h(co)q(e\016cien)o(t)g(matrix)d Fy(A)450 1540 y FC(that)f(an)h(ellipse)f(en)o(v)o(eloping)g(the)h(sp)q (ectrum)g(can)f(b)q(e)h(iden)o(ti\014ed;)g(ho)o(w)o(ev)o(er)g(this)f (di\016cult)o(y)g(can)450 1590 y(b)q(e)h(o)o(v)o(ercome)f(via)g(an)g(adaptiv) o(e)g(construction)i(dev)o(elop)q(ed)f(b)o(y)f(Man)o(teu\013el)h([143)o(],)f 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(that)g(encloses)450 2135 y(the)g(sp)q(ectrum)h(of)e Fy(A)h FC(and)g(for)f(whic)o(h)h(the)h(rate)f Fy(r)h FC(of)e(con)o(v)o(ergence)j(is) d(minima)o(l:)554 2263 y Fy(r)f FC(=)634 2235 y Fy(a)d FC(+)707 2200 y Fu(p)p 741 2200 128 2 v 741 2235 a Fy(a)763 2223 y Fv(2)791 2235 y Fu(\000)h Fy(c)851 2223 y Fv(2)p 634 2253 236 2 v 634 2296 a Fy(d)f FC(+)707 2262 y Fu(p)p 741 2262 128 2 v 741 2296 a Fy(d)763 2284 y Fv(2)791 2296 y Fu(\000)g Fy(c)850 2284 y Fv(2)874 2263 y Fy(;)1007 b FC(\(2.15\))450 2382 y(where)15 b Fy(a)f FC(is)g(the)g(length)g(of)f(the)i Fy(x)p FC(-axis)e(of)g(the)h (ellipse.)512 2434 y(W)m(e)g(pro)o(vide)g(co)q(de)h(in)f(whic)o(h)g(it)g(is)g (assumed)g(that)g Fy(c)g FC(and)g Fy(d)g FC(are)g(kno)o(wn.)19 b(F)m(or)14 b(co)q(de)h(includ-)450 2483 y(ing)g(the)h(adaptiv)o(e)f (detemination)f(of)h(these)i(iteration)f(parameters)f(the)i(reader)f(is)g (referred)h(to)450 2533 y(Ash)o(b)o(y)f([7].)24 b(The)17 b(Cheb)o(yshev)g 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y(F)m(or)g(symmetric)f(p)q(ositiv) o(e)h(de\014nite)h(systems)g(the)g(\\ellipse")e(en)o(v)o(eloping)h(the)h(sp)q (ectrum)g(de-)150 543 y(generates)h(to)d(the)i(in)o(terv)n(al)e([)p Fy(\025)641 549 y Fv(min)698 543 y Fy(;)7 b(\025)741 549 y Fv(max)804 543 y FC(])13 b(on)h(the)h(p)q(ositiv)o(e)f Fy(x)p FC(-axis,)e(where)j Fy(\025)1390 549 y Fv(min)1461 543 y FC(and)f Fy(\025)1566 549 y Fv(max)1644 543 y FC(are)150 593 y(the)k(smallest)e(and)i (largest)g(eigen)o(v)n(alues)f(of)g Fy(M)925 577 y Fw(\000)p Fv(1)969 593 y Fy(A)p FC(.)30 b(In)17 b(circumstances)i(where)f(the)g(compu-) 150 642 y(tation)f(of)f(inner)i(pro)q(ducts)h(is)e(a)g(b)q(ottlenec)o(k,)h (it)f(ma)o(y)f(b)q(e)h(adv)n(an)o(tageous)g(to)g(start)h(with)f(CG,)150 692 y(compute)d(estimates)f(of)h(the)h(extremal)e(eigen)o(v)n(alues)h(from)e (the)j(CG)e(co)q(e\016cien)o(ts,)i(and)f(then)h(af-)150 742 y(ter)i(su\016cien)o(t)f(con)o(v)o(ergence)i(of)e(these)h(appro)o(ximations)d (switc)o(h)i(to)g(Cheb)o(yshev)h(Iteration.)25 b(A)150 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b(the)g(computation)e(of)h(inner)h(pro)q(ducts,)i(as)d(is)h (necessary)i(in)d(metho)q(ds)g(lik)o(e)g(GMRES)g(or)h(CG,)150 1839 y(is)16 b(a)o(v)o(oided.)24 b(This)16 b(a)o(v)o(oids)f(the)i(sync)o (hronization)f(p)q(oin)o(ts)g(required)h(of)e(CG-t)o(yp)q(e)h(metho)q(ds,)g (so)150 1888 y(mac)o(hines)c(with)g(hierarc)o(hical)g(or)h(distributed)g (memory)d(ma)o(y)h(ac)o(hiev)o(e)h(higher)h(p)q(erformance)f(\(it)150 1938 y(also)i(suggests)j(strong)e(parallelization)e(prop)q(erties;)j(for)f(a) g(discussion)g(of)g(this)g(see)h(Saad)e([181)o(],)150 1988 y(and)f(Dongarra,)e Fq(et)j(al.)e FC([68)o(]\).)18 b(Sp)q(eci\014cally)m(,)12 b(as)h(so)q(on)f(as)h(some)f(segmen)o(t)h(of)f Fy(w)h FC(is)g(computed,)f(w)o (e)150 2038 y(ma)o(y)g(b)q(egin)i(computing,)e(in)h(sequence,)j(corresp)q (onding)e(segmen)o(ts)g(of)g Fy(p)p FC(,)f Fy(x)p FC(,)g(and)h Fy(r)q FC(.)212 2090 y(The)e(pseudo)q(co)q(de)i(for)d(the)i(Preconditioned)f (Cheb)o(yshev)h(Metho)q(d)f(with)g(preconditioner)g Fy(M)150 2140 y 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Fu(\017)20 b FC(Requires)c(matrix-v)o(ector)e(pro)q(ducts)i(with)f (the)h(co)q(e\016cien)o(t)g(matrix)e(and)h(its)g(trans-)645 2196 y(p)q(ose.)25 b(This)16 b(disquali\014es)f(the)i(metho)q(d)e(for)g (cases)j(where)f(the)f(matrix)e(is)i(only)f(im-)645 2246 y(plicitly)g(giv)o (en)g(as)i(an)e(op)q(erator,)i(and)f(the)h(transp)q(ose)g(therefore)h(not)e (a)o(v)n(ailable)d(at)645 2296 y(all.)604 2359 y Fu(\017)20 b FC(P)o(arallelization)10 b(prop)q(erties)i(are)g(similar)d(to)i(those)i (for)e(CG;)f(the)i(t)o(w)o(o)f(matrix)e(v)o(ector)645 2409 y(pro)q(ducts)j(\(as)g(w)o(ell)e(as)h(the)h(preconditioning)f(steps\))h(are)g (indep)q(enden)o(t,)g(so)f(they)h(can)645 2459 y(b)q(e)j(done)f(in)f (parallel,)f(or)i(their)h(comm)o(uni)o(cation)c(stages)k(can)f(b)q(e)g(pac)o (k)n(aged.)501 2539 y(7.)20 b(Quasi-Minimal)11 b(Residual)i(\(QMR\))604 2619 y Fu(\017)20 b FC(Applicable)14 b(to)f(nonsymmetric)f(matrices.)604 2683 y Fu(\017)20 b FC(Designed)12 b(to)f(a)o(v)o(oid)f(the)i(irregular)f (con)o(v)o(ergence)i(b)q(eha)o(vior)e(of)f(BiCG,)h(it)g(a)o(v)o(oids)f(one) 645 2733 y(of)j(the)i(t)o(w)o(o)e(breakdo)o(wn)h(situations)g(of)f(BiCG.)p eop %%Page: 33 45 44 bop 150 275 a Fr(2.4.)31 b(SUMMAR)m(Y)13 b(OF)i(THE)f(METHODS)787 b FC(33)304 391 y Fu(\017)20 b FC(If)11 b(BiCG)g(mak)o(es)f(signi\014can)o(t) h(progress)i(in)e(one)g(iteration)g(step,)i(then)f(QMR)f(deliv)o(ers)345 441 y(ab)q(out)i(the)h(same)e(result)i(at)f(the)h(same)e(step.)19 b(But)14 b(when)f(BiCG)g(temp)q(orarily)e(stag-)345 491 y(nates)18 b(or)f(div)o(erges,)g(QMR)g(ma)o(y)e(still)h(further)i(reduce)h(the)e (residual,)g(alb)q(eit)g(v)o(ery)345 541 y(slo)o(wly)m(.)304 606 y Fu(\017)j FC(Computational)10 b(costs)15 b(p)q(er)f(iteration)f(are)g (similar)e(to)i(BiCG,)f(but)i(sligh)o(tly)e(higher.)345 656 y(The)i(metho)q(d)f(requires)j(the)e(transp)q(ose)h(matrix-v)o(ector)e(pro)q (duct.)304 721 y Fu(\017)20 b FC(P)o(arallelization)12 b(prop)q(erties)j(are) g(as)f(for)f(BiCG.)201 803 y(8.)20 b(Conjugate)13 b(Gradien)o(t)h(Squared)g (\(CGS\))304 884 y Fu(\017)20 b FC(Applicable)14 b(to)f(nonsymmetric)f (matrices.)304 949 y Fu(\017)20 b FC(Con)o(v)o(erges)15 b(\(div)o(erges\))f (t)o(ypically)f(ab)q(out)h(t)o(wice)g(as)g(fast)g(as)g(BiCG.)304 1015 y Fu(\017)20 b FC(Con)o(v)o(ergence)d(b)q(eha)o(vior)e(is)g(often)g (quite)h(irregular,)f(whic)o(h)g(ma)o(y)e(lead)i(to)g(a)h(loss)f(of)345 1064 y(accuracy)h(in)e(the)i(up)q(dated)f(residual.)21 b(T)m(ends)15 b(to)g(div)o(erge)g(if)e(the)j(starting)f(guess)g(is)345 1114 y(close)f(to)g(the)h(solution.)304 1179 y Fu(\017)20 b FC(Computational)d (costs)j(p)q(er)g(iteration)f(are)h(similar)d(to)i(BiCG,)f(but)i(the)g(metho) q(d)345 1229 y(do)q(esn't)14 b(require)h(the)g(transp)q(ose)g(matrix.)304 1294 y Fu(\017)20 b FC(Unlik)o(e)15 b(BiCG,)f(the)i(t)o(w)o(o)f(matrix-v)o (ector)f(pro)q(ducts)j(are)e(not)h(indep)q(enden)o(t,)g(so)g(the)345 1344 y(n)o(um)o(b)q(er)d(of)h(sync)o(hronization)f(p)q(oin)o(ts)h(in)g(a)f (parallel)g(en)o(vironmen)o(t)g(is)g(larger.)201 1426 y(9.)20 b(Biconjugate)14 b(Gradien)o(t)f(Stabilized)h(\(Bi-CGST)m(AB\))304 1508 y Fu(\017)20 b FC(Applicable)14 b(to)f(nonsymmetric)f(matrices.)304 1573 y Fu(\017)20 b FC(Computational)13 b(costs)18 b(p)q(er)f(iteration)f (are)g(similar)e(to)i(BiCG)g(and)g(CGS,)f(but)i(the)345 1623 y(metho)q(d)c(do)q(esn't)i(require)f(the)h(transp)q(ose)g(matrix.)304 1688 y Fu(\017)20 b FC(An)d(alternativ)o(e)g(for)g(CGS)g(that)g(a)o(v)o(oids) f(the)i(irregular)f(con)o(v)o(ergence)h(patterns)h(of)345 1738 y(CGS)d(while)h(main)o(taini)o(ng)d(ab)q(out)i(the)i(same)e(sp)q(eed)i(of)e (con)o(v)o(ergence;)j(as)e(a)f(result)345 1787 y(w)o(e)e(observ)o(e)h(often)f (less)h(loss)f(of)f(accuracy)i(in)e(the)i(up)q(dated)f(residual.)180 1869 y(10.)20 b(Cheb)o(yshev)15 b(Iteration)304 1951 y Fu(\017)20 b FC(Applicable)15 b(to)f(nonsymmetric)f(matrices)i(\(but)g(presen)o(ted)i (in)e(this)g(b)q(o)q(ok)f(only)h(for)345 2001 y(the)g(symmetric)d(case\).)304 2066 y Fu(\017)20 b FC(This)15 b(metho)q(d)f(requires)i(some)e(explicit)g (kno)o(wledge)h(of)f(the)i(sp)q(ectrum)f(\(or)g(\014eld)g(of)345 2116 y(v)n(alues\);)e(in)g(the)i(symmetric)d(case)i(the)g(iteration)g (parameters)f(are)h(easily)f(obtained)345 2165 y(from)g(the)j(t)o(w)o(o)f (extremal)f(eigen)o(v)n(alues,)h(whic)o(h)f(can)i(b)q(e)f(estimated)g(either) h(directly)345 2215 y(from)8 b(the)i(matrix,)e(or)h(from)e(applying)h(a)h (few)h(iterations)f(of)g(the)h(Conjugate)f(Gradien)o(t)345 2265 y(Metho)q(d.)304 2330 y Fu(\017)20 b FC(The)g(computational)d(structure) 22 b(is)d(similar)e(to)i(that)h(of)f(CG,)f(but)i(there)h(are)e(no)345 2380 y(sync)o(hronization)14 b(p)q(oin)o(ts.)304 2445 y Fu(\017)20 b FC(The)11 b(Adaptiv)o(e)g(Cheb)o(yshev)h(metho)q(d)e(can)h(b)q(e)h(used)f (in)g(com)o(bination)d(with)j(metho)q(ds)345 2495 y(as)17 b(CG)f(or)g(GMRES,) g(to)g(con)o(tin)o(ue)h(the)g(iteration)f(once)i(suitable)e(b)q(ounds)h(on)g (the)345 2545 y(sp)q(ectrum)d(ha)o(v)o(e)g(b)q(een)h(obtained)f(from)e(these) j(metho)q(ds.)212 2633 y(Selecting)d(the)g(\\b)q(est")f(metho)q(d)g(for)f(a)h (giv)o(en)g(class)g(of)g(problems)f(is)h(largely)f(a)h(matter)g(of)f(trial) 150 2683 y(and)15 b(error.)22 b(It)15 b(also)f(dep)q(ends)j(on)e(ho)o(w)f(m)o (uc)o(h)g(storage)i(one)f(has)g(a)o(v)n(ailable)e(\(GMRES\),)h(on)h(the)150 2733 y(a)o(v)n(ailabili)o(t)o(y)8 b(of)i Fy(A)434 2717 y Fx(T)472 2733 y FC(\(BiCG)g(and)h(QMR\),)f(and)h(on)f(ho)o(w)h(exp)q(ensiv)o(e)h(the)f (matrix)e(v)o(ector)j(pro)q(ducts)p eop %%Page: 34 46 45 bop 450 275 a FC(34)755 b Fr(CHAPTER)15 b(2.)31 b(ITERA)m(TIVE)14 b(METHODS)450 391 y FC(\(and)i(Solv)o(e)g(steps)h(with)f Fy(M)5 b FC(\))15 b(are)i(in)e(comparison)g(to)h FA(SAXPY)p FC(s)f(and)h(inner)g (pro)q(ducts.)26 b(If)16 b(these)450 441 y(matrix)f(v)o(ector)j(pro)q(ducts)g (are)f(relativ)o(ely)f(exp)q(ensiv)o(e,)i(and)f(if)f(su\016cien)o(t)h (storage)h(is)e(a)o(v)n(ailable)450 491 y(then)f(it)e(ma)o(y)f(b)q(e)j (attractiv)o(e)f(to)f(use)i(GMRES)f(and)f(dela)o(y)h(restarting)g(as)g(m)o (uc)o(h)f(as)h(p)q(ossible.)512 541 y(T)m(able)19 b(2.1)g(sho)o(ws)h(the)g(t) o(yp)q(e)g(of)f(op)q(erations)g(p)q(erformed)h(p)q(er)g(iteration.)35 b(Based)20 b(on)g(the)450 591 y(particular)12 b(problem)e(or)i(data)f (structure,)j(the)f(user)g(ma)o(y)c(observ)o(e)k(that)f(a)g(particular)f(op)q (eration)450 641 y(could)j(b)q(e)g(p)q(erformed)g(more)f(e\016cien)o(tly)m(.) 450 780 y Fp(2.5)70 b(A)22 b(short)i(history)f(of)g(Krylo)n(v)h(metho)r(ds) 1656 758 y FC(1)450 871 y(Metho)q(ds)18 b(based)g(on)f(orthogonalization)e(w) o(ere)j(dev)o(elop)q(ed)g(b)o(y)f(a)g(n)o(um)o(b)q(er)g(of)f(authors)i(in)f (the)450 921 y(early)g('50s.)26 b(Lanczos')17 b(metho)q(d)f([139)n(])h(w)o (as)f(based)i(on)e(t)o(w)o(o)g(m)o(utually)f(orthogonal)g(v)o(ector)j(se-)450 971 y(quences,)k(and)d(his)h(motiv)n(atio)o(n)d(came)h(from)g(eigen)o(v)n (alue)h(problems.)34 b(In)19 b(that)h(con)o(text,)h(the)450 1021 y(most)14 b(prominen)o(t)g(feature)i(of)e(the)i(metho)q(d)e(is)h(that)h (it)e(reduces)k(the)d(original)f(matrix)f(to)i(tridi-)450 1070 y(agonal)d(form.)k(Lanczos)d(later)g(applied)g(his)g(metho)q(d)f(to)h (solving)f(linear)g(systems,)h(in)f(particular)450 1120 y(symmetric)h(ones)i ([140)n(].)k(An)c(imp)q(ortan)o(t)d(prop)q(ert)o(y)k(for)e(pro)o(ving)f(con)o (v)o(ergence)j(of)e(the)h(metho)q(d)450 1170 y(when)i(solving)e(linear)g (systems)i(is)f(that)g(the)h(iterates)g(are)g(related)f(to)g(the)h(initial)d (residual)i(b)o(y)450 1220 y(m)o(ultiplicatio)o(n)11 b(with)j(a)g(p)q (olynomial)c(in)j(the)i(co)q(e\016cien)o(t)g(matrix.)512 1270 y(The)j(join)o(t)f(pap)q(er)h(b)o(y)g(Hestenes)i(and)d(Stiefel)h([121)n(],)g (after)g(their)g(indep)q(enden)o(t)h(disco)o(v)o(ery)450 1320 y(of)14 b(the)h(same)f(metho)q(d,)f(is)h(the)i(classical)e(description)h(of)f (the)h(conjugate)g(gradien)o(t)f(metho)q(d)g(for)450 1370 y(solving)i(linear) h(systems.)28 b(Although)17 b(error-reduction)i(prop)q(erties)f(are)g(pro)o (v)o(ed,)g(and)f(exp)q(eri-)450 1419 y(men)o(ts)12 b(sho)o(wing)g(premature)h (con)o(v)o(ergence)i(are)e(rep)q(orted,)h(the)g(conjugate)f(gradien)o(t)f (metho)q(d)g(is)450 1469 y(presen)o(ted)k(here)f(as)f(a)g(direct)g(metho)q (d,)f(rather)i(than)f(an)f(iterativ)o(e)h(metho)q(d.)512 1519 y(This)22 b(Hestenes/Stiefel)i(metho)q(d)e(is)g(closely)g(related)h(to)f(a)f (reduction)i(of)f(the)g(Lanczos)450 1569 y(metho)q(d)17 b(to)h(symmetric)e (matrices,)i(reducing)g(the)h(t)o(w)o(o)e(m)o(utually)e(orthogonal)i (sequences)j(to)450 1619 y(one)c(orthogonal)f(sequence,)j(but)f(there)g(is)f (an)g(imp)q(ortan)o(t)e(algorithmic)g(di\013erence.)26 b(Whereas)450 1669 y(Lanczos)13 b(used)f(three-term)h(recurrences,)i(the)d(metho)q(d)f(b)o (y)h(Hestenes)i(and)d(Stiefel)h(uses)h(coupled)450 1719 y(t)o(w)o(o-term)j (recurrences)q(.)31 b(By)18 b(com)o(bining)c(the)k(t)o(w)o(o)f(t)o(w)o (o-term)f(recurrence)q(s)k(\(eliminating)15 b(the)450 1769 y(\\searc)o(h)g(directions"\))f(the)h(Lanczos)f(metho)q(d)f(is)h(obtained.) 512 1819 y(A)33 b(pap)q(er)g(b)o(y)f(Arnoldi)f([6])g(further)i(discusses)i (the)e(Lanczos)g(biorthogonalization)450 1868 y(metho)q(d,)17 b(but)g(it)g(also)f(presen)o(ts)j(a)e(new)h(metho)q(d,)e(com)o(bining)f (features)j(of)f(the)h(Lanczos)f(and)450 1918 y(Hestenes/Stiefel)k(metho)q (ds.)31 b(Lik)o(e)18 b(the)h(Lanczos)h(metho)q(d)d(it)h(is)h(applied)f(to)g (nonsymmetric)450 1968 y(systems,)j(and)e(it)g(do)q(es)i(not)e(use)h(searc)o (h)h(directions.)36 b(Lik)o(e)19 b(the)h(Hestenes/Stiefel)i(metho)q(d,)450 2018 y(it)d(generates)i(only)d(one,)j(self-orthogonal)d(sequence.)36 b(This)19 b(last)g(fact,)i(com)o(bined)d(with)h(the)450 2068 y(asymmetry)j(of)h(the)h(co)q(e\016cien)o(t)h(matrix)d(means)h(that)g(the)i (metho)q(d)e(no)g(longer)h(e\013ects)h(a)450 2118 y(reduction)16 b(to)g(tridiagonal)d(form,)h(but)i(instead)f(one)h(to)f(upp)q(er)i(Hessen)o (b)q(erg)h(form.)j(Presen)o(ted)450 2167 y(as)16 b(\\minim)o(i)o(zed)e (iterations)i(in)f(the)h(Galerkin)f(metho)q(d")f(this)i(algorithm)d(has)j(b)q (ecome)g(kno)o(wn)450 2217 y(as)e(the)g Fq(A)o(rnoldi)h(algorithm)p FC(.)512 2267 y(The)22 b(conjugate)e(gradien)o(t)h(metho)q(d)f(receiv)o(ed)i (little)e(atten)o(tion)h(as)g(a)f(practical)h(metho)q(d)450 2317 y(for)15 b(some)f(time,)g(partly)h(b)q(ecause)h(of)f(a)g(misp)q(erceiv)o (ed)g(imp)q(ortance)f(of)h(the)h(\014nite)f(termination)450 2367 y(prop)q(ert)o(y)m(.)i(Reid)10 b([174)o(])g(p)q(oin)o(ted)g(out)g(that)h (the)g(most)e(imp)q(ortan)o(t)f(application)h(area)i(la)o(y)e(in)h(sparse)450 2417 y(de\014nite)15 b(systems,)e(and)h(this)g(renew)o(ed)h(the)g(in)o (terest)g(in)e(the)i(metho)q(d.)512 2467 y(Sev)o(eral)i(metho)q(ds)f(ha)o(v)o (e)g(b)q(een)i(dev)o(elop)q(ed)f(in)f(later)g(y)o(ears)h(that)f(emplo)o(y)m (,)f(most)g(often)h(im-)450 2517 y(plicitly)m(,)g(the)i(upp)q(er)h(Hessen)o (b)q(erg)h(matrix)c(of)h(the)h(Arnoldi)f(metho)q(d.)28 b(F)m(or)17 b(an)h(o)o(v)o(erview)f(and)450 2566 y(c)o(haracterization)12 b(of)f(these)i(orthogonal)e(pro)r(jection)h(metho)q(ds)f(for)g(nonsymmetric)f (systems)i(see)450 2616 y(Ash)o(b)o(y)m(,)h(Man)o(teu\013el)i(and)e(Sa)o (ylor)g([10)o(],)g(Saad)h(and)f(Sc)o(h)o(ultz)i([184)n(],)e(and)h(Jea)g(and)g (Y)m(oung)f([124)o(].)p 450 2655 620 2 v 496 2681 a Fo(1)514 2693 y Fn(F)m(or)g(a)g(more)g(detailed)e(accoun)o(t)h(of)h(the)g(early)g (history)f(of)h(CG)h(metho)q(ds,)e(w)o(e)i(refer)f(the)g(reader)f(to)h(Golub) 450 2733 y(and)e(O'Leary)f([107)o(])i(and)e(Hestenes)h([122)n(].)p eop %%Page: 35 47 46 bop 150 275 a Fr(2.6.)31 b(SUR)-5 b(VEY)14 b(OF)h(RECENT)f(KR)m(YLO)o(V)f (METHODS)555 b FC(35)212 391 y(Fletc)o(her)18 b([94)o(])e(prop)q(osed)i(an)f (implem)o(en)o(tation)d(of)i(the)h(Lanczos)h(metho)q(d,)e(similar)e(to)j(the) 150 441 y(Conjugate)12 b(Gradien)o(t)g(metho)q(d,)g(with)g(t)o(w)o(o)g (coupled)h(t)o(w)o(o-term)e(recurrences)q(,)k(whic)o(h)d(he)h(named)150 491 y(the)h Fq(bi-c)n(onjugate)i(gr)n(adient)e(metho)n(d)19 b FC(\(BiCG\).)150 638 y Fp(2.6)70 b(Surv)n(ey)23 b(of)h(recen)n(t)d(Krylo)n (v)j(metho)r(ds)150 733 y FC(Researc)o(h)14 b(in)o(to)f(the)h(design)f(of)f (Krylo)o(v)h(subspace)i(metho)q(ds)e(for)f(solving)g(nonsymmetric)g(linear) 150 782 y(systems)g(is)f(an)h(activ)o(e)f(\014eld)h(of)f(researc)o(h)i(and)f (new)g(metho)q(ds)f(are)h(still)f(emerging.)16 b(In)11 b(this)h(b)q(o)q(ok,) 150 832 y(w)o(e)k(ha)o(v)o(e)f(included)h(only)e(the)j(b)q(est)f(kno)o(wn)f (and)h(most)e(p)q(opular)h(metho)q(ds,)g(and)g(in)g(particular)150 882 y(those)d(for)f(whic)o(h)g(extensiv)o(e)i(computational)c(exp)q(erience)k (has)f(b)q(een)g(gathered.)18 b(In)12 b(this)f(section,)150 932 y(w)o(e)20 b(shall)f(brie\015y)g(highligh)o(t)f(some)h(of)g(the)h(recen)o (t)h(dev)o(elopmen)o(ts)e(and)h(other)g(metho)q(ds)f(not)150 982 y(treated)d(here.)23 b(A)16 b(surv)o(ey)g(of)e(metho)q(ds)h(up)g(to)g(ab) q(out)g(1991)f(can)i(b)q(e)f(found)g(in)g(F)m(reund,)g(Golub)150 1031 y(and)k(Nac)o(h)o(tigal)f([98)o(].)34 b(Tw)o(o)18 b(more)h(recen)o(t)h (rep)q(orts)h(b)o(y)e(Meier-Y)m(ang)g([148)o(])g(and)g(T)m(ong)f([192)o(])150 1081 y(ha)o(v)o(e)12 b(extensiv)o(e)i(n)o(umerical)d(comparisons)g(among)f(v) n(arious)i(metho)q(ds,)g(including)f(sev)o(eral)i(more)150 1131 y(recen)o(t)i(ones)g(that)f(ha)o(v)o(e)g(not)f(b)q(een)i(discussed)h(in) d(detail)h(in)f(this)h(b)q(o)q(ok.)212 1183 y(Sev)o(eral)f(suggestions)h(ha)o (v)o(e)e(b)q(een)i(made)e(to)h(reduce)h(the)g(increase)g(in)e(memory)f(and)h (compu-)150 1233 y(tational)f(costs)j(in)f(GMRES.)f(An)h(ob)o(vious)f(one)h (is)g(to)f(restart)i(\(this)g(one)f(is)f(included)h(in)g Fu(x)p FC(2.3.4\):)150 1282 y(GMRES\()p Fy(m)p FC(\).)27 b(Another)17 b(approac)o(h)g(is)g(to)f(restrict)j(the)e(GMRES)f(searc)o(h)i(to)f(a)f (suitable)h(sub-)150 1332 y(space)c(of)d(some)h(higher-dimensional)e(Krylo)o (v)i(subspace.)19 b(Metho)q(ds)13 b(based)f(on)f(this)h(idea)f(can)h(b)q(e) 150 1382 y(view)o(ed)g(as)h(preconditioned)g(GMRES)e(metho)q(ds.)17 b(The)c(simplest)e(ones)i(exploit)f(a)g(\014xed)g(p)q(olyno-)150 1432 y(mial)e(preconditioner)j(\(see)h(Johnson,)f(Micc)o(helli)f(and)g(P)o (aul)g([125)o(],)g(Saad)g([178)o(],)g(and)g(Nac)o(h)o(tigal,)150 1482 y(Reic)o(hel)j(and)g(T)m(refethen)i([155)n(]\).)22 b(In)16 b(more)e(sophisticated)i(approac)o(hes,)g(the)g(p)q(olynomial)c(pre-)150 1531 y(conditioner)e(is)h(adapted)f(to)h(the)g(iterations)f(\(Saad)g([183)o (]\),)g(or)h(the)g(preconditioner)g(ma)o(y)d(ev)o(en)k(b)q(e)150 1581 y(some)h(other)i(\(iterativ)o(e\))g(metho)q(d)e(of)h(c)o(hoice)h(\(V)m (an)e(der)i(V)m(orst)g(and)f(V)m(uik)f([204)o(],)h(Axelsson)g(and)150 1631 y(V)m(assilevski)d([23)o(]\).)16 b(Stagnation)11 b(is)g(prev)o(en)o(ted) h(in)f(the)h(GMRESR)e(metho)q(d)g(\(V)m(an)h(der)h(V)m(orst)f(and)150 1681 y(V)m(uik)i([204)o(]\))h(b)o(y)g(including)f(LSQR)h(steps)h(in)f(some)f (phases)i(of)f(the)g(pro)q(cess.)21 b(In)14 b(Desturler)h(and)150 1731 y(F)m(okk)o(ema)e([61)o(],)h(part)h(of)g(the)h(optimali)o(t)o(y)c(of)j 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y(exp)q(ended)d(in)f(forming)d(the)k(Krylo)o(v)e(space)i(with)f Fy(A)982 2116 y Fx(T)1022 2131 y FC(for)f(further)i(reduction)g(of)e(the)h (residual.)212 2183 y(As)g(discussed)h(b)q(efore,)f(the)f(BiCG)g(metho)q(d)g (can)g(ha)o(v)o(e)g(t)o(w)o(o)g(kinds)g(of)g(breakdo)o(wn:)18 b Fq(L)n(anczos)150 2233 y(br)n(e)n(akdown)h FC(\(the)e(underlying)f(Lanczos) h(pro)q(cess)h(breaks)f(do)o(wn\),)f(and)g Fq(pivot)h(br)n(e)n(akdown)i FC(\(the)150 2282 y(tridiagonal)8 b(matrix)h Fy(T)16 b FC(implicitly)7 b(generated)12 b(in)d(the)i(underlying)f(Lanczos)h(pro)q(cess)h(encoun)o (ters)150 2332 y(a)e(zero)h(piv)o(ot)e(when)h(Gaussian)g(elimination)d (without)i(piv)o(oting)g(is)h(used)g(to)g(factor)g(it\).)17 b(Although)150 2382 y(suc)o(h)h(exact)g(breakdo)o(wns)g(are)g(v)o(ery)f(rare) h(in)f(practice,)i(near)f(breakdo)o(wns)g(can)f(cause)i(sev)o(ere)150 2432 y(n)o(umerical)12 b(stabilit)o(y)h(problems.)212 2483 y(The)18 b(piv)o(ot)f(breakdo)o(wn)g(is)g(the)h(easier)h(one)e(to)g(o)o(v)o 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b(has)g(b)q(een)h(giv)o(en)f(to)g(analyzing)f(the)i (nature)f(of)g(the)h(Lanczos)g(breakdo)o(wn)f(\(see)h(P)o(ar-)450 491 y(lett)20 b([168)o(],)g(and)f(Gutknec)o(h)o(t)h([112)o(,)f(115]\),)h(as)f (w)o(ell)g(as)h(v)n(arious)f(lo)q(ok-ahead)g(tec)o(hniques)i(for)450 541 y(remedying)13 b(it)g(\(see)i(Brezinski)g(and)e(Sadok)g([38],)f (Brezinski,)i(Zaglia)f(and)g(Sadok)h([39)o(,)f(40)o(],)g(F)m(re-)450 591 y(und)e(and)g(Nac)o(h)o(tigal)f([100)o(],)g(P)o(arlett)i([168)o(],)e(Nac) o(h)o(tigal)g([156)o(],)h(F)m(reund,)g(Gutknec)o(h)o(t)h(and)f(Nac)o(h)o(ti-) 450 640 y(gal)f([105)o(],)g(Joub)q(ert)i([129)o(],)f(F)m(reund,)g(Golub)f (and)h(Nac)o(h)o(tigal)e([104)o(],)i(and)f(Gutknec)o(h)o(t)i([112)n(,)f(115)o (]\).)450 690 y(Ho)o(w)o(ev)o(er,)h(the)f(resulting)g(algorithms)e(are)i (usually)g(to)q(o)f(complicated)g(to)h(giv)o(e)f(in)h(template)f(form)450 740 y(\(some)h(co)q(des)i(of)e(F)m(reund)h(and)f(Nac)o(h)o(tigal)f(are)i(a)o (v)n(ailable)e(on)h FA(netlib)p FC(.\))16 b(Moreo)o(v)o(er,)c(it)f(is)h (still)e(not)450 790 y(p)q(ossible)16 b(to)g(eliminate)f(breakdo)o(wns)h (that)g(require)h(lo)q(ok-ahead)f(steps)h(of)f(arbitrary)g(size)h(\(in-)450 840 y(curable)e(breakdo)o(wns\).)j(So)c(far,)f(these)j(metho)q(ds)d(ha)o(v)o (e)h(not)g(y)o(et)g(receiv)o(ed)i(m)o(uc)o(h)c(practical)i(use)450 889 y(but)f(some)f(form)f(of)i(lo)q(ok-ahead)f(ma)o(y)f(pro)o(v)o(e)i(to)g(b) q(e)g(a)g(crucial)g(comp)q(onen)o(t)f(in)h(future)g(metho)q(ds.)512 939 y(In)18 b(the)h(BiCG)e(metho)q(d,)h(the)h(need)g(for)f(matrix-v)o(ector)e (m)o(ultiplies)g(with)i Fy(A)1766 924 y Fx(T)1810 939 y FC(can)g(b)q(e)h(in-) 450 989 y(con)o(v)o(enien)o(t)14 b(as)g(w)o(ell)f(as)g(doubling)g(the)h(n)o (um)o(b)q(er)f(of)g(matrix-v)o(ector)f(m)o(ultiplies)g(compared)h(with)450 1039 y(CG)h(for)g(eac)o(h)g(increase)i(in)d(the)i(degree)h(of)d(the)i (underlying)f(Krylo)o(v)g(subspace.)20 b(Sev)o(eral)15 b(recen)o(t)450 1089 y(metho)q(ds)g(ha)o(v)o(e)g(b)q(een)i(prop)q(osed)f(to)g(o)o(v)o(ercome) 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y(iterates)h(corresp)q(onding)f(to)f(a)g(pro)q(duct)i(of)e(the)h(BiCG)f (p)q(olynomial)d(with)j(another)h(p)q(olynomial)450 1487 y(of)c(the)h(same)e (degree,)j(c)o(hosen)g(to)e(ha)o(v)o(e)g(certain)h(desirable)g(prop)q(erties) h(but)e(computable)g(without)450 1537 y(recourse)j(to)d Fy(A)687 1522 y Fx(T)714 1537 y FC(.)17 b(The)11 b(Bi-CGST)m(AB)f(metho)q(d)g(of)g(V)m (an)g(der)i(V)m(orst)e([202)o(])g(is)h(suc)o(h)h(an)e(example,)g(in)450 1587 y(whic)o(h)g(the)i(auxiliary)c(p)q(olynomial)g(is)i(de\014ned)i(b)o(y)e (a)g(lo)q(cal)g(minim)o(ization)d(c)o(hosen)12 b(to)e(smo)q(oth)g(the)450 1637 y(con)o(v)o(ergence)16 b(b)q(eha)o(vior.)k(Gutknec)o(h)o(t)15 b([114)o(])f(noted)h(that)g(Bi-CGST)m(AB)f(could)g(b)q(e)h(view)o(ed)g(as)f (a)450 1686 y(pro)q(duct)f(of)e(BiCG)g(and)h(GMRES\(1\),)f(and)g(he)i (suggested)g(com)o(bining)c(BiCG)i(with)h(GMRES\(2\))450 1736 y(for)i(the)h(ev)o(en)g(n)o(um)o(b)q(ered)f(iteration)f(steps.)21 b(This)14 b(w)o(as)g(an)o(ticipated)g(to)g(lead)g(to)g(b)q(etter)i(con)o(v)o (er-)450 1786 y(gence)f(for)e(the)h(case)g(where)h(the)f(eigen)o(v)n(alues)f (of)g Fy(A)h FC(are)f(complex.)k(A)d(more)e(e\016cien)o(t)i(and)f(more)450 1836 y(robust)h(v)n(arian)o(t)f(of)g(this)h(approac)o(h)f(has)h(b)q(een)h (suggested)g(b)o(y)f(Sleijp)q(en)f(and)h(F)m(okk)o(ema)d(in)i([186)o(],)450 1886 y(where)g(they)e(describ)q(e)j(ho)o(w)d(to)g(easily)g(com)o(bine)f(BiCG) h(with)g(an)o(y)g(GMRES\()p Fy(m)p FC(\),)g(for)g(mo)q(dest)g Fy(m)p FC(.)512 1936 y(Man)o(y)19 b(other)h(basic)f(metho)q(ds)f(can)i(also)e (b)q(e)i(squared.)34 b(F)m(or)19 b(example,)f(b)o(y)h(squaring)g(the)450 1985 y(Lanczos)g(pro)q(cedure,)j(Chan,)d(de)g(Pillis)f(and)g(V)m(an)g(der)i (V)m(orst)f([44)o(])f(obtained)g(transp)q(ose-free)450 2035 y(implemen)o(tatio)o(ns)g(of)i(BiCG)g(and)h(QMR.)f(By)h(squaring)f(the)h(QMR) f(metho)q(d,)h(F)m(reund)g(and)450 2085 y(Szeto)14 b([102)o(])f(deriv)o(ed)i (a)e(transp)q(ose-free)j(QMR)d(squared)i(metho)q(d)d(whic)o(h)i(is)g(quite)f (comp)q(etitiv)o(e)450 2135 y(with)d(CGS)g(but)h(with)g(m)o(uc)o(h)e(smo)q (other)h(con)o(v)o(ergence.)19 b(Unfortunately)m(,)10 b(these)i(metho)q(ds)e (require)450 2185 y(an)j(extra)g(matrix-v)o(ector)f(pro)q(duct)i(p)q(er)g (step)g(\(three)g(instead)g(of)e(t)o(w)o(o\))h(whic)o(h)f(mak)o(es)g(them)h (less)450 2234 y(e\016cien)o(t.)512 2284 y(In)18 b(addition)e(to)h(Bi-CGST)m (AB,)g(sev)o(eral)h(recen)o(t)h(pro)q(duct)f(metho)q(ds)g(ha)o(v)o(e)f(b)q (een)i(designed)450 2334 y(to)g(smo)q(oth)f(the)i(con)o(v)o(ergence)i(of)c (CGS.)h(One)h(idea)f(is)g(to)h(use)g(the)g(quasi-minim)o(a)o(l)c(residual)450 2384 y(\(QMR\))e(principle)g(to)g(obtain)f(smo)q(othed)h(iterates)h(from)d (the)j(Krylo)o(v)e(subspace)j(generated)f(b)o(y)450 2434 y(other)i(pro)q (duct)h(metho)q(ds.)25 b(F)m(reund)17 b([103)o(])f(prop)q(osed)h(suc)o(h)h(a) e(QMR)g(v)o(ersion)h(of)f(CGS,)f(whic)o(h)450 2483 y(he)h(called)g(TF)o(QMR.) f(Numerical)g(exp)q(erimen)o(ts)h(sho)o(w)g(that)g(TF)o(QMR)f(in)h(most)e (cases)k(retains)450 2533 y(the)e(desirable)g(con)o(v)o(ergence)i(features)e (of)g(CGS)f(while)g(correcting)i(its)e(erratic)i(b)q(eha)o(vior.)23 b(The)450 2583 y(transp)q(ose)16 b(free)g(nature)g(of)e(TF)o(QMR,)h(its)g(lo) o(w)f(computational)e(cost)k(and)f(its)g(smo)q(oth)f(con)o(v)o(er-)450 2633 y(gence)19 b(b)q(eha)o(vior)e(mak)o(e)f(it)h(an)g(attractiv)o(e)h (alternativ)o(e)f(to)h(CGS.)e(On)i(the)g(other)h(hand,)e(since)450 2683 y(the)g(BiCG)g(p)q(olynomial)c(is)j(still)g(used,)i(TF)o(QMR)f(breaks)g (do)o(wn)g(whenev)o(er)h(CGS)e(do)q(es.)27 b(One)450 2733 y(p)q(ossible)16 b(remedy)e(w)o(ould)h(b)q(e)h(to)f(com)o(bine)f(TF)o(QMR)h(with)g(a)g(lo)q (ok-ahead)f(Lanczos)i(tec)o(hnique)p eop %%Page: 37 49 48 bop 150 275 a Fr(2.6.)31 b(SUR)-5 b(VEY)14 b(OF)h(RECENT)f(KR)m(YLO)o(V)f (METHODS)555 b FC(37)150 391 y(but)17 b(this)g(app)q(ears)h(to)f(b)q(e)g (quite)g(complicated)f(and)h(no)f(metho)q(ds)h(of)f(this)h(kind)g(ha)o(v)o(e) f(y)o(et)i(ap-)150 441 y(p)q(eared)d(in)f(the)g(literature.)20 b(Recen)o(tly)m(,)13 b(Chan)h Fq(et.)h(al.)e FC([45)o(])h(deriv)o(ed)g(a)g (similar)e(QMR)i(v)o(ersion)g(of)150 491 y(V)m(an)f(der)i(V)m(orst's)f (Bi-CGST)m(AB)f(metho)q(d,)g(whic)o(h)g(is)h(called)g(QMR)o(CGST)m(AB.)f (These)i(metho)q(ds)150 541 y(o\013er)g(smo)q(other)e(con)o(v)o(ergence)i(o)o (v)o(er)f(CGS)g(and)g(Bi-CGST)m(AB)f(with)g(little)h(additional)e(cost.)212 591 y(There)h(is)e(no)g(clear)g(b)q(est)i(Krylo)o(v)d(subspace)j(metho)q(d)e (at)g(this)g(time,)f(and)h(there)i(will)c(nev)o(er)k(b)q(e)150 640 y(a)g(b)q(est)i Fq(over)n(al)r(l)i FC(Krylo)o(v)c(subspace)i(metho)q(d.)i (Eac)o(h)d(of)f(the)h(metho)q(ds)f(is)g(a)h(winner)f(in)g(a)h(sp)q(eci\014c) 150 690 y(problem)e(class,)i(and)f(the)h(main)d(problem)h(is)i(to)f(iden)o (tify)g(these)h(classes)h(and)e(to)h(construct)h(new)150 740 y(metho)q(ds)h(for)g(unco)o(v)o(ered)i(classes.)27 b(The)17 b(pap)q(er)g(b)o(y)f(Nac)o(h)o(tigal,)g(Reddy)g(and)g(T)m(refethen)i([154)o (])150 790 y(sho)o(ws)e(that)h(for)f(an)o(y)f(of)h(a)g(group)g(of)f(metho)q (ds)h(\(CG,)f(BiCG,)g(GMRES,)g(CGNE,)h(and)g(CGS\),)150 840 y(there)e(is)f(a)f(class)h(of)g(problems)e(for)i(whic)o(h)g(a)f(giv)o(en)g (metho)q(d)g(is)h(the)g(winner)g(and)g(another)g(one)g(is)150 889 y(the)h(loser.)19 b(This)13 b(sho)o(ws)h(clearly)g(that)g(there)h(will)d (b)q(e)i(no)g(ultimate)e(metho)q(d.)17 b(The)d(b)q(est)h(w)o(e)f(can)150 939 y(hop)q(e)f(for)f(is)g(some)g(exp)q(ert)h(system)f(that)h(guides)f(the)h (user)h(in)e(his/her)h(c)o(hoice.)18 b(Hence,)13 b(iterativ)o(e)150 989 y(metho)q(ds)i(will)f(nev)o(er)i(reac)o(h)h(the)f(robustness)h(of)e (direct)h(metho)q(ds,)f(nor)g(will)f(they)i(b)q(eat)g(direct)150 1039 y(metho)q(ds)e(for)h(all)f(problems.)20 b(F)m(or)15 b(some)f(problems)g (iterativ)o(e)h(sc)o(hemes)g(and)g(for)f(others)i(direct)150 1089 y(metho)q(ds)i(\(or)g(m)o(ultigrid\))e(will)h(b)q(e)i(most)e(attractiv)o (e.)31 b(W)m(e)18 b(hop)q(e)h(to)f(\014nd)h(suitable)f(metho)q(ds)150 1139 y(\(and)12 b(preconditioners\))h(for)e(classes)i(of)e(v)o(ery)h(large)g (problems)e(that)i(w)o(e)g(are)g(y)o(et)g(unable)g(to)f(solv)o(e)150 1188 y(b)o(y)k(an)o(y)g(kno)o(wn)g(metho)q(d,)g(b)q(ecause)i(of)d (CPU-restrictions,)j(memory)m(,)c(con)o(v)o(ergence)k(problems,)150 1238 y(ill-conditioning,)10 b(et)15 b(cetera.)p eop %%Page: 38 50 49 bop 450 275 a FC(38)755 b Fr(CHAPTER)15 b(2.)31 b(ITERA)m(TIVE)14 b(METHODS)p eop %%Page: 39 51 50 bop 150 703 a Fz(Chapter)34 b(3)150 910 y FB(Preconditione)q(rs)150 1151 y Fp(3.1)70 b(The)22 b(wh)n(y)h(and)h(ho)n(w)150 1242 y FC(As)14 b(w)o(e)f(ha)o(v)o(e)h(seen,)g(the)g(con)o(v)o(ergence)h(rate)f (of)e(iterativ)o(e)i(metho)q(ds)f(dep)q(ends)i(on)e(sp)q(ectral)h(prop-)150 1292 y(erties)f(of)e(the)h(co)q(e\016cien)o(t)h(matrix.)i(Hence)f(one)e(ma)o (y)e(attempt)h(to)g(transform)g(the)h(linear)f(system)150 1342 y(in)o(to)17 b(one)g(that)h(is)f(equiv)n(alen)o(t)g(in)g(the)h(sense)h(that)f (it)f(has)g(the)h(same)f(solution,)g(but)h(that)f(has)150 1391 y(more)f(fa)o(v)o(orable)f(sp)q(ectral)j(prop)q(erties.)28 b(A)16 b Fq(pr)n(e)n(c)n(onditioner)21 b FC(is)c(a)f(matrix)f(that)h (e\013ects)j(suc)o(h)e(a)150 1441 y(transformation.)212 1491 y(F)m(or)g(instance,)h(if)e(a)h(matrix)e Fy(M)21 b FC(appro)o(ximates)16 b(the)h(co)q(e\016cien)o(t)h(matrix)d Fy(A)i FC(in)g(some)f(w)o(a)o(y)m(,)150 1541 y(the)e(transformed)g(system)254 1618 y Fy(M)299 1600 y Fw(\000)p Fv(1)343 1618 y Fy(Ax)e FC(=)f Fy(M)498 1600 y Fw(\000)p Fv(1)543 1618 y Fy(b)150 1694 y FC(has)i(the)h(same)e(solution)g (as)i(the)f(original)e(system)i Fy(Ax)f FC(=)f Fy(b)p FC(,)i(but)g(the)h(sp)q (ectral)g(prop)q(erties)h(of)d(its)150 1744 y(co)q(e\016cien)o(t)j(matrix)d Fy(M)524 1729 y Fw(\000)p Fv(1)568 1744 y Fy(A)i FC(ma)o(y)e(b)q(e)j(more)d (fa)o(v)o(orable.)212 1794 y(In)f(devising)e(a)i(preconditioner,)g(w)o(e)f (are)h(faced)g(with)f(a)g(c)o(hoice)h(b)q(et)o(w)o(een)g(\014nding)f(a)g (matrix)e Fy(M)150 1844 y FC(that)k(appro)o(ximates)d Fy(A)p FC(,)j(and)f(for)g(whic)o(h)h(solving)e(a)h(system)g(is)h(easier)g(than)f (solving)g(one)g(with)h Fy(A)p FC(,)150 1894 y(or)17 b(\014nding)f(a)h (matrix)e Fy(M)21 b FC(that)c(appro)o(ximates)e Fy(A)965 1878 y Fw(\000)p Fv(1)1010 1894 y FC(,)i(so)g(that)g(only)f(m)o(ultiplicatio)o(n)e (b)o(y)j Fy(M)k FC(is)150 1943 y(needed.)e(The)c(ma)r(jorit)o(y)c(of)j (preconditioners)h(falls)d(in)i(the)g(\014rst)h(category;)e(a)h(notable)g (example)150 1993 y(of)f(the)i(second)g(category)f(will)e(b)q(e)j(discussed)h (in)d Fu(x)p FC(3.5.)150 2108 y Fl(3.1.1)55 b(Cost)19 b(trade-o\013)150 2185 y FC(Since)11 b(applying)e(a)h(preconditioner)i(incurs)f(some)e(extra)i (cost,)h(b)q(oth)e(initially)e(and)j(p)q(er)g(iteration,)150 2234 y(there)k(is)f(a)g(trade-o\013)h(b)q(et)o(w)o(een)g(the)g(cost)g(of)e (constructing)i(and)f(applying)f(the)i(preconditioner,)150 2284 y(and)20 b(the)h(gain)f(in)g(con)o(v)o(ergence)i(sp)q(eed.)39 b(Certain)21 b(preconditioners)h(need)f(no)f(construction)150 2334 y(phase)12 b(at)g(all)e(\(for)h(instance)i(the)f(SSOR)f (preconditioner\),)i(but)f(for)f(others,)h(suc)o(h)h(as)e(incomplete)150 2384 y(factorizations,)17 b(there)h(can)f(b)q(e)h(substan)o(tial)e(w)o(ork)h (in)o(v)o(olv)o(ed.)26 b(Although)16 b(the)i(w)o(ork)f(in)f(scalar)150 2434 y(terms)d(ma)o(y)f(b)q(e)i(comparable)e(to)i(a)f(single)g(iteration,)g (the)h(construction)g(of)f(the)h(preconditioner)150 2483 y(ma)o(y)e(not)j(b)q (e)g(v)o(ectorizable/parallelizable)e(ev)o(en)i(if)e(application)g(of)h(the)h (preconditioner)g(is.)k(In)150 2533 y(that)13 b(case,)g(the)h(initial)d(cost) i(has)g(to)g(b)q(e)g(amortized)f(o)o(v)o(er)g(the)i(iterations,)e(or)h(o)o(v) o(er)g(rep)q(eated)h(use)150 2583 y(of)f(the)i(same)e(preconditioner)h(in)g (m)o(ultiple)d(linear)j(systems.)212 2633 y(Most)h(preconditioners)g(tak)o(e) g(in)f(their)h(application)d(an)j(amoun)o(t)d(of)i(w)o(ork)g(prop)q(ortional) f(to)150 2683 y(the)18 b(n)o(um)o(b)q(er)e(of)h(v)n(ariables.)26 b(This)17 b(implies)e(that)j(they)f(m)o(ultiply)d(the)k(w)o(ork)f(p)q(er)h (iteration)e(b)o(y)150 2733 y(a)g(constan)o(t)h(factor.)25 b(On)16 b(the)h(other)g(hand,)f(the)h(n)o(um)o(b)q(er)e(of)h(iterations)g(as) g(a)g(function)g(of)g(the)904 2838 y(39)p eop %%Page: 40 52 51 bop 450 275 a FC(40)802 b Fr(CHAPTER)14 b(3.)32 b(PRECONDITIONERS)450 391 y FC(matrix)13 b(size)j(is)g(usually)e(only)h(impro)o(v)o(ed)e(b)o(y)i(a) g(constan)o(t.)23 b(Certain)15 b(preconditioners)i(are)e(able)450 441 y(to)e(impro)o(v)o(e)e(on)h(this)h(situation,)f(most)g(notably)g(the)i (mo)q(di\014ed)d(incomplete)h(factorizations)g(and)450 491 y(preconditioners)j(based)g(on)e(m)o(ultigrid)e(tec)o(hniques.)512 541 y(On)20 b(parallel)e(mac)o(hines)g(there)j(is)e(a)g(further)h (trade-o\013)g(b)q(et)o(w)o(een)g(the)g(e\016cacy)g(of)f(a)g(pre-)450 591 y(conditioner)d(in)g(the)g(classical)g(sense,)i(and)d(its)h(parallel)f (e\016ciency)m(.)25 b(Man)o(y)16 b(of)f(the)i(traditional)450 640 y(preconditioners)e(ha)o(v)o(e)f(a)f(large)h(sequen)o(tial)g(comp)q(onen) o(t.)450 754 y Fl(3.1.2)55 b(Theoretical)17 b(prerequisites)f(on)j (preconditioners)450 831 y FC(The)g(ab)q(o)o(v)o(e)f(transformation)e(of)i (the)g(linear)g(system)g Fy(A)h Fu(!)f Fy(M)1493 816 y Fw(\000)p Fv(1)1538 831 y Fy(A)g FC(is)g(not)g(what)g(is)h(used)g(in)450 881 y(practice.)i(A)15 b(more)f(correct)i(w)o(a)o(y)e(of)g(in)o(tro)q(ducing) g(the)h(preconditioner)h(w)o(ould)e(b)q(e)h(to)f(split)h(the)450 930 y(preconditioner)g(as)f Fy(M)i FC(=)c Fy(M)916 936 y Fv(1)935 930 y Fy(M)975 936 y Fv(2)1007 930 y FC(and)i(to)g(transform)f(the)h(system)g (as)554 1003 y Fy(M)599 985 y Fw(\000)p Fv(1)594 1014 y(1)643 1003 y Fy(AM)719 985 y Fw(\000)p Fv(1)714 1014 y(2)764 1003 y FC(\()p Fy(M)820 1009 y Fv(2)839 1003 y Fy(x)p FC(\))d(=)h Fy(M)979 985 y Fw(\000)p Fv(1)974 1014 y(1)1023 1003 y Fy(b:)450 1075 y FC(The)i(matrices)g Fy(M)742 1081 y Fv(1)774 1075 y FC(and)g Fy(M)895 1081 y Fv(2)928 1075 y FC(are)g(called)g(the)g Fq(left-)f FC(and)h Fq(right)g(pr)n(e)n(c)n(onditioners)p FC(,)f(resp)q (ectiv)o(ely)m(.)512 1124 y(An)h(iterativ)o(e)g(metho)q(d)f(can)h(b)q(e)h (preconditioned)f(according)g(to)g(the)g(follo)o(wing)e(sc)o(heme:)501 1197 y(1.)20 b(T)m(ransform)12 b(the)i(righ)o(t)g(hand)g(side)g(v)o(ector)g Fy(b)e Fu( )f Fy(M)1366 1179 y Fw(\000)p Fv(1)1361 1208 y(1)1410 1197 y Fy(b)p FC(.)501 1274 y(2.)20 b(Apply)13 b(the)i(\(unpreconditioned\))f (iterativ)o(e)g(metho)q(d,)f(replacing)g(the)h(co)q(e\016cien)o(t)h(matrix) 554 1324 y Fy(A)f FC(b)o(y)f Fy(M)701 1306 y Fw(\000)p Fv(1)696 1335 y(1)746 1324 y Fy(AM)822 1306 y Fw(\000)p Fv(1)817 1335 y(2)866 1324 y FC(;)h(call)f(the)h(resulting)g(solution)f Fy(y)q FC(.)501 1401 y(3.)20 b(Compute)13 b Fy(x)e FC(=)h Fy(M)858 1384 y Fw(\000)p Fv(1)853 1413 y(2)902 1401 y Fy(y)q FC(.)450 1474 y(The)g(imp)q(ortan)o(t)e(theoretical)j(p)q(oin)o(t)e(is)h(that)g(the)g (transformed)g(co)q(e\016cien)o(t)g(matrix)e Fy(M)1834 1456 y Fw(\000)p Fv(1)1829 1485 y(1)1879 1474 y Fy(AM)1955 1456 y Fw(\000)p Fv(1)1950 1485 y(2)450 1523 y FC(preserv)o(es)21 b(some)d(theoretical)i(prop)q(erties)g(of)f Fy(A)g FC(and)g Fy(M)5 b FC(:)27 b(if)19 b Fy(A)g FC(is)f(symmetric)g(and)g(p)q(ositiv)o(e) 450 1573 y(de\014nite)g(and)f Fy(M)725 1579 y Fv(1)760 1573 y FC(=)g Fy(M)854 1558 y Fx(T)849 1583 y Fv(2)880 1573 y FC(,)h(then)f(the)h (transformed)e(co)q(e\016cien)o(t)i(matrix)d(is)i(again)f(symmetric)450 1623 y(and)e(p)q(ositiv)o(e)f(de\014nite.)512 1673 y(Since)h(symmetry)e(and)h (de\014niteness)i(are)f(crucial)f(to)h(the)g(success)i(of)c(some)h(iterativ)o (e)g(meth-)450 1723 y(o)q(ds,)h(this)f(transformation)f(is)i(to)f(b)q(e)h (preferred)i(o)o(v)o(er)e Fy(M)1359 1708 y Fw(\000)p Fv(1)1403 1723 y Fy(A)p FC(,)f(whic)o(h)h(is)f(not)h(guaran)o(teed)g(to)g(b)q(e)450 1772 y(either)h(symmetric)d(or)i(de\014nite,)g(ev)o(en)g(if)f Fy(A)h FC(and)g Fy(M)19 b FC(are.)512 1822 y(It)13 b(is)g(a)f(remark)n(able)g (prop)q(ert)o(y)h(of)f(man)o(y)f(iterativ)o(e)i(metho)q(ds)f(that)h(the)h (splitting)d(of)i Fy(M)k FC(is)c(in)450 1872 y(practice)i(not)g(needed.)21 b(By)14 b(rewriting)h(the)g(steps)g(of)f(the)h(metho)q(d)f(\(see)h(for)g (instance)g(Axelsson)450 1922 y(and)f(Bark)o(er)h([14)o(,)e(pgs.)h(16,29])e (or)i(Golub)f(and)h(V)m(an)f(Loan)h([108)o(,)f Fu(x)q FC(10.3]\))f(it)i(is)g (usually)f(p)q(ossible)450 1972 y(to)h(rein)o(tro)q(duce)h(a)f(computational) d(step)554 2044 y(solv)o(e)i Fy(u)h FC(from)e Fy(M)5 b(u)11 b FC(=)h Fy(v)r(;)450 2116 y FC(that)i(is,)f(a)h(step)h(that)f(applies)f(the) i(preconditioner)f(in)g(its)g(en)o(tiret)o(y)m(.)450 2251 y Fp(3.2)70 b(Jacobi)22 b(Preconditioning)450 2342 y FC(The)14 b(simplest)f(preconditioner)i(consists)g(of)e(just)h(the)h(diagonal)d(of)h (the)h(matrix:)554 2436 y Fy(m)590 2442 y Fx(i;j)641 2436 y FC(=)685 2378 y Ft(\032)737 2411 y Fy(a)759 2417 y Fx(i;i)835 2411 y FC(if)f Fy(i)f FC(=)g Fy(j)737 2461 y FC(0)77 b(otherwise)q Fy(:)450 2533 y FC(This)14 b(is)g(kno)o(wn)f(as)h(the)g(\(p)q(oin)o(t\))g (Jacobi)g(preconditioner.)512 2583 y(It)k(is)f(p)q(ossible)h(to)f(use)i(this) e(preconditioner)i(without)e(using)g(an)o(y)g(extra)h(storage)g(b)q(ey)o(ond) 450 2633 y(that)e(of)e(the)j(matrix)c(itself.)23 b(Ho)o(w)o(ev)o(er,)16 b(division)e(op)q(erations)i(are)g(usually)e(quite)i(costly)m(,)f(so)h(in)450 2683 y(practice)g(storage)f(is)f(allo)q(cated)h(for)f(the)h(recipro)q(cals)h (of)e(the)h(matrix)e(diagonal.)19 b(This)14 b(strategy)450 2733 y(applies)g(to)f(man)o(y)f(preconditioners)j(b)q(elo)o(w.)p eop %%Page: 41 53 52 bop 150 275 a Fr(3.3.)31 b(SSOR)14 b(PRECONDITIONING)874 b FC(41)150 391 y Fl(3.2.1)55 b(Blo)r(c)n(k)18 b(Jacobi)g(Metho)r(ds)150 472 y FC(Blo)q(c)o(k)g(v)o(ersions)g(of)f(the)h(Jacobi)g(preconditioner)g (can)g(b)q(e)g(deriv)o(ed)h(b)o(y)e(a)g(partitioning)g(of)g(the)150 522 y(v)n(ariables.)i(If)14 b(the)g(index)h(set)g Fy(S)g FC(=)e Fu(f)p FC(1)p Fy(;)7 b(:)g(:)g(:)t(;)g(n)p Fu(g)13 b FC(is)i(partitioned)f (as)g Fy(S)h FC(=)1303 491 y Ft(S)1337 534 y Fx(i)1358 522 y Fy(S)1383 528 y Fx(i)1411 522 y FC(with)f(the)h(sets)h Fy(S)1685 528 y Fx(i)150 572 y FC(m)o(utually)11 b(disjoin)o(t,)i(then)254 686 y Fy(m)290 692 y Fx(i;j)341 686 y FC(=)385 628 y Ft(\032)437 661 y Fy(a)459 667 y Fx(i;j)539 661 y FC(if)g Fy(i)h FC(and)g Fy(j)i FC(are)f(in)e(the)i(same)e(index)g(subset)437 711 y(0)81 b(otherwise)q Fy(:)150 798 y FC(The)14 b(preconditioner)h(is)f(no)o(w)f(a)h (blo)q(c)o(k-diagonal)d(matrix.)212 851 y(Often,)j(natural)g(c)o(hoices)g (for)g(the)g(partitioning)f(suggest)i(themselv)o(es:)212 940 y Fu(\017)21 b FC(In)14 b(problems)f(with)g(m)o(ultiple)f(ph)o(ysical)h(v)n (ariables)g(p)q(er)i(no)q(de,)f(blo)q(c)o(ks)g(can)g(b)q(e)h(formed)d(b)o(y) 254 990 y(grouping)h(the)h(equations)g(p)q(er)h(no)q(de.)212 1082 y Fu(\017)21 b FC(In)9 b(structured)j(matrices,)d(suc)o(h)h(as)g(those)g (from)d(partial)i(di\013eren)o(tial)g(equations)g(on)g(regular)254 1132 y(grids,)21 b(a)f(partitioning)f(can)h(b)q(e)h(based)g(on)f(the)g(ph)o (ysical)g(domain.)35 b(Examples)19 b(are)h(a)254 1182 y(partitioning)15 b(along)g(lines)h(in)g(the)h(2D)e(case,)j(or)e(planes)h(in)f(the)g(3D)g (case.)27 b(This)16 b(will)f(b)q(e)254 1231 y(discussed)g(further)g(in)f Fu(x)p FC(3.4.3.)212 1323 y Fu(\017)21 b FC(On)f(parallel)g(computers)g(it)g (is)g(natural)g(to)g(let)g(the)h(partitioning)e(coincide)i(with)f(the)254 1373 y(division)12 b(of)i(v)n(ariables)f(o)o(v)o(er)h(the)g(pro)q(cessors.) 150 1501 y Fl(3.2.2)55 b(Discussion)150 1582 y FC(Jacobi)9 b(preconditioners)i(need)g(v)o(ery)f(little)f(storage,)h(ev)o(en)g(in)g(the)g (blo)q(c)o(k)f(case,)i(and)f(they)g(are)g(easy)150 1632 y(to)15 b(implemen)o(t.)21 b(Additionally)m(,)13 b(on)j(parallel)e(computers)i(they)g (don't)f(presen)o(t)i(an)o(y)e(particular)150 1681 y(problems.)212 1733 y(On)24 b(the)g(other)g(hand,)h(more)d(sophisticated)i(preconditioners)h (usually)d(yield)h(a)g(larger)150 1783 y(impro)o(v)o(emen)o(t.)400 1768 y Fv(1)150 1932 y Fp(3.3)70 b(SSOR)22 b(preconditioning)150 2027 y FC(The)d(SSOR)f(preconditioner)627 2012 y Fv(2)665 2027 y FC(lik)o(e)f(the)i(Jacobi)g(preconditioner,)g(can)g(b)q(e)g(deriv)o(ed)g (from)e(the)150 2077 y(co)q(e\016cien)o(t)e(matrix)d(without)h(an)o(y)h(w)o (ork.)212 2129 y(If)g(the)g(original,)e(symmetric,)f(matrix)i(is)g(decomp)q (osed)h(as)254 2221 y Fy(A)d FC(=)h Fy(D)f FC(+)f Fy(L)f FC(+)h Fy(L)534 2204 y Fx(T)p 150 2260 620 2 v 196 2287 a Fo(1)214 2299 y Fn(Under)i(certain)g(conditions,)f(one)i(can)g(sho)o(w)g(that)g(the)g (p)q(oin)o(t)f(Jacobi)g(algorithm)f(is)i(optimal,)f(or)h(close)g(to)150 2338 y(optimal,)d(in)i(the)f(sense)g(of)h(reducing)d(the)j(condition)d(n)o (um)o(b)q(er,)h(among)g(all)i(preconditio)o(ner)o(s)d(of)j(diagonal)e(form.) 150 2377 y(This)g(w)o(as)g(sho)o(wn)f(b)o(y)g(F)m(orsythe)f(and)h(Strauss)f (for)h(matrices)f(with)h(Prop)q(ert)o(y)f(A)i([96)o(],)g(and)f(b)o(y)g(v)n (an)g(der)g(Sluis)g([193)n(])150 2417 y(for)15 b(general)e(sparse)h (matrices.)24 b(F)m(or)15 b(extensions)e(to)h(blo)q(c)o(k)g(Jacobi)g (precondition)o(ers,)f(see)i(Demmel)e([63)o(])i(and)150 2456 y(Elsner)c([91)o(].)196 2484 y Fo(2)214 2496 y Fn(The)k(SOR)h(and)e (Gauss-Seidel)f(matrices)g(are)i(nev)o(er)f(used)h(as)g(preconditio)o(ner)o (s,)f(for)h(a)g(rather)f(tec)o(hnical)150 2535 y(reason.)f(SOR-preconditio)o (nin)o(g)7 b(with)j(optimal)d Fh(!)k Fn(maps)e(the)g(eigen)o(v)n(alues)e(of)i (the)g(co)q(e\016cien)o(t)f(matrix)g(to)h(a)h(circle)150 2575 y(in)j(the)g(complex)e(plane;)h(see)h(Hageman)f(and)g(Y)m(oung)h([119)o(,)g Fg(x)p Fn(9.3].)20 b(In)13 b(this)f(case)h(no)g(p)q(olynomial)d(acceleration) 150 2614 y(is)j(p)q(ossible,)f Ff(i.e.)p Fn(,)i(the)e(accelerating)e(p)q (olynomia)o(l)h(reduces)g(to)i(the)f(trivial)g(p)q(olynomial)e Fh(P)1394 2618 y Fi(n)1415 2614 y Fn(\()p Fh(x)p Fn(\))i(=)h Fh(x)1535 2603 y Fi(n)1556 2614 y Fn(,)h(and)e(the)150 2654 y(resulting)c(metho)q(d)g(is)i(simply)f(the)g(stationary)f(SOR)i(metho)q(d.)j (Recen)o(t)c(researc)o(h)f(b)o(y)i(Eiermann)d(and)j(V)m(arga)f([81)o(])150 2693 y(has)k(sho)o(wn)g(that)f(p)q(olynomial)e(acceleration)g(of)j(SOR)h (with)f(sub)q(optimal)e Fh(!)j Fn(will)g(yield)e(no)h(impro)o(v)o(emen)n(t)e (o)o(v)o(er)150 2733 y(simple)f(SOR)i(with)f(optimal)e Fh(!)q Fn(.)p eop %%Page: 42 54 53 bop 450 275 a FC(42)802 b Fr(CHAPTER)14 b(3.)32 b(PRECONDITIONERS)450 391 y FC(in)13 b(its)h(diagonal,)e(lo)o(w)o(er,)h(and)h(upp)q(er)h (triangular)d(part,)i(the)h(SSOR)e(matrix)f(is)i(de\014ned)h(as)554 482 y Fy(M)h FC(=)c(\()p Fy(D)f FC(+)e Fy(L)p FC(\))p Fy(D)835 465 y Fw(\000)p Fv(1)881 482 y FC(\()p Fy(D)i FC(+)e Fy(L)p FC(\))1027 465 y Fx(T)1054 482 y Fy(;)450 571 y FC(or,)k(parametrized)h(b)o (y)g Fy(!)554 679 y(M)5 b FC(\()p Fy(!)q FC(\))12 b(=)758 650 y(1)p 719 669 99 2 v 719 707 a(2)c Fu(\000)i Fy(!)822 679 y FC(\()847 650 y(1)p 843 669 28 2 v 843 707 a Fy(!)876 679 y(D)g FC(+)g Fy(L)p FC(\)\()1031 650 y(1)p 1027 669 V 1027 707 a Fy(!)1060 679 y(D)q FC(\))1111 661 y Fw(\000)p Fv(1)1156 679 y FC(\()1180 650 y(1)p 1177 669 V 1177 707 a Fy(!)1210 679 y(D)g FC(+)g Fy(L)p FC(\))1340 661 y Fx(T)1366 679 y Fy(:)450 782 y FC(The)17 b(optimal)d(v)n(alue)i(of)g(the)i Fy(!)g FC(parameter,)f(lik) o(e)f(the)h(parameter)f(in)h(the)g(SOR)g(metho)q(d,)f(will)450 832 y(reduce)h(the)f(n)o(um)o(b)q(er)f(of)g(iterations)h(to)f(a)g(lo)o(w)o (er)g(order.)24 b(Sp)q(eci\014cally)m(,)15 b(the)h(sp)q(ectral)h(condition) 450 882 y(n)o(um)o(b)q(er)h Fy(\024)p FC(\()p Fy(M)691 867 y Fw(\000)p Fv(1)686 892 y Fx(!)707 896 y Fk(opt)752 882 y Fy(A)p FC(\))i(=)g Fy(O)q FC(\()920 846 y Ft(p)p 962 846 88 2 v 36 x Fy(\024)p FC(\()p Fy(A)p FC(\)\))f(is)f(attainable,)h(see)h (Axelsson)f(and)g(Bark)o(er)g([14)o(,)h Fu(x)p FC(1.4].)450 932 y(In)e(practice,)i(ho)o(w)o(ev)o(er,)e(the)h(sp)q(ectral)g(information)c (needed)k(to)f(calculate)g(the)h(optimal)c Fy(!)k FC(is)450 982 y(prohibitiv)o(ely)12 b(exp)q(ensiv)o(e)j(to)f(compute.)512 1033 y(The)k(SSOR)g(matrix)d(is)j(giv)o(en)f(in)g(factored)h(form,)e(so)i (this)f(preconditioner)h(shares)h(man)o(y)450 1083 y(prop)q(erties)14 b(of)e(other)h(factorization-based)f(metho)q(ds)g(\(see)i(b)q(elo)o(w\).)k(F) m(or)12 b(instance,)h(its)f(suitabil-)450 1133 y(it)o(y)i(for)g(v)o(ector)i (pro)q(cessors)g(or)f(parallel)e(arc)o(hitectures)k(dep)q(ends)f(strongly)f (on)f(the)h(ordering)g(of)450 1183 y(the)i(v)n(ariables.)27 b(On)17 b(the)g(other)h(hand,)f(since)g(this)g(factorization)f(is)h(giv)o(en) f Fq(a)i(priori)t FC(,)e(there)i(is)450 1233 y(no)e(p)q(ossibilit)o(y)f(of)h (breakdo)o(wn)h(as)f(in)g(the)h(construction)h(phase)f(of)f(incomplete)f (factorization)450 1282 y(metho)q(ds.)450 1430 y Fp(3.4)70 b(Incomplete)20 b(F)-6 b(actorization)22 b(Preconditioners)450 1524 y FC(A)11 b(broad)h(class)g(of)e(preconditioners)j(is)e(based)h(on)f (incomplete)g(factorizations)g(of)f(the)i(co)q(e\016cien)o(t)450 1574 y(matrix.)19 b(W)m(e)c(call)f(a)h(factorization)f(incomplete)g(if)g (during)g(the)i(factorization)e(pro)q(cess)j(certain)450 1624 y Fq(\014l)r(l)e FC(elemen)o(ts,)d(zero)g(p)q(ositions)g(that)f(w)o(ould)g(b) q(e)h(nonzero)g(in)f(an)h(exact)g(factorization,)f(ha)o(v)o(e)g(b)q(een)450 1674 y(ignored.)21 b(Suc)o(h)15 b(a)g(preconditioner)g(is)g(then)g(giv)o(en)g (in)f(factored)i(form)d Fy(M)18 b FC(=)13 b Fy(LU)20 b FC(with)15 b Fy(L)g FC(lo)o(w)o(er)450 1723 y(and)h Fy(U)21 b FC(upp)q(er)c(triangular.) 24 b(The)17 b(e\016cacy)f(of)g(the)h(preconditioner)g(dep)q(ends)g(on)f(ho)o (w)g(w)o(ell)f Fy(M)450 1773 y FC(appro)o(ximates)d Fy(A)p FC(.)450 1899 y Fl(3.4.1)55 b(Creating)19 b(an)g(incomplete)c(factorization) 450 1980 y FC(Incomplete)f(factorizations)g(are)h(the)h(\014rst)f (preconditioners)h(w)o(e)f(ha)o(v)o(e)f(encoun)o(tered)j(so)d(far)h(for)450 2029 y(whic)o(h)e(there)h(is)f(a)g(non-trivial)e(creation)j(stage.)k (Incomplete)12 b(factorizations)h(ma)o(y)e(break)j(do)o(wn)450 2079 y(\(attempted)k(division)e(b)o(y)i(zero)g(piv)o(ot\))g(or)f(result)i(in) e(inde\014nite)h(matrices)g(\(negativ)o(e)f(piv)o(ots\))450 2129 y(ev)o(en)i(if)f(the)h(full)e(factorization)g(of)h(the)h(same)f(matrix)e (is)i(guaran)o(teed)h(to)g(exist)f(and)h(yield)e(a)450 2179 y(p)q(ositiv)o(e)d(de\014nite)g(matrix.)512 2231 y(An)e(incomplete)e (factorization)h(is)g(guaran)o(teed)h(to)f(exist)h(for)f(man)o(y)e (factorization)i(strategies)450 2280 y(if)16 b(the)h(original)e(matrix)g(is)h (an)h Fy(M)5 b FC(-matrix.)24 b(This)17 b(w)o(as)f(originally)f(pro)o(v)o(ed) h(b)o(y)h(Meijerink)f(and)450 2330 y(V)m(an)i(der)i(V)m(orst)f([149)n(];)i (see)f(further)f(Beau)o(w)o(ens)h(and)f(Quenon)g([32)o(],)g(Man)o(teu\013el)h ([144)n(],)f(and)450 2380 y(V)m(an)13 b(der)i(V)m(orst)f([195)o(].)512 2432 y(In)20 b(cases)h(where)g(piv)o(ots)e(are)h(zero)g(or)g(negativ)o(e,)g (strategies)h(ha)o(v)o(e)e(b)q(een)i(prop)q(osed)f(suc)o(h)450 2482 y(as)e(substituting)h(an)f(arbitrary)g(p)q(ositiv)o(e)g(n)o(um)o(b)q(er) g(\(see)i(Kersha)o(w)f([132)o(]\),)f(or)h(restarting)g(the)450 2531 y(factorization)13 b(on)h Fy(A)9 b FC(+)h Fy(\013I)17 b FC(for)c(some)g(p)q(ositiv)o(e)h(v)n(alue)f(of)g Fy(\013)h FC(\(see)h(Man)o(teu\013el)g([144)n(]\).)512 2583 y(An)d(imp)q(ortan)o(t)e (consideration)h(for)g(incomplete)g(factorization)f(preconditioners)j(is)e (the)i(cost)450 2633 y(of)i(the)i(factorization)e(pro)q(cess.)27 b(Ev)o(en)16 b(if)f(the)i(incomplete)e(factorization)g(exists,)i(the)g(n)o (um)o(b)q(er)450 2683 y(of)e(op)q(erations)h(in)o(v)o(olv)o(ed)f(in)g (creating)h(it)f(is)h(at)f(least)h(as)g(m)o(uc)o(h)f(as)h(for)f(solving)g(a)g (system)h(with)450 2733 y(suc)o(h)g(a)e(co)q(e\016cien)o(t)i(matrix,)d(so)h (the)i(cost)f(ma)o(y)e(equal)i(that)g(of)f(one)h(or)g(more)f(iterations)g(of) h(the)p eop %%Page: 43 55 54 bop 150 275 a Fr(3.4.)31 b(INCOMPLETE)15 b(F)-5 b(A)o(CTORIZA)m(TION)14 b(PRECONDITIONERS)328 b FC(43)150 391 y(iterativ)o(e)15 b(metho)q(d.)20 b(On)c(parallel)e(computers)h(this)g(problem)e(is)i(aggra)o(v)n(ated)f(b)o(y) h(the)h(generally)150 441 y(p)q(o)q(or)e(parallel)f(e\016ciency)h(of)g(the)g (factorization.)212 491 y(Suc)o(h)f(factorization)e(costs)i(can)f(b)q(e)g (amortized)f(if)g(the)i(iterativ)o(e)f(metho)q(d)f(tak)o(es)h(man)o(y)e (itera-)150 541 y(tions,)i(or)g(if)f(the)i(same)e(preconditioner)i(will)d(b)q (e)j(used)g(for)e(sev)o(eral)i(linear)f(systems,)g(for)f(instance)150 591 y(in)i(successiv)o(e)k(time)12 b(steps)j(or)f(Newton)h(iterations.)150 707 y Fl(3.4.2)55 b(P)n(oin)n(t)20 b(incomplete)15 b(factorizations)150 783 y FC(The)c(most)f(common)d(t)o(yp)q(e)12 b(of)e(incomplete)f (factorization)h(is)h(based)g(on)g(taking)e(a)i(set)g Fy(S)j FC(of)c(matrix)150 833 y(p)q(ositions,)j(and)g(k)o(eeping)g(all)f(p)q (ositions)g(outside)i(this)f(set)h(equal)f(to)g(zero)h(during)f(the)h (factoriza-)150 883 y(tion.)j(The)e(resulting)f(factorization)f(is)h (incomplete)f(in)g(the)h(sense)i(that)e(\014ll)f(is)h(supressed.)212 933 y(The)k(set)h Fy(S)h FC(is)e(usually)e(c)o(hosen)j(to)e(encompass)h(all)e (p)q(ositions)h(\()p Fy(i;)7 b(j)r FC(\))19 b(for)e(whic)o(h)g Fy(a)1560 939 y Fx(i;j)1617 933 y Fu(6)p FC(=)h(0.)150 983 y(A)12 b(p)q(osition)f(that)h(is)f(zero)i(in)e Fy(A)h FC(but)g(not)f(so)h(in) f(an)h(exact)g(factorization)f(is)h(called)f(a)h Fq(\014l)r(l)j FC(p)q(osition,)150 1032 y(and)h(if)g(it)g(is)h(outside)g Fy(S)r FC(,)g(the)g(\014ll)f(there)i(is)e(said)h(to)f(b)q(e)h(\\discarded".)27 b(Often,)17 b Fy(S)i FC(is)e(c)o(hosen)g(to)150 1082 y(coincide)f(with)f(the) h(set)g(of)f(nonzero)h(p)q(ositions)f(in)g Fy(A)p FC(,)h(discarding)f(all)f (\014ll.)22 b(This)15 b(factorization)150 1132 y(t)o(yp)q(e)k(is)f(called)g (the)h Fy(I)s(LU)5 b FC(\(0\))20 b(factorization:)26 b(the)19 b(Incomplete)f Fy(LU)23 b FC(factorization)18 b(of)g(degree)150 1182 y(zero)223 1167 y Fv(3)243 1182 y FC(.)212 1232 y(W)m(e)c(can)g(describ) q(e)h(an)f(incomplete)f(factorization)g(formally)e(as)254 1340 y(for)i(eac)o(h)i Fy(k)q FC(,)e Fy(i;)7 b(j)14 b(>)e(k)q FC(:)41 b Fy(a)665 1346 y Fx(i;j)715 1340 y Fu( )768 1281 y Ft(\032)820 1313 y Fy(a)842 1319 y Fx(i;j)890 1313 y Fu(\000)10 b Fy(a)954 1319 y Fx(i;k)996 1313 y Fy(a)1018 1296 y Fw(\000)p Fv(1)1018 1326 y Fx(k)q(;k)1066 1313 y Fy(a)1088 1319 y Fx(k)q(;j)1175 1313 y FC(if)k(\()p Fy(i;)7 b(j)r FC(\))12 b Fu(2)f Fy(S)820 1366 y(a)842 1372 y Fx(i;j)1175 1366 y FC(otherwise)q Fy(:)150 1450 y FC(Meijerink)e(and)h(V)m(an)f(der)h(V)m(orst)g([149)n(])f(pro)o(v)o (ed)h(that,)g(if)f Fy(A)g FC(is)h(an)f Fy(M)c FC(-matrix,)j(suc)o(h)i(a)f (factorization)150 1499 y(exists)21 b(for)g(an)o(y)f(c)o(hoice)h(of)f Fy(S)r FC(,)i(and)f(giv)o(es)f(a)h(symmetric)d(p)q(ositiv)o(e)j(de\014nite)g (matrix)e(if)h Fy(A)g FC(is)150 1549 y(symmetric)8 b(p)q(ositiv)o(e)h (de\014nite.)17 b(Guidelines)9 b(for)g(allo)o(wing)e(lev)o(els)i(of)g(\014ll) f(w)o(ere)i(giv)o(en)f(b)o(y)g(Meijerink)150 1599 y(and)14 b(V)m(an)f(der)i(V)m(orst)f(in)f([150)o(].)212 1649 y(F)m(or)21 b(the)g Fy(I)s(LU)5 b FC(\(0\))21 b(metho)q(d,)g(the)h(incomplete)d (factorization)h(nev)o(er)i(alters)f(the)g(nonzero)150 1699 y(elemen)o(ts)14 b(of)g(the)h(original)e(matrix,)f(so)j(that)f(w)o(e)h(ha)o (v)o(e)g(the)g(follo)o(wing)d(situation.)19 b(If)14 b(the)h(co)q(e\016-)150 1749 y(cien)o(t)g(matrix)e(is)h(split)g(in)o(to)g(its)h(diagonal,)d(lo)o(w)o (er)i(triangular,)f(and)i(upp)q(er)g(triangular)f(parts)h(as)150 1798 y Fy(A)d FC(=)f Fy(D)d FC(+)e Fy(L)g FC(+)g Fy(U)f FC(,)14 b(the)f(preconditioner)h(can)e(b)q(e)h(written)g(as)g Fy(M)j FC(=)c(\()p Fy(D)c FC(+)e Fy(L)p FC(\))p Fy(D)1389 1783 y Fw(\000)p Fv(1)1435 1798 y FC(\()p Fy(D)i FC(+)e Fy(U)f FC(\))13 b(where)150 1848 y Fy(D)i FC(is)f(the)h(diagonal)d(matrix)g(con)o(taining)g(the)j(piv)o (ots)e(generated.)212 1898 y(Remark:)24 b(the)18 b(resulting)f Fy(L)h FC(and)f Fy(U)23 b FC(factors)17 b(of)g(the)h(preconditioner)h(ha)o(v) o(e)e(only)f(nonzero)150 1948 y(elemen)o(ts)h(in)g(the)h(set)g Fy(S)r FC(,)g(but)f(this)h(fact)f(is)g(in)f(general)i(not)f(true)h(for)f(the) h(preconditioner)f Fy(M)150 1998 y FC(itself.)212 2047 y(The)12 b(fact)f(that)g(the)h Fy(I)s(LU)5 b FC(\(0\))12 b(preconditioner)f(con)o (tains)g(the)h(o\013-diagonal)d(parts)j(of)e(the)i(orig-)150 2097 y(inal)g(matrix)g(w)o(as)i(used)g(b)o(y)g(Eisenstat)g([87)o(])g(to)f (deriv)o(e)h(at)g(a)f(more)g(e\016cien)o(t)h(implem)o(en)o(tation)d(of)150 2147 y(preconditioned)k(CG.)e(This)i(new)g(implem)o(en)o(tation)d(merges)i (the)h(application)e(of)h(the)h(tridiago-)150 2197 y(nal)e(factors)g(of)g (the)h(matrix)d(and)i(the)h(preconditioner,)f(thereb)o(y)i(sa)o(ving)d(a)h (substan)o(tial)g(n)o(um)o(b)q(er)150 2247 y(of)g(op)q(erations)h(p)q(er)h (iteration.)150 2355 y Fs(Sp)q(ecial)f(cases:)22 b(cen)o(tral)13 b(di\013erences)150 2431 y FC(Incomplete)c(factorizations)g(of)g Fy(I)s(LU)c FC(\(0\))10 b(t)o(yp)q(e)g(are)g(particularly)e(simple)g(if)h (the)h(matrix)d(is)j(deriv)o(ed)150 2481 y(from)f(cen)o(tral)j(di\013erences) i(on)d(a)g(Cartesian)g(pro)q(duct)h(grid.)17 b(As)12 b(remark)o(ed)e(ab)q(o)o (v)o(e,)h(w)o(e)h(only)e(ha)o(v)o(e)150 2531 y(to)i(calculate)g(the)h(piv)o (ots)f(of)f(the)i(factorization;)e(other)i(elemen)o(ts)f(in)f(the)i (triangular)e(factors)i(are)150 2581 y(equal)h(to)f(o\013-diagonal)f(elemen)o (ts)i(of)f Fy(A)p FC(.)p 150 2615 620 2 v 196 2642 a Fo(3)214 2654 y Fn(The)d(zero)f(refers)g(to)h(the)g(fact)f(that)h(only)f(\\lev)o(el)g (zero")g(\014ll)g(is)i(p)q(ermitted,)c(that)j(is,)g(nonzero)e(elemen)o(ts)g (of)i(the)150 2693 y(original)g(matrix.)k(Fill)d(lev)o(els)f(are)h(de\014ned) f(b)o(y)h(calling)f(an)h(elemen)o(t)f(of)h(lev)o(el)g Fh(k)e Fn(+)f(1)k(if)f(it)h(is)g(caused)e(b)o(y)h(elemen)o(ts)150 2733 y(at)g(least)g(one)f(of)i(whic)o(h)e(is)i(of)f(lev)o(el)g Fh(k)q Fn(.)k(The)d(\014rst)f(\014ll)g(lev)o(el)f(is)h(that)g(caused)f(b)o(y) h(the)f(original)g(matrix)g(elemen)o(ts.)p eop %%Page: 44 56 55 bop 450 275 a FC(44)802 b Fr(CHAPTER)14 b(3.)32 b(PRECONDITIONERS)512 391 y FC(In)14 b(the)g(follo)o(wing)d(w)o(e)j(will)e(assume)i(a)f(natural,)g (line-b)o(y-line,)f(ordering)h(of)g(the)i(grid)e(p)q(oin)o(ts.)512 441 y(Letting)i Fy(i)p FC(,)p Fy(j)i FC(b)q(e)e(co)q(ordinates)h(in)e(a)g (regular)h(2D)f(grid,)g(it)g(is)h(easy)g(to)g(see)g(that)g(the)h(piv)o(ot)d (on)450 491 y(grid)f(p)q(oin)o(t)g(\()p Fy(i;)7 b(j)r FC(\))14 b(is)e(only)g(determined)h(b)o(y)f(piv)o(ots)g(on)h(p)q(oin)o(ts)f(\()p Fy(i)7 b Fu(\000)g FC(1)p Fy(;)g(j)r FC(\))13 b(and)g(\()p Fy(i;)7 b(j)i Fu(\000)e FC(1\).)18 b(If)12 b(there)450 541 y(are)j Fy(n)f FC(p)q(oin)o(ts)g(on)g(eac)o(h)h(of)e Fy(m)i FC(grid)f(lines,)g(w)o(e)g(get)h(the)g(follo)o(wing)c(generating)k(relations) f(for)g(the)450 591 y(piv)o(ots:)554 769 y Fy(d)576 775 y Fx(i;i)622 769 y FC(=)666 634 y Ft(8)666 671 y(>)666 684 y(>)666 696 y(>)666 709 y(>)666 721 y(<)666 796 y(>)666 808 y(>)666 821 y(>)666 833 y(>)666 846 y(:)724 666 y Fy(a)746 672 y Fv(1)p Fx(;)p Fv(1)1320 666 y FC(if)f Fy(i)f FC(=)f(1)724 717 y Fy(a)746 723 y Fx(i;i)790 717 y Fu(\000)f Fy(a)854 723 y Fx(i;i)p Fw(\000)p Fv(1)932 717 y Fy(d)954 699 y Fw(\000)p Fv(1)954 729 y Fx(i)p Fw(\000)p Fv(1)1010 717 y Fy(a)1032 723 y Fx(i)p Fw(\000)p Fv(1)p Fx(;i)1320 717 y FC(if)j(1)e Fy(<)h(i)g Fu(\024)g Fy(n)724 769 y(a)746 775 y Fx(i;i)790 769 y Fu(\000)e Fy(a)854 775 y Fx(i;i)p Fw(\000)p Fx(n)936 769 y Fy(d)958 751 y Fw(\000)p Fv(1)958 780 y Fx(i)p Fw(\000)p Fx(n)1018 769 y Fy(a)1040 775 y Fx(i)p Fw(\000)p Fx(n;i)1320 769 y FC(if)j Fy(i)f FC(=)f Fy(k)q(n)f FC(+)f(1)14 b(with)f Fy(k)g(>)e FC(1)745 820 y Fy(a)767 826 y Fx(i;i)843 820 y Fu(\000)42 b Fy(a)939 826 y Fx(i;i)p Fw(\000)p Fv(1)1017 820 y Fy(d)1039 802 y Fw(\000)p Fv(1)1039 832 y Fx(i)p Fw(\000)p Fv(1)1095 820 y Fy(a)1117 826 y Fx(i)p Fw(\000)p Fv(1)p Fx(;i)926 872 y Fu(\000)10 b Fy(a)990 878 y Fx(i;i)p Fw(\000)p Fx(n)1072 872 y Fy(d)1094 854 y Fw(\000)p Fv(1)1094 883 y Fx(i)p Fw(\000)p Fx(n)1154 872 y Fy(a)1176 878 y Fx(i)p Fw(\000)p Fx(n;i)1320 845 y FC(otherwise)q Fy(:)450 943 y FC(Con)o(v)o(ersely)m(,)j(w)o(e)h(can)g(describ)q(e)i(the)f (factorization)e(algorithmical)o(ly)e(as)581 1021 y(Initially:)16 b Fy(d)776 1027 y Fx(i;i)823 1021 y FC(=)c Fy(a)889 1027 y Fx(i;i)938 1021 y FC(for)i(all)e Fy(i)581 1071 y Fs(for)h Fy(i)f FC(=)g(1)p Fy(::nm)h FC(do:)653 1090 y Ft(8)653 1127 y(<)653 1202 y(:)711 1122 y Fy(d)733 1128 y Fx(i)p Fv(+1)p Fx(;i)p Fv(+1)864 1122 y FC(=)e Fy(d)929 1128 y Fx(i)p Fv(+1)p Fx(;i)p Fv(+1)1058 1122 y Fu(\000)e Fy(a)1121 1128 y Fx(i)p Fv(+1)p Fx(;i)1199 1122 y Fy(d)1221 1104 y Fw(\000)p Fv(1)1221 1134 y Fx(i;i)1265 1122 y Fy(a)1287 1128 y Fx(i;i)p Fv(+1)1430 1122 y FC(if)k(there)i(is)f(no)f Fy(k)1430 1174 y FC(suc)o(h)h(that)g Fy(i)e FC(=)g Fy(k)q(n)711 1226 y(d)733 1232 y Fx(i)p Fv(+)p Fx(n;i)p Fv(+)p Fx(n)872 1226 y FC(=)f Fy(d)937 1232 y Fx(i)p Fv(+)p Fx(n;i)p Fv(+)p Fx(n)1074 1226 y Fu(\000)e Fy(a)1137 1232 y Fx(i)p Fv(+)p Fx(n;i)1219 1226 y Fy(d)1241 1208 y Fw(\000)p Fv(1)1241 1237 y Fx(i;i)1285 1226 y Fy(a)1307 1232 y Fx(i;i)p Fv(+)p Fx(n)1430 1226 y FC(if)k Fy(i)c FC(+)h Fy(n)h Fu(\024)h Fy(nm)512 1304 y FC(In)17 b(the)g(ab)q(o)o(v)o(e)f(w)o(e)g(ha)o(v)o(e)h (assumed)f(that)g(the)h(v)n(ariables)f(in)g(the)h(problem)e(are)i(ordered)g (ac-)450 1354 y(cording)f(to)f(the)h(so-called)g(\\natural)f(ordering":)21 b(a)16 b(sequen)o(tial)f(n)o(um)o(b)q(ering)g(of)g(the)h(grid)f(lines)450 1403 y(and)h(the)h(p)q(oin)o(ts)g(within)f(eac)o(h)h(grid)f(line.)25 b(Belo)o(w)17 b(w)o(e)f(will)f(encoun)o(ter)j(di\013eren)o(t)g(orderings)f (of)450 1453 y(the)d(v)n(ariables.)450 1560 y Fs(Mo)q(di\014ed)g(incomplete)f (factorizations)450 1637 y FC(One)18 b(mo)q(di\014cation)d(to)j(the)g(basic)f (idea)h(of)f(incomplete)f(factorizations)h(is)g(as)h(follo)o(ws:)24 b(If)17 b(the)450 1686 y(pro)q(duct)j Fy(a)633 1692 y Fx(i;k)675 1686 y Fy(a)697 1669 y Fw(\000)p Fv(1)697 1699 y Fx(k)q(;k)746 1686 y Fy(a)768 1692 y Fx(k)q(;j)833 1686 y FC(is)f(nonzero,)i(and)e(\014ll)f (is)h(not)g(allo)o(w)o(ed)f(in)h(p)q(osition)g(\()p Fy(i;)7 b(j)r FC(\),)20 b(instead)g(of)450 1736 y(simply)12 b(discarding)i(this)f (\014ll)g(quan)o(tit)o(y)g(subtract)j(it)d(from)f(diagonal)g(elemen)o(t)i Fy(a)1734 1742 y Fx(i;i)1769 1736 y FC(.)512 1786 y(Mathematically)i(this)h (corresp)q(onds)j(to)e(forcing)f(the)h(preconditioner)h(to)e(ha)o(v)o(e)h (the)g(same)450 1836 y(ro)o(wsums)13 b(as)h(the)h(original)d(matrix;)f(in)j (applications)e(of)i(computational)d(\015uid)j(mec)o(hanics)f(this)450 1886 y(idea)18 b(is)h(justi\014ed)g(with)f(the)h(argumen)o(t)f(that)h(the)g (preconditioner)g(do)q(es)h(not)e(in)o(tro)q(duce)i(an)o(y)450 1936 y(arti\014cial)9 b(di\013usion)g(in)o(to)f(the)j(system;)f(see)h(Appley) o(ard)e(and)h(Cheshire)g([4])f(and)g(Dup)q(on)o(t,)h(Kendall)450 1985 y(and)k(Rac)o(hford)f([78)o(])g(for)h(early)g(applications)f(of)g(this)h (idea.)512 2035 y(Suc)o(h)e(a)f(factorization)f(sc)o(heme)i(is)f(usually)f (called)h(a)g(\\mo)q(di\014ed)f(incomplete)g(factorization".)450 2085 y(In)h(this)f(case)i(there)g(is)e(a)g(danger)h(of)f(breakdo)o(wn,)h(esp) q(ecially)f(when)h(the)h(v)n(ariables)d(are)i(n)o(um)o(b)q(ered)450 2135 y(other)i(than)f(in)f(the)i(natural)f(ro)o(w-b)o(y-ro)o(w)f(ordering.)18 b(This)12 b(w)o(as)g(noted)g(b)o(y)g(Chan)g(and)g(Kuo)g([49)o(],)450 2185 y(and)i(a)f(full)g(analysis)g(w)o(as)h(giv)o(en)f(b)o(y)h(Eijkhout)f ([83)o(])h(and)f(Nota)o(y)h([157)n(].)512 2234 y(One)19 b(reason)f(for)f (considering)g(mo)q(di\014ed)g(incomplete)f(factorizations)h(is)g(the)i(b)q (eha)o(vior)e(of)450 2284 y(the)e(sp)q(ectral)g(condition)f(n)o(um)o(b)q(er)f (of)h(the)g(preconditioned)h(system.)k(It)14 b(w)o(as)g(men)o(tioned)f(ab)q (o)o(v)o(e)450 2334 y(that)19 b(the)g(condition)e(n)o(um)o(b)q(er)h(of)g(the) h(co)q(e\016cien)o(t)h(matrix)c(is)j Fy(O)q FC(\()p Fy(h)1549 2319 y Fw(\000)p Fv(2)1593 2334 y FC(\))g(as)f(a)g(function)h(of)f(the)450 2384 y(discretization)13 b(mesh)f(width.)18 b(This)12 b(order)h(of)f (magnitude)f(is)i(preserv)o(ed)i(b)o(y)d(simple)f(incomplete)450 2434 y(factorizations,)i(although)g(usually)g(a)h(reduction)g(b)o(y)g(a)f (large)h(constan)o(t)h(factor)e(is)h(obtained.)512 2483 y(Mo)q(di\014ed)h (factorizations)g(are)g(of)f(in)o(terest)i(b)q(ecause,)g(in)f(com)o(bination) d(with)i(small)f(p)q(ertur-)450 2533 y(bations,)f(the)h(sp)q(ectral)h (condition)e(n)o(um)o(b)q(er)g(of)g(the)h(preconditioned)h(system)e(can)h(b)q (e)g(of)f(a)h(lo)o(w)o(er)450 2583 y(order.)36 b(It)19 b(w)o(as)h(\014rst)g (pro)o(v)o(ed)g(b)o(y)f(Dup)q(on)o(t,)h(Kendall)f(and)h(Rac)o(hford)f([78)o (])g(that)g(a)h(mo)q(di\014ed)450 2633 y(incomplete)d(factorization)f(of)h Fy(A)12 b FC(+)g Fy(O)q FC(\()p Fy(h)1119 2618 y Fv(2)1138 2633 y FC(\))p Fy(D)1188 2639 y Fx(A)1233 2633 y FC(giv)o(es)17 b Fy(\024)p FC(\()p Fy(M)1423 2618 y Fw(\000)p Fv(1)1467 2633 y Fy(A)p FC(\))h(=)g Fy(O)q FC(\()p Fy(h)1655 2618 y Fw(\000)p Fv(1)1700 2633 y FC(\))f(for)h(the)g(cen)o(tral)450 2683 y(di\013erence)k (case.)39 b(More)21 b(general)f(pro)q(ofs)g(are)h(giv)o(en)f(b)o(y)g (Gustafsson)h([111)n(],)g(Axelsson)g(and)450 2733 y(Bark)o(er)15 b([14)o(,)e Fu(x)q FC(7.2],)f(and)h(Beau)o(w)o(ens)i([30)o(,)f(31)o(].)p eop %%Page: 45 57 56 bop 150 275 a Fr(3.4.)31 b(INCOMPLETE)15 b(F)-5 b(A)o(CTORIZA)m(TION)14 b(PRECONDITIONERS)328 b FC(45)212 391 y(A)16 b(sligh)o(t)f(v)n(arian)o(t)f (of)h(mo)q(di\014ed)f(incomplete)g(factorizations)i(consists)g(of)f(the)h (class)g(of)f(\\re-)150 441 y(laxed)21 b(incomplete)g(factorizations".)40 b(Here)23 b(the)f(\014ll)e(is)i(m)o(ultipli)o(ed)d(b)o(y)j(a)f(parameter)g(0) j Fy(<)150 491 y(\013)17 b(<)g FC(1)g(b)q(efore)g(it)g(is)g(subtracted)i (from)c(the)j(diagonal;)e(see)j(Ashcraft)f(and)f(Grimes)e([11)o(],)i(Ax-)150 541 y(elsson)f(and)g(Lindsk)o(og)f([18)o(,)g(19],)g(Chan)g([43)o(],)h (Eijkhout)f([83)o(],)g(Nota)o(y)g([158)o(],)g(Stone)h([190)o(],)f(and)150 591 y(V)m(an)g(der)g(V)m(orst)h([199)n(].)21 b(F)m(or)15 b(the)h(dangers)g (of)e(MILU)h(in)g(the)h(presence)h(of)e(rounding)f(error,)i(see)150 640 y(V)m(an)d(der)i(V)m(orst)f([201)o(].)150 745 y Fs(P)o(aralleli)o(sm)f (asp)q(ects)150 822 y FC(A)o(t)19 b(\014rst)h(it)e(ma)o(y)f(app)q(ear)j(that) f(the)g(sequen)o(tial)g(time)f(of)g(solving)g(a)g(factorization)h(is)f(of)h (the)150 872 y(order)g(of)f(the)h(n)o(um)o(b)q(er)f(of)f(v)n(ariables,)i(but) f(things)g(are)h(not)g(quite)f(that)g(bad.)32 b(Consider)19 b(the)150 921 y(sp)q(ecial)c(case)h(of)e(cen)o(tral)h(di\013erences)i(on)d(a) g(regular)h(domain)d(of)i Fy(n)c Fu(\002)g Fy(n)15 b FC(p)q(oin)o(ts.)20 b(The)15 b(v)n(ariables)150 971 y(on)h(an)o(y)g(diagonal,)e(that)i(is,)g(in)g (lo)q(cations)f(\()p Fy(i;)7 b(j)r FC(\))17 b(with)f Fy(i)11 b FC(+)g Fy(j)17 b FC(=)f Fy(k)q FC(,)g(dep)q(end)h(only)f(on)g(those)g(on) 150 1021 y(the)g(previous)g(diagonal,)e(that)i(is,)f(with)h Fy(i)10 b FC(+)h Fy(j)17 b FC(=)e Fy(k)c Fu(\000)g FC(1.)23 b(Therefore)17 b(it)e(is)h(p)q(ossible)g(to)f(ha)o(v)o(e)g(a)150 1071 y(v)o(ector)d(computer)g Fq(pip)n(eline)i FC(the)e(op)q(erations)g(on)g (eac)o(h)g(diagonal,)e(and)h(a)g(parallel)g(computer)g(can)150 1121 y(pro)q(cess)16 b(the)e(elemen)o(ts)g(of)f(a)h(diagonal)e(sim)o (ultaneously;)f(see)k(V)m(an)f(der)g(V)m(orst)g([198)o(,)f(200].)212 1171 y(Another)19 b(w)o(a)o(y)e(of)g(v)o(ectorizing)h(the)h(solution)e(of)g (the)i(triangular)e(factors)h(is)g(to)f(use)i(some)150 1220 y(expansion.)h(If)14 b(the)h(lo)o(w)o(er)f(triangular)g(factor)g(is)h (normalized)e(to)h(the)h(form)e Fy(I)g Fu(\000)d Fy(L)15 b FC(\(where)h Fy(L)e FC(is)150 1270 y(strictly)h(lo)o(w)o(er)g(triangular\),)g (then)g(its)h(in)o(v)o(erse)f(can)h(b)q(e)g(giv)o(en)e(as)i(either)g(of)e (the)i(follo)o(wing)d(t)o(w)o(o)150 1320 y(series:)254 1403 y(\()p Fy(I)g Fu(\000)c Fy(L)p FC(\))386 1386 y Fw(\000)p Fv(1)443 1403 y FC(=)486 1345 y Ft(\032)525 1380 y Fy(I)j FC(+)e Fy(L)f FC(+)h Fy(L)704 1365 y Fv(2)732 1380 y FC(+)f Fy(L)801 1365 y Fv(3)829 1380 y FC(+)h Fu(\001)d(\001)g(\001)525 1429 y FC(\()p Fy(I)12 b FC(+)e Fy(L)p FC(\)\()p Fy(I)j FC(+)d Fy(L)774 1414 y Fv(2)793 1429 y FC(\)\()p Fy(I)j FC(+)c Fy(L)925 1414 y Fv(4)944 1429 y FC(\))e Fu(\001)g(\001)g(\001)150 1497 y FC(\(The)16 b(\014rst)g(series)g(is)g(called)f(a)f(\\Neumann)h(expansion",)f(the)i (second)h(an)e(\\Euler)g(expansion".)150 1547 y(Both)f(series)h(are)f (\014nite,)f(but)h(their)g(length)g(prohibits)f(practical)h(use)g(of)f(this)h (fact.\))k(P)o(arallel)12 b(or)150 1596 y(v)o(ectorizable)g(preconditioners)g (can)g(b)q(e)f(deriv)o(ed)h(from)e(an)h(incomplete)f(factorization)g(b)o(y)h (taking)150 1646 y(a)h(small)e(n)o(um)o(b)q(er)i(of)g(terms)h(in)f(either)h (series.)19 b(Exp)q(erimen)o(ts)12 b(indicate)h(that)g(a)f(small)e(n)o(um)o (b)q(er)i(of)150 1696 y(terms,)j(while)g(giving)f(high)h(execution)h(rates,)h (yields)e(almost)f(the)i(full)e(precision)i(of)f(the)h(more)150 1746 y(recursiv)o(e)d(triangular)e(solution)g(\(see)j(Axelsson)e(and)g (Eijkhout)f([15)o(])h(and)f(V)m(an)h(der)g(V)m(orst)g([196)o(]\).)212 1796 y(More)h(radical)f(approac)o(hes)i(for)e(increasing)h(the)g(parallelism) d(in)i(incomplete)g(factorizations)150 1845 y(are)18 b(based)g(on)f(a)h(ren)o (um)o(b)q(ering)e(of)h(the)h(problem)f(v)n(ariables.)28 b(F)m(or)17 b(instance,)i(on)e(rectangular)150 1895 y(domains)11 b(one)i(could)f(start)i (n)o(um)o(b)q(ering)d(the)i(v)n(ariables)f(from)f(all)h(four)g(corners)i(sim) o(ultaneously)m(,)150 1945 y(thereb)o(y)k(creating)f(four-fold)e(parallelism) f(\(see)j(Dongarra,)f Fq(et)i(al.)e FC([68)o(],)g(V)m(an)g(der)h(V)m(orst)g ([197)o(,)150 1995 y(199)o(]\).)h(The)d(most)e(extreme)h(case)h(is)f(the)g (red/blac)o(k)g(ordering)h(\(or)f(for)f(more)g(general)i(matrices)150 2045 y(the)21 b(m)o(ulti-color)d(ordering\))j(whic)o(h)f(giv)o(es)g(the)i (absolute)e(minim)n(um)c(n)o(um)o(b)q(er)k(of)g(sequen)o(tial)150 2094 y(steps.)212 2144 y(Multi-coloring)e(is)h(also)g(an)h(attractiv)o(e)g (metho)q(d)e(for)i(v)o(ector)g(computers.)35 b(Since)20 b(p)q(oin)o(ts)150 2194 y(of)d(one)g(color)g(are)g(uncoupled,)h(they)g(can)g(b)q(e)f(pro)q (cessed)j(as)d(one)h(v)o(ector;)h(see)f(Doi)e([65)o(],)h(Mel-)150 2244 y(hem)c([151)o(],)g(and)g(P)o(o)q(ole)h(and)g(Ortega)g([171)o(].)212 2294 y(Ho)o(w)o(ev)o(er,)e(for)e(suc)o(h)h(ordering)g(strategies)h(there)g (is)f(usually)f(a)g(trade-o\013)h(b)q(et)o(w)o(een)h(the)g(degree)150 2344 y(of)17 b(parallelism)e(and)i(the)h(resulting)g(n)o(um)o(b)q(er)e(of)h (iterations.)29 b(The)18 b(reason)g(for)f(this)h(is)f(that)g(a)150 2393 y(di\013eren)o(t)f(ordering)f(ma)o(y)f(giv)o(e)g(rise)i(to)f(a)g (di\013eren)o(t)h(error)g(matrix,)e(in)g(particular)h(the)h(norm)e(of)150 2443 y(the)i(error)h(matrix)d(ma)o(y)g(v)n(ary)h(considerably)h(b)q(et)o(w)o (een)h(orderings.)24 b(See)17 b(exp)q(erimen)o(tal)e(results)150 2493 y(b)o(y)f(Du\013)f(and)h(Meuran)o(t)h([76)o(])e(and)h(a)f(partial)g (explanation)g(of)g(them)g(b)o(y)h(Eijkhout)f([82)o(].)150 2606 y Fl(3.4.3)55 b(Blo)r(c)n(k)18 b(factorization)g(metho)r(ds)150 2683 y FC(W)m(e)12 b(can)h(also)f(consider)i(blo)q(c)o(k)e(v)n(arian)o(ts)g (of)g(preconditioners)i(for)f(accelerated)h(metho)q(ds.)j(Blo)q(c)o(k)150 2733 y(metho)q(ds)e(are)h(normally)d(feasible)i(if)g(the)h(problem)e(domain)f (is)j(a)f(Cartesian)h(pro)q(duct)g(grid;)g(in)p eop %%Page: 46 58 57 bop 450 275 a FC(46)802 b Fr(CHAPTER)14 b(3.)32 b(PRECONDITIONERS)450 391 y FC(that)13 b(case)h(a)f(natural)g(division)e(in)i(lines)g(\(or)g (planes)h(in)e(the)i(3-dimensional)c(case\),)k(can)g(b)q(e)f(used)450 441 y(for)e(blo)q(c)o(king,)g(though)h(incomplete)f(factorizations)g(are)h (not)g(as)g(e\013ectiv)o(e)h(in)f(the)g(3-dimensional)450 491 y(case;)k(see)h(for)d(instance)i(Kettler)h([134)o(].)k(In)15 b(suc)o(h)h(a)f(blo)q(c)o(king)f(sc)o(heme)h(for)g(Cartesian)g(pro)q(duct)450 541 y(grids,)e(b)q(oth)i(the)f(size)h(and)f(n)o(um)o(b)q(er)f(of)g(the)i(blo) q(c)o(ks)f(increases)h(with)f(the)h(o)o(v)o(erall)d(problem)h(size.)450 591 y(Another)i(t)o(yp)q(e)f(of)f(blo)q(c)o(k)h(metho)q(d)f(will)f(b)q(e)j (men)o(tioned)e(in)g Fu(x)q FC(3.4.3.)450 707 y Fs(The)j(idea)e(b)q(ehind)g (blo)q(c)o(k)h(factorizations)450 787 y FC(The)c(starting)g(p)q(oin)o(t)g (for)g(an)f(incomplete)g(blo)q(c)o(k)h(factorization)f(is)h(a)g(partitioning) e(of)i(the)h(matrix,)450 836 y(as)e(men)o(tioned)f(in)g Fu(x)q FC(3.2.1.)15 b(Then)10 b(an)g(incomplete)f(factorization)g(is)h(p)q(erformed) g(using)f(the)i(matrix)450 886 y(blo)q(c)o(ks)g(as)g(basic)g(en)o(tities)g (\(see)h(Axelsson)f([12])f(and)g(Concus,)i(Golub)e(and)g(Meuran)o(t)i([55)o (])e(as)h(basic)450 936 y(references\).)512 987 y(The)17 b(most)f(imp)q (ortan)o(t)f(di\013erence)k(with)d(p)q(oin)o(t)h(metho)q(ds)f(arises)h(in)g (the)g(in)o(v)o(ersion)f(of)h(the)450 1037 y(piv)o(ot)11 b(blo)q(c)o(ks.)18 b(Whereas)13 b(in)o(v)o(erting)f(a)g(scalar)g(is)g(easily)g(done,)g(in)g(the) g(blo)q(c)o(k)g(case)h(t)o(w)o(o)f(problems)450 1087 y(arise.)17 b(First,)11 b(in)o(v)o(erting)f(the)h(piv)o(ot)e(blo)q(c)o(k)h(is)g(lik)o (ely)f(to)h(b)q(e)h(a)f(costly)h(op)q(eration.)16 b(Second,)c(initially)450 1137 y(all)k(diagonal)e(blo)q(c)o(ks)j(of)f(the)i(matrix)d(ma)o(y)g(b)q(e)i (sparse)h(and)f(w)o(e)g(w)o(ould)f(lik)o(e)g(to)g(main)o(tain)e(this)450 1187 y(t)o(yp)q(e)g(of)g(structure.)20 b(Hence)15 b(the)g(need)g(for)e(appro) o(ximations)e(of)j(in)o(v)o(erses)h(arises.)512 1238 y(As)e(in)f(the)h(case)g (of)f(incomplete)g(p)q(oin)o(t)f(factorizations,)h(the)h(existence)h(of)e (incomplete)g(blo)q(c)o(k)450 1288 y(metho)q(ds)j(is)g(guaran)o(teed)h(if)e (the)i(co)q(e\016cien)o(t)g(matrix)e(is)h(an)g Fy(M)5 b FC(-matrix.)20 b(F)m(or)15 b(a)g(general)g(pro)q(of,)450 1338 y(see)g(Axelsson)g([13)o(].) 450 1454 y Fs(Appro)o(ximate)e(in)o(v)o(erses)450 1534 y FC(In)21 b(blo)q(c)o(k)f(factorizations)h(a)f(piv)o(ot)g(blo)q(c)o(k)h(is)f(generally) h(forced)g(to)g(b)q(e)g(sparse,)i(t)o(ypically)d(of)450 1584 y(banded)f(form,)e(and)h(that)h(w)o(e)f(need)i(an)e(appro)o(ximation)d(to)j (its)h(in)o(v)o(erse)g(that)f(has)h(a)f(similar)450 1633 y(structure.)26 b(F)m(urthermore,)16 b(this)g(appro)o(ximation)d(should)j(b)q(e)g(easily)g (computable,)f(so)h(w)o(e)g(rule)450 1683 y(out)e(the)g(option)f(of)h (calculating)f(the)h(full)f(in)o(v)o(erse)h(and)g(taking)f(a)h(banded)g(part) g(of)f(it.)512 1735 y(The)i(simplest)d(appro)o(ximation)f(to)j Fy(A)1118 1720 y Fw(\000)p Fv(1)1177 1735 y FC(is)f(the)i(diagonal)d(matrix)g Fy(D)j FC(of)e(the)i(recipro)q(cals)g(of)450 1784 y(the)f(diagonal)f(of)g Fy(A)p FC(:)18 b Fy(d)818 1790 y Fx(i;i)864 1784 y FC(=)12 b(1)p Fy(=a)972 1790 y Fx(i;i)1007 1784 y FC(.)512 1836 y(Other)18 b(p)q(ossibilities)f(w)o(ere)h(considered)g(b)o(y)f(Axelsson)g(and)g (Eijkhout)f([15)o(],)h(Axelsson)g(and)450 1886 y(P)o(olman)12 b([20)o(],)h(and)g(Concus,)h(Golub)f(and)h(Meuran)o(t)g([55)o(].)512 1937 y(Banded)j(appro)o(ximations)c(to)j(the)g(in)o(v)o(erse)h(of)e(banded)h (matrices)g(ha)o(v)o(e)f(a)h(theoretical)g(jus-)450 1987 y(ti\014cation.)27 b(In)17 b(the)g(con)o(text)h(of)f(partial)f(di\013eren)o(tial)g(equations)h (the)h(diagonal)d(blo)q(c)o(ks)i(of)g(the)450 2037 y(co)q(e\016cien)o(t)h (matrix)e(are)i(usually)e(strongly)i(diagonally)d(dominan)o(t.)26 b(F)m(or)17 b(suc)o(h)h(matrices,)g(the)450 2087 y(elemen)o(ts)e(of)g(the)g (in)o(v)o(erse)h(ha)o(v)o(e)f(a)g(size)h(that)f(is)g(exp)q(onen)o(tially)g (decreasing)h(in)f(their)g(distance)450 2136 y(from)f(the)j(main)d(diagonal.) 26 b(See)18 b(Demk)o(o,)e(Moss)h(and)g(Smith)f([62)o(])g(for)h(a)g(general)g (pro)q(of,)g(and)450 2186 y(Eijkhout)c(and)h(P)o(olman)e([86)o(])h(for)h(a)f (more)g(detailed)h(analysis)f(in)g(the)i Fy(M)5 b FC(-matrix)12 b(case.)450 2303 y Fs(The)k(sp)q(ecial)e(case)i(of)f(blo)q(c)o(k)g (tridiagonali)o(t)n(y)450 2382 y FC(In)k(man)o(y)e(applications,)h(a)h(blo)q (c)o(k)g(tridiagonal)d(structure)22 b(can)d(b)q(e)g(found)f(in)h(the)g(co)q (e\016cien)o(t)450 2432 y(matrix.)d(Examples)d(are)h(problems)f(on)g(a)h(2D)f (regular)h(grid)f(if)g(the)h(blo)q(c)o(ks)g(corresp)q(ond)i(to)d(lines)450 2482 y(of)h(grid)h(p)q(oin)o(ts,)f(and)h(problems)e(on)i(a)g(regular)f(3D)h (grid,)f(if)g(the)h(blo)q(c)o(ks)g(corresp)q(ond)h(to)f(planes)450 2532 y(of)g(grid)g(p)q(oin)o(ts.)23 b(Ev)o(en)16 b(if)f(suc)o(h)i(a)e(blo)q (c)o(k)g(tridiagonal)f(structure)k(do)q(es)e(not)g(arise)f(naturally)m(,)g (it)450 2581 y(can)f(b)q(e)h(imp)q(osed)d(b)o(y)i(ren)o(um)o(b)q(ering)f(the) i(v)n(ariables)e(in)g(a)h(Cuthill-McKee)g(ordering)g([57)o(].)512 2633 y(Suc)o(h)h(a)e(matrix)f(has)i(incomplete)f(blo)q(c)o(k)h (factorizations)f(of)h(a)f(particularly)h(simple)e(nature:)450 2683 y(since)19 b(no)e(\014ll)g(can)i(o)q(ccur)g(outside)f(the)h(diagonal)d (blo)q(c)o(ks)i(\()p Fy(A)1448 2689 y Fx(i;i)1483 2683 y FC(\),)h(all)e(prop) q(erties)i(follo)o(w)d(from)450 2733 y(our)d(treatmen)o(t)g(of)f(the)i(piv)o (ot)e(blo)q(c)o(ks.)18 b(The)c(generating)f(recurrence)j(for)d(the)g(piv)o (ot)g(blo)q(c)o(ks)g(also)p eop %%Page: 47 59 58 bop 150 275 a Fr(3.4.)31 b(INCOMPLETE)15 b(F)-5 b(A)o(CTORIZA)m(TION)14 b(PRECONDITIONERS)328 b FC(47)150 391 y(tak)o(es)12 b(a)g(simple)f(form.)k (Let)e Fy(A)f FC(b)q(e)h(the)f(co)q(e\016cien)o(t)h(matrix,)d(blo)q(c)o(k)i (indexed,)g(and)g(let)g Fu(f)p Fy(X)1568 397 y Fx(i)1582 391 y Fu(g)1603 397 y Fx(i)p Fv(=1)p Fx(::n)150 441 y FC(b)q(e)j(the)f(sequence)i (of)d(piv)o(ots,)g(then)281 535 y Fy(X)315 541 y Fv(1)346 535 y FC(=)f Fy(A)421 541 y Fv(1)p Fx(;)p Fv(1)281 585 y Fs(for)h Fy(i)f Fu(\025)g FC(1)353 637 y(let)i Fy(Y)437 643 y Fx(i)463 637 y Fu(\031)e Fy(X)544 619 y Fw(\000)p Fv(1)541 648 y Fx(i)353 686 y Fy(X)387 692 y Fx(i)p Fv(+1)455 686 y FC(=)g Fy(A)530 692 y Fx(i)p Fv(+1)p Fx(;i)p Fv(+1)659 686 y Fu(\000)d Fy(A)731 692 y Fx(i)p Fv(+1)p Fx(;i)809 686 y Fy(Y)833 692 y Fx(i)847 686 y Fy(A)878 692 y Fx(i;i)p Fv(+1)150 778 y FC(The)k(sequence)i Fu(f)p Fy(Y)450 784 y Fx(i)464 778 y Fu(g)485 784 y Fx(i)p Fv(=1)p Fx(::n)593 778 y FC(consists)f(of)e(appro)o(ximations)f(to)h(the)i (in)o(v)o(erses)f(of)g(the)g(piv)o(ots)f(in)h(the)150 828 y(manner)g (outlined)g(ab)q(o)o(v)o(e.)150 950 y Fs(Tw)o(o)j(t)o(yp)q(es)f(of)g (incomplete)e(blo)q(c)o(k)i(factorizations)150 1032 y FC(One)c(reason)g(that) g(blo)q(c)o(k)f(metho)q(ds)g(are)h(of)e(in)o(terest)j(is)e(that)h(they)g(are) g(p)q(oten)o(tially)e(more)g(suitable)150 1082 y(for)14 b(v)o(ector)g (computers)g(and)g(parallel)f(arc)o(hitectures.)20 b(Consider)14 b(the)h(blo)q(c)o(k)e(factorization)254 1176 y Fy(A)e FC(=)h(\()p Fy(D)f FC(+)f Fy(L)p FC(\))p Fy(D)522 1159 y Fw(\000)p Fv(1)567 1176 y FC(\()p Fy(D)h FC(+)f Fy(U)5 b FC(\))11 b(=)h(\()p Fy(D)f FC(+)e Fy(L)p FC(\)\()p Fy(I)14 b FC(+)9 b Fy(D)1044 1159 y Fw(\000)p Fv(1)1089 1176 y Fy(U)c FC(\))150 1268 y(where)15 b Fy(D)g FC(is)f(the)h(blo)q(c)o(k)e(diagonal)f(matrix)g(of)i(piv)o(ot)f(blo) q(c)o(ks.)1121 1252 y Fv(4)212 1320 y FC(Making)i(the)g(transition)g(to)g(an) g(incomplete)g(factorization)f(w)o(e)i(can)f(replace)h(the)g(diagonal)150 1370 y(of)g(piv)o(ots)g Fy(D)h FC(b)o(y)f(either)i(the)f(diagonal)d(of)i (incomplete)f(factorization)h(piv)o(ots)f Fy(X)20 b FC(=)c(diag)o(\()p Fy(X)1657 1376 y Fx(i)1672 1370 y FC(\),)150 1420 y(or)f(the)h(in)o(v)o(erse) g(of)e Fy(Y)24 b FC(=)14 b(diag)o(\()p Fy(Y)672 1426 y Fx(i)686 1420 y FC(\),)h(the)h(diagonal)e(of)g(appro)o(ximations)f(to)i(the)h(in)o(v)o (erses)g(of)f(the)150 1470 y(piv)o(ots.)j(In)13 b(the)i(\014rst)g(case)g(w)o (e)f(\014nd)g(for)f(the)i(incomplete)e(factorization)254 1564 y Fy(C)h FC(=)e(\()p Fy(X)h FC(+)c Fy(L)p FC(\)\()p Fy(I)14 b FC(+)9 b Fy(X)616 1547 y Fw(\000)p Fv(1)661 1564 y Fy(U)c FC(\))150 1655 y(and)14 b(in)f(the)i(second)g(case)254 1749 y Fy(C)f FC(=)e(\()p Fy(Y)391 1732 y Fw(\000)p Fv(1)445 1749 y FC(+)e Fy(L)p FC(\)\()p Fy(I)j FC(+)c Fy(Y)h(U)5 b FC(\))p Fy(:)212 1841 y FC(W)m(e)11 b(see)i(that)e(for)g(factorizations)g(of)g(the)h (\014rst)g(t)o(yp)q(e)g(\(whic)o(h)f(co)o(v)o(ers)h(all)e(metho)q(ds)h(in)g (Concus,)150 1891 y(Golub)16 b(and)h(Meuran)o(t)g([55)o(]\))g(solving)e(a)i (systems)g(means)f(solving)g(smaller)f(systems)i(with)g(the)150 1940 y Fy(X)184 1946 y Fx(i)215 1940 y FC(matrices.)26 b(F)m(or)16 b(the)h(second)h(t)o(yp)q(e)f(\(whic)o(h)g(w)o(as)f(discussed)j(b)o(y)d (Meuran)o(t)h([152)o(],)g(Axelsson)150 1990 y(and)i(P)o(olman)e([20)o(])h (and)h(Axelsson)h(and)f(Eijkhout)g([15)o(]\))f(solving)g(a)h(system)g(with)g Fy(C)j FC(en)o(tails)150 2040 y Fq(multiplying)14 b FC(b)o(y)g(the)h Fy(Y)519 2046 y Fx(i)547 2040 y FC(blo)q(c)o(ks.)k(Therefore,)c(the)g(second) g(t)o(yp)q(e)g(has)f(a)g(m)o(uc)o(h)f(higher)i(p)q(oten)o(tial)150 2090 y(for)f(v)o(ectorizabilit)o(y)m(.)150 2213 y Fs(Blo)q(c)o(king)g(o)o(v)o (er)h(systems)g(of)g(partial)f(di\013eren)o(ti)o(al)e(equations)150 2294 y FC(If)g(the)h(ph)o(ysical)e(problem)g(has)h(sev)o(eral)h(v)n(ariables) f(p)q(er)h(grid)e(p)q(oin)o(t,)h(that)g(is,)g(if)g(there)h(are)g(sev)o(eral) 150 2344 y(coupled)i(partial)e(di\013eren)o(tial)h(equations,)g(it)f(is)h(p)q (ossible)h(to)f(in)o(tro)q(duce)h(blo)q(c)o(king)e(in)h(a)g(natural)150 2394 y(w)o(a)o(y)m(.)212 2447 y(Blo)q(c)o(king)i(of)g(the)h(equations)f (\(whic)o(h)g(giv)o(es)h(a)f(small)e(n)o(um)o(b)q(er)h(of)h(v)o(ery)h(large)f (blo)q(c)o(ks\))g(w)o(as)150 2496 y(used)11 b(b)o(y)f(Axelsson)h(and)f (Gustafsson)g([17)o(])g(for)g(the)h(equations)f(of)g(linear)f(elasticit)o(y)m (,)h(and)g(blo)q(c)o(king)150 2546 y(of)15 b(the)i(v)n(ariables)e(p)q(er)i (no)q(de)f(\(whic)o(h)g(giv)o(es)g(man)o(y)e(v)o(ery)i(small)e(blo)q(c)o (ks\))i(w)o(as)g(used)h(b)o(y)f(Aarden)150 2596 y(and)c(Karlsson)g([1)o(])f (for)g(the)i(semiconductor)e(equations.)18 b(A)12 b(systematic)f(comparison)f (of)h(the)i(t)o(w)o(o)150 2646 y(approac)o(hes)i(w)o(as)e(made)g(b)o(y)h (Bank,)f Fq(et)i(al.)e FC([26)o(].)p 150 2694 620 2 v 196 2721 a Fo(4)214 2733 y Fn(W)m(riting)d(\()p Fh(I)g Fn(+)f Fh(LD)472 2721 y Fe(\000)p Fo(1)513 2733 y Fn(\)\()p Fh(D)f Fn(+)g Fh(U)t Fn(\))j(is)h(equally)d(v)n(alid,)h(but)h(in)g(practice)e(is)j(harder)e(to)h (implemen)o(t.)p eop %%Page: 48 60 59 bop 450 275 a FC(48)802 b Fr(CHAPTER)14 b(3.)32 b(PRECONDITIONERS)450 391 y Fl(3.4.4)55 b(Incomplete)16 b(LQ)j(factorizations)450 468 y FC(Saad)d([180)o(])f(prop)q(oses)j(to)e(construct)i(an)e(incomplete)f (LQ)h(factorization)g(of)g(a)g(general)g(sparse)450 518 y(matrix.)35 b(The)21 b(idea)f(is)g(to)g(orthogonalize)g(the)h(ro)o(ws)f(of)g(the)h (matrix)d(b)o(y)i(a)g(Gram-Sc)o(hmidt)450 568 y(pro)q(cess)d(\(note)f(that)g (in)e(sparse)j(matrices,)e(most)f(ro)o(ws)i(are)f(t)o(ypically)f(orthogonal)h (already)m(,)f(so)450 617 y(that)e(standard)f(Gram-Sc)o(hmidt)e(ma)o(y)g(b)q (e)j(not)f(so)h(bad)f(as)h(in)f(general\).)17 b(Saad)11 b(suggest)i(dropping) 450 667 y(strategies)k(for)e(the)i(\014ll-in)d(pro)q(duced)j(in)e(the)h (orthogonalization)e(pro)q(cess.)25 b(It)16 b(turns)h(out)e(that)450 717 y(the)h(resulting)f(incomplete)f(L)i(factor)f(can)g(b)q(e)h(view)o(ed)f (as)h(the)g(incomplete)e(Choleski)h(factor)g(of)450 767 y(the)k(matrix)e Fy(AA)728 752 y Fx(T)754 767 y FC(.)32 b(Exp)q(erimen)o(ts)18 b(sho)o(w)h(that)f(using)g Fy(L)h FC(in)f(a)g(CG)g(pro)q(cess)i(for)e(the)h (normal)450 817 y(equations:)f Fy(L)680 802 y Fw(\000)p Fv(1)725 817 y Fy(AA)787 802 y Fx(T)813 817 y Fy(L)841 802 y Fw(\000)p Fx(T)894 817 y Fy(y)13 b FC(=)f Fy(b)h FC(is)h(e\013ectiv)o(e)h(for)f(some)f (relev)n(an)o(t)h(problems.)450 954 y Fp(3.5)70 b(P)n(olynomial)21 b(preconditioners)450 1045 y FC(So)16 b(far,)g(w)o(e)h(ha)o(v)o(e)f(describ)q (ed)j(preconditioners)e(in)f(only)g(one)h(of)e(t)o(w)o(o)h(classes:)25 b(those)17 b(that)g(ap-)450 1095 y(pro)o(ximate)12 b(the)h(co)q(e\016cien)o (t)h(matrix,)e(and)h(where)h(linear)f(systems)g(with)g(the)h(preconditioner)g (as)450 1145 y(co)q(e\016cien)o(t)i(matrix)e(are)i(easier)g(to)g(solv)o(e)f (than)g(the)h(original)e(system.)23 b Fq(Polynomial)d FC(precondi-)450 1194 y(tioners)e(can)f(b)q(e)h(considered)g(as)f(mem)o(b)q(ers)f(of)h(the)g (second)i(class)e(of)g(preconditioners:)25 b(direct)450 1244 y(appro)o(ximations)11 b(of)j(the)g(in)o(v)o(erse)h(of)e(the)h(co)q (e\016cien)o(t)h(matrix.)512 1294 y(Supp)q(ose)d(that)f(the)g(co)q(e\016cien) o(t)h(matrix)d Fy(A)i FC(of)f(the)h(linear)g(system)f(is)h(normalized)e(to)i (the)g(form)450 1344 y Fy(A)h FC(=)f Fy(I)t Fu(\000)q Fy(B)r FC(,)g(and)e(that)h(the)g(sp)q(ectral)h(radius)e(of)g Fy(B)j FC(is)e(less)g(than)f(one.)17 b(Using)10 b(the)g(Neumann)e(series,)450 1394 y(w)o(e)k(can)f(write)h(the)g(in)o(v)o(erse)g(of)f Fy(A)h FC(as)f Fy(A)1057 1379 y Fw(\000)p Fv(1)1113 1394 y FC(=)1157 1362 y Ft(P)1201 1373 y Fw(1)1201 1406 y Fx(k)q Fv(=0)1270 1394 y Fy(B)1303 1379 y Fx(k)1324 1394 y FC(,)h(so)f(an)h(appro)o(ximation)c (ma)o(y)i(b)q(e)i(deriv)o(ed)450 1443 y(b)o(y)j(truncating)h(this)g (in\014nite)g(series.)24 b(Since)17 b(the)f(iterativ)o(e)g(metho)q(ds)f(w)o (e)h(are)g(considering)g(are)450 1493 y(already)10 b(based)g(on)g(the)h(idea) f(of)f(applying)g(p)q(olynomial)o(s)f(in)h(the)i(co)q(e\016cien)o(t)g(matrix) d(to)i(the)g(initial)450 1543 y(residual,)16 b(there)i(are)e(analytic)g (connections)h(b)q(et)o(w)o(een)h(the)e(basic)h(metho)q(d)e(and)h(p)q (olynomiall)o(y)450 1593 y(accelerated)g(one.)512 1643 y(Dub)q(ois,)10 b(Green)o(baum)f(and)g(Ro)q(drigue)h([74)o(])f(in)o(v)o(estigated)h(the)g (relationship)f(b)q(et)o(w)o(een)j(a)d(basic)450 1692 y(metho)q(d)i(using)g (a)h(splitting)e Fy(A)i FC(=)g Fy(M)d Fu(\000)c Fy(N)g FC(,)12 b(and)f(a)h(p)q(olynomial)o(ly)c(preconditioned)13 b(metho)q(d)e(with)554 1815 y Fy(M)599 1798 y Fw(\000)p Fv(1)594 1825 y Fx(p)655 1815 y FC(=)g(\()714 1761 y Fx(p)p Fw(\000)p Fv(1)714 1775 y Ft(X)717 1864 y Fx(i)p Fv(=0)775 1815 y FC(\()p Fy(I)h Fu(\000)e Fy(M)908 1798 y Fw(\000)p Fv(1)952 1815 y Fy(A)p FC(\))999 1798 y Fx(i)1013 1815 y FC(\))p Fy(M)1074 1798 y Fw(\000)p Fv(1)1119 1815 y Fy(:)450 1934 y FC(The)17 b(basic)g(result)g(is)f(that)h(for)f(classical)g (metho)q(ds,)g Fy(k)h FC(steps)h(of)e(the)h(p)q(olynomiall)o(y)d(precondi-) 450 1984 y(tioned)c(metho)q(d)g(are)g(exactly)h(equiv)n(alen)o(t)e(to)h Fy(k)q(p)g FC(steps)i(of)d(the)i(original)d(metho)q(d;)j(for)f(accelerated) 450 2034 y(metho)q(ds,)h(sp)q(eci\014cally)g(the)h(Cheb)o(yshev)g(metho)q(d,) e(the)i(preconditioned)f(iteration)g(can)g(impro)o(v)o(e)450 2083 y(the)j(n)o(um)o(b)q(er)g(of)f(iterations)h(b)o(y)f(at)h(most)f(a)g (factor)h(of)g Fy(p)p FC(.)512 2133 y(Although)f(there)i(is)e(no)g(gain)g(in) g(the)h(n)o(um)o(b)q(er)f(of)g(times)f(the)i(co)q(e\016cien)o(t)g(matrix)e (is)h(applied,)450 2183 y(p)q(olynomial)f(preconditioning)i(do)q(es)i (eliminate)d(a)h(large)h(fraction)g(of)f(the)i(inner)f(pro)q(ducts)h(and)450 2233 y(up)q(date)f(op)q(erations,)e(so)h(there)h(ma)o(y)d(b)q(e)j(an)e(o)o(v) o(erall)g(increase)i(in)f(e\016ciency)m(.)512 2283 y(Let)f(us)g(de\014ne)h(a) e(p)q(olynomial)d(preconditioner)k(more)e(abstractly)i(as)g(an)o(y)f(p)q (olynomial)d Fy(M)16 b FC(=)450 2333 y Fy(P)477 2339 y Fx(n)499 2333 y FC(\()p Fy(A)p FC(\))e(normalized)e(to)h Fy(P)6 b FC(\(0\))11 b(=)h(1.)18 b(No)o(w)13 b(the)h(c)o(hoice)g(of)f(the)h(b)q(est)g(p)q (olynomial)c(preconditioner)450 2382 y(b)q(ecomes)16 b(that)f(of)g(c)o(ho)q (osing)g(the)h(b)q(est)h(p)q(olynomial)12 b(that)k(minim)o(izes)d Fu(k)p Fy(I)h Fu(\000)c Fy(M)1734 2367 y Fw(\000)p Fv(1)1779 2382 y Fy(A)p Fu(k)p FC(.)22 b(F)m(or)16 b(the)450 2432 y(c)o(hoice)g(of)f (the)i(in\014nit)o(y)e(norm)f(w)o(e)i(th)o(us)g(obtain)f(Cheb)o(yshev)i(p)q (olynomial)o(s,)d(and)h(they)h(require)450 2482 y(estimates)g(of)g(b)q(oth)h (a)f(lo)o(w)o(er)g(and)h(upp)q(er)g(b)q(ound)g(on)f(the)h(sp)q(ectrum)h(of)d Fy(A)p FC(.)27 b(These)17 b(estimates)450 2532 y(ma)o(y)12 b(b)q(e)i(deriv)o(ed)h(from)d(the)i(conjugate)g(gradien)o(t)g(iteration)g (itself;)f(see)i Fu(x)p FC(5.1.)512 2582 y(Since)21 b(an)g(accurate)h(lo)o(w) o(er)e(b)q(ound)h(on)f(the)h(sp)q(ectrum)h(of)e Fy(A)g FC(ma)o(y)f(b)q(e)i (hard)g(to)f(obtain,)450 2631 y(Johnson,)d(Micc)o(helli)f(and)g(P)o(aul)g ([126)n(])g(and)g(Saad)g([178)o(])g(prop)q(ose)h(least)g(squares)g(p)q (olynomials)450 2681 y(based)e(on)g(sev)o(eral)g(w)o(eigh)o(t)f(functions.)21 b(These)16 b(functions)f(only)f(require)h(an)g(upp)q(er)h(b)q(ound)e(and)450 2731 y(this)j(is)g(easily)g(computed,)g(using)g(for)f(instance)i(the)g (\\Gersc)o(hgorin)f(b)q(ound")g(max)1805 2737 y Fx(i)1826 2700 y Ft(P)1870 2743 y Fx(j)1894 2731 y Fu(j)p Fy(A)1937 2737 y Fx(i;j)1976 2731 y Fu(j)p FC(;)p eop %%Page: 49 61 60 bop 150 275 a Fr(3.6.)31 b(OTHER)14 b(PRECONDITIONERS)828 b FC(49)150 391 y(see)15 b([206)o(,)e Fu(x)p FC(1.4].)k(Exp)q(erimen)o(ts)d (comparing)d(Cheb)o(yshev)k(and)f(least)g(squares)g(p)q(olynomials)d(can)150 441 y(b)q(e)k(found)e(in)g(Ash)o(b)o(y)m(,)h(Man)o(teu\013el)g(and)g(Otto)g ([8)o(].)212 491 y(Application)f(of)f(p)q(olynomial)e(preconditioning)j(to)g (symmetric)f(inde\014nite)h(problems)g(is)g(de-)150 541 y(scrib)q(ed)j(b)o(y) f(Ash)o(b)o(y)m(,)g(Man)o(teu\013el)h(and)f(Sa)o(ylor)f([9)o(].)21 b(There)16 b(the)g(p)q(olynomial)c(is)j(c)o(hosen)h(so)f(that)150 591 y(it)f(transforms)f(the)h(system)g(in)o(to)f(a)h(de\014nite)g(one.)150 726 y Fp(3.6)70 b(Preconditioners)27 b(from)h(prop)r(erties)g(of)g(the)g (di\013er-)307 801 y(en)n(tial)21 b(equation)150 892 y FC(A)12 b(n)o(um)o(b)q(er)g(of)f(preconditioners)j(exist)e(that)g(deriv)o(e)h(their)f (justi\014cation)g(from)e(prop)q(erties)k(of)e(the)150 942 y(underlying)f(partial)g(di\013eren)o(tial)g(equation.)17 b(W)m(e)12 b(will)e(co)o(v)o(er)i(some)f(of)g(them)g(here)i(\(see)g(also)e Fu(x)p FC(5.5)150 991 y(and)j Fu(x)q FC(5.4\).)19 b(These)c(preconditioners)h (usually)e(in)o(v)o(olv)o(e)f(more)g(w)o(ork)i(than)f(the)h(t)o(yp)q(es)h (discussed)150 1041 y(ab)q(o)o(v)o(e,)d(ho)o(w)o(ev)o(er,)h(they)g(allo)o(w)f (for)g(sp)q(ecialized)i(faster)f(solution)f(metho)q(ds.)150 1156 y Fl(3.6.1)55 b(Preconditioning)18 b(b)n(y)g(the)h(symme)o(tric)c(part) 150 1232 y FC(In)f Fu(x)q FC(2.3.4)e(w)o(e)i(p)q(oin)o(ted)h(out)f(that)g (conjugate)g(gradien)o(t)g(metho)q(ds)g(for)f(non-selfadjoin)o(t)g(systems) 150 1282 y(require)21 b(the)g(storage)f(of)g(previously)g(calculated)g(v)o (ectors.)38 b(Therefore)22 b(it)d(is)h(somewhat)g(re-)150 1332 y(mark)n(able)15 b(that)j(preconditioning)e(b)o(y)h(the)h(symmetric)e(part)h (\()p Fy(A)12 b FC(+)g Fy(A)1293 1317 y Fx(T)1319 1332 y FC(\))p Fy(=)p FC(2)17 b(of)f(the)i(co)q(e\016cien)o(t)150 1382 y(matrix)11 b Fy(A)i FC(leads)g(to)g(a)f(metho)q(d)g(that)h(do)q(es)h(not)f(need)h(this)f (extended)h(storage.)k(Suc)o(h)13 b(a)g(metho)q(d)150 1432 y(w)o(as)h(prop)q(osed)h(b)o(y)e(Concus)i(and)e(Golub)g([54)o(])h(and)f (Widlund)g([211)o(].)212 1481 y(Ho)o(w)o(ev)o(er,)i(solving)f(a)h(system)g (with)f(the)i(symmetric)d(part)i(of)f(a)h(matrix)e(ma)o(y)g(b)q(e)j(no)e (easier)150 1531 y(than)c(solving)g(a)g(system)g(with)g(the)h(full)e(matrix.) 15 b(This)10 b(problem)f(ma)o(y)g(b)q(e)i(tac)o(kled)f(b)o(y)g(imp)q(osing)e (a)150 1581 y(nested)15 b(iterativ)o(e)e(metho)q(d,)f(where)j(a)e (preconditioner)h(based)g(on)f(the)h(symmetric)d(part)j(is)f(used.)150 1631 y(V)m(assilevski)k([207)o(])g(pro)o(v)o(ed)g(that)h(the)g(e\016ciency)h (of)e(this)g(preconditioner)i(for)e(the)h(symmetric)150 1681 y(part)c(carries)h(o)o(v)o(er)f(to)g(the)g(outer)h(metho)q(d.)150 1795 y Fl(3.6.2)55 b(The)19 b(use)f(of)h(fast)g(solv)n(ers)150 1872 y FC(In)d(man)o(y)e(applications,)h(the)i(co)q(e\016cien)o(t)g(matrix)d (is)i(symmetric)e(and)i(p)q(ositiv)o(e)g(de\014nite.)25 b(The)150 1922 y(reason)14 b(for)f(this)g(is)g(usually)f(that)h(the)h(partial)e (di\013eren)o(tial)h(op)q(erator)h(from)d(whic)o(h)i(it)g(is)g(deriv)o(ed)150 1972 y(is)i(self-adjoin)o(t,)e(co)q(erciv)o(e,)j(and)f(b)q(ounded)g(\(see)h (Axelsson)g(and)e(Bark)o(er)i([14)o(,)f Fu(x)p FC(3.2]\).)20 b(It)15 b(follo)o(ws)150 2021 y(that)f(for)f(the)i(co)q(e\016cien)o(t)g (matrix)d Fy(A)i FC(the)g(follo)o(wing)d(relation)j(holds)f(for)h(an)o(y)f (matrix)f Fy(B)k FC(from)c(a)150 2071 y(similar)f(di\013eren)o(tial)j (equation:)254 2172 y Fy(c)272 2178 y Fv(1)302 2172 y Fu(\024)352 2144 y Fy(x)376 2129 y Fx(T)402 2144 y Fy(Ax)p 351 2162 108 2 v 351 2200 a(x)375 2188 y Fx(T)401 2200 y Fy(B)r(x)474 2172 y Fu(\024)e Fy(c)536 2178 y Fv(2)596 2172 y FC(for)i(all)e Fy(x;)150 2259 y FC(where)i Fy(c)287 2265 y Fv(1)306 2259 y FC(,)f Fy(c)349 2265 y Fv(2)381 2259 y FC(do)g(not)g(dep)q(end)h(on)f(the)h (matrix)e(size.)18 b(The)c(imp)q(ortance)e(of)h(this)g(is)g(that)h(the)f(use) 150 2309 y(of)j Fy(B)i FC(as)e(a)g(preconditioner)h(giv)o(es)f(an)g(iterativ) o(e)g(metho)q(d)f(with)h(a)g(n)o(um)o(b)q(er)g(of)f(iterations)h(that)150 2358 y(do)q(es)f(not)e(dep)q(end)j(on)d(the)i(matrix)d(size.)212 2408 y(Th)o(us)g(w)o(e)g(can)g(precondition)f(our)h(original)e(matrix)g(b)o (y)h(one)h(deriv)o(ed)g(from)e(a)h(di\013eren)o(t)i(PDE,)150 2458 y(if)e(one)i(can)f(b)q(e)g(found)g(that)g(has)g(attractiv)o(e)h(prop)q (erties)g(as)f(preconditioner.)19 b(One)12 b(c)o(hoice)h(w)o(ould)150 2508 y(b)q(e)i(to)g(tak)o(e)g(a)g(matrix)e(from)g(a)h Fq(sep)n(ar)n(able)k FC(PDE.)c(A)h(system)g(in)o(v)o(olving)d(suc)o(h)k(a)f(matrix)e(can)i(b)q(e) 150 2558 y(solv)o(ed)d(with)g(v)n(arious)g(so-called)g(\\fast)g(solv)o(ers",) h(suc)o(h)g(as)f(FFT)h(metho)q(ds,)f(cyclic)g(reduction,)h(or)150 2607 y(the)h(generalized)h(marc)o(hing)d(algorithm)f(\(see)k(Dorr)f([72)o(],) f(Sw)o(arztraub)q(er)i([191)o(],)e(Bank)h([24)o(])f(and)150 2657 y(Bank)h(and)g(Rose)g([27)o(]\).)j(F)m(or)d(instance,)g(if)f(the)i (original)d(matrix)g(arises)i(from)254 2733 y Fu(\000)p FC(\()p Fy(a)p FC(\()p Fy(x;)7 b(y)q FC(\))p Fy(u)444 2739 y Fx(x)465 2733 y FC(\))481 2739 y Fx(x)511 2733 y Fu(\000)j FC(\()p Fy(b)p FC(\()p Fy(x;)d(y)q FC(\))p Fy(u)707 2739 y Fx(y)727 2733 y FC(\))743 2739 y Fx(y)774 2733 y FC(=)12 b Fy(f)r(;)p eop %%Page: 50 62 61 bop 450 275 a FC(50)802 b Fr(CHAPTER)14 b(3.)32 b(PRECONDITIONERS)450 391 y FC(then)15 b(the)f(preconditioner)h(can)f(b)q(e)g(formed)f(from)554 474 y Fu(\000)p FC(\()q(~)-22 b Fy(a)p FC(\()p Fy(x)p FC(\))p Fy(u)704 480 y Fx(x)725 474 y FC(\))741 480 y Fx(x)771 474 y Fu(\000)10 b FC(\()828 463 y(~)829 474 y Fy(b)p FC(\()p Fy(y)q FC(\))p Fy(u)924 480 y Fx(y)944 474 y FC(\))960 480 y Fx(y)992 474 y FC(=)i Fy(f)r(:)450 557 y FC(An)i(extension)h(to)e(the)i(non-self)e (adjoin)o(t)g(case)i(is)f(considered)h(b)o(y)f(Elman)e(and)h(Sc)o(h)o(ultz)i ([90)o(].)512 607 y(F)m(ast)d(solv)o(ers)g(are)g(attractiv)o(e)g(in)f(that)g (the)i(n)o(um)o(b)q(er)e(of)g(op)q(erations)g(they)i(require)f(is)f(\(sligh)o (tly)450 657 y(higher)k(than\))g(of)f(the)i(order)f(of)g(the)g(n)o(um)o(b)q (er)f(of)h(v)n(ariables.)20 b(Coupled)15 b(with)f(the)i(fact)f(that)g(the)450 707 y(n)o(um)o(b)q(er)e(of)g(iterations)h(in)f(the)i(resulting)f (preconditioned)g(iterativ)o(e)g(metho)q(ds)f(is)h(indep)q(enden)o(t)450 757 y(of)g(the)i(matrix)d(size,)j(suc)o(h)f(metho)q(ds)g(are)g(close)h(to)e (optimal.)19 b(Ho)o(w)o(ev)o(er,)c(fast)g(solv)o(ers)h(are)f(usu-)450 806 y(ally)f(only)h(applicable)g(if)g(the)h(ph)o(ysical)f(domain)f(is)h(a)h (rectangle)g(or)g(other)g(Cartesian)g(pro)q(duct)450 856 y(structure.)k(\(F)m (or)11 b(a)h(domain)d(consisting)j(of)f(a)h(n)o(um)o(b)q(er)f(of)g(suc)o(h)i (pieces,)g(domain)d(decomp)q(osition)450 906 y(metho)q(ds)j(can)i(b)q(e)f (used;)g(see)h Fu(x)q FC(5.4\).)450 1022 y Fl(3.6.3)55 b(Alternating)18 b(Direction)f(Implicit)f(metho)r(ds)450 1099 y FC(The)k(P)o(oisson)f (di\013eren)o(tial)g(op)q(erator)g(can)h(b)q(e)f(split)g(in)g(a)g(natural)g (w)o(a)o(y)f(as)h(the)h(sum)e(of)h(t)o(w)o(o)450 1149 y(op)q(erators:)554 1236 y Fu(L)11 b FC(=)h Fu(L)667 1242 y Fv(1)694 1236 y FC(+)e Fu(L)765 1242 y Fv(2)783 1236 y Fy(;)89 b FC(where)16 b Fu(L)1034 1242 y Fv(1)1063 1236 y FC(=)c Fu(\000)1154 1220 y Fx(@)1174 1208 y Fk(2)p 1144 1227 55 2 v 1144 1251 a Fx(@)r(x)1183 1242 y Fk(2)1204 1236 y FC(,)h Fu(L)1258 1242 y Fv(2)1288 1236 y FC(=)f Fu(\000)1378 1220 y Fx(@)1398 1208 y Fk(2)p 1369 1227 54 2 v 1369 1251 a Fx(@)r(y)1407 1242 y Fk(2)1428 1236 y Fy(:)450 1319 y FC(No)o(w)f(let)g Fy(L)627 1325 y Fv(1)646 1319 y FC(,)h Fy(L)698 1325 y Fv(2)728 1319 y FC(b)q(e)g(discretized)h(represen)o(tations)g (of)e Fu(L)1344 1325 y Fv(1)1362 1319 y FC(,)g Fu(L)1414 1325 y Fv(2)1432 1319 y FC(.)18 b(Based)12 b(on)f(the)h(observ)n(ation)f(that)450 1369 y Fy(L)478 1375 y Fv(1)506 1369 y FC(+)f Fy(L)576 1375 y Fv(2)606 1369 y FC(=)i(\()p Fy(I)h FC(+)c Fy(L)766 1375 y Fv(1)785 1369 y FC(\)\()p Fy(I)k FC(+)d Fy(L)918 1375 y Fv(2)936 1369 y FC(\))g Fu(\000)f Fy(I)k Fu(\000)d Fy(L)1104 1375 y Fv(1)1122 1369 y Fy(L)1150 1375 y Fv(2)1169 1369 y FC(,)k(iterativ)o(e)g(sc)o (hemes)g(suc)o(h)h(as)554 1452 y(\(1)9 b(+)g Fy(\013L)696 1458 y Fv(1)715 1452 y FC(\)\(1)g(+)h Fy(\013L)874 1458 y Fv(2)892 1452 y FC(\))p Fy(u)932 1435 y Fv(\()p Fx(m)p Fv(+1\))1043 1452 y FC(=)i([\(1)d(+)g Fy(\014)r(L)1239 1458 y Fv(1)1259 1452 y FC(\)\(1)g(+)h Fy(\014)r(L)1416 1458 y Fv(2)1435 1452 y FC(\)])d Fy(u)1494 1435 y Fv(\()p Fx(m)p Fv(\))450 1535 y FC(with)14 b(suitable)f(c)o(hoices)i(of)e Fy(\013)h FC(and)g Fy(\014)i FC(ha)o(v)o(e)e(b)q(een)h(prop)q(osed.)512 1585 y(This)d Fq(alternating)g(dir)n(e)n(ction)g(implicit)p FC(,)e(or)i Fq(ADI)p FC(,)f(metho)q(d)f(w)o(as)i(\014rst)g(prop)q(osed)g(as)g(a)f(solution)450 1635 y(metho)q(d)e(for)h(parab)q(olic)g(equations.)17 b(The)11 b Fy(u)1142 1620 y Fv(\()p Fx(m)p Fv(\))1209 1635 y FC(are)g(then)g(appro)o (ximations)c(on)j(subsequen)o(t)j(time)450 1685 y(steps.)28 b(Ho)o(w)o(ev)o(er,)17 b(it)g(can)g(also)f(b)q(e)h(used)h(for)e(the)h(steady) h(state,)f(that)g(is,)g(for)f(solving)g(elliptic)450 1735 y(equations.)32 b(In)18 b(that)h(case,)h(the)f Fy(u)1022 1719 y Fv(\()p Fx(m)p Fv(\))1098 1735 y FC(b)q(ecome)f(subsequen)o(t)j(iterates;)g(see)f(D'Y)m(ak)o (ono)o(v)c([79)o(],)450 1784 y(F)m(airw)o(eather,)g(Gourla)o(y)f(and)i(Mitc)o (hell)e([93)o(],)h(Hadjidimos)e([118)o(],)i(and)g(P)o(eaceman)g(and)g(Rac)o (h-)450 1834 y(ford)11 b([169)o(].)17 b(Generalization)11 b(of)g(this)h(sc)o (heme)g(to)g(v)n(ariable)e(co)q(e\016cien)o(ts)j(or)f(fourth)g(order)g (elliptic)450 1884 y(problems)h(is)h(relativ)o(ely)f(straigh)o(tforw)o(ard.) 512 1934 y(The)i(ab)q(o)o(v)o(e)e(metho)q(d)g(is)h(implicit)d(since)k(it)f (requires)h(systems)f(solutions,)f(and)h(it)f(alternates)450 1984 y(the)d Fy(x)g FC(and)f Fy(y)j FC(\(and)d(if)g(necessary)j Fy(z)r FC(\))e(directions.)17 b(It)10 b(is)f(attractiv)o(e)h(from)e(a)i (practical)f(p)q(oin)o(t)g(of)g(view)450 2033 y(\(although)14 b(mostly)g(on)g(tensor)i(pro)q(duct)g(grids\),)f(since)h(solving)e(a)h (system)g(with,)f(for)h(instance,)450 2083 y(a)j(matrix)e Fy(I)g FC(+)c Fy(\013L)761 2089 y Fv(1)798 2083 y FC(en)o(tails)18 b(only)f(a)h(n)o(um)o(b)q(er)f(of)h(uncoupled)g(tridiagonal)f(solutions.)30 b(These)450 2133 y(need)15 b(v)o(ery)g(little)f(storage)g(o)o(v)o(er)h(that)f (needed)i(for)e(the)h(matrix,)d(and)i(they)h(can)f(b)q(e)h(executed)i(in)450 2183 y(parallel,)12 b(or)i(one)g(can)g(v)o(ectorize)h(o)o(v)o(er)f(them.)512 2233 y(Ho)o(w)o(ev)o(er,)20 b(there)g(is)f(a)f(problem)f(of)h(data)g (distribution.)32 b(F)m(or)18 b(v)o(ector)i(computers,)f(either)450 2282 y(the)d(system)g(solution)f(with)g Fy(L)946 2288 y Fv(1)981 2282 y FC(or)g(with)h Fy(L)1158 2288 y Fv(2)1192 2282 y FC(will)f(in)o(v)o (olv)o(e)f(v)o(ery)i(large)g(strides:)22 b(if)15 b(columns)g(of)450 2332 y(v)n(ariables)i(in)g(the)h(grid)g(are)g(stored)h(con)o(tiguously)m(,)d (only)h(the)i(solution)d(with)i Fy(L)1755 2338 y Fv(1)1791 2332 y FC(will)f(in)o(v)o(olv)o(e)450 2382 y(con)o(tiguous)d(data.)j(F)m(or)d (the)g Fy(L)941 2388 y Fv(2)974 2382 y FC(the)h(stride)f(equals)g(the)h(n)o (um)o(b)q(er)e(of)g(v)n(ariables)g(in)h(a)f(column.)512 2432 y(On)g(parallel)e(mac)o(hines)g(the)i(same)e(problem)g(o)q(ccurs,)i(and)f(it) g(requires)i(a)d(global)g(data)h(trans-)450 2482 y(p)q(osition)h(in)h(b)q(et) o(w)o(een)h(the)f Fy(L)916 2488 y Fv(1)949 2482 y FC(and)g Fy(L)1058 2488 y Fv(2)1091 2482 y FC(system)f(solution.)512 2532 y(A)j(theoretical)g(reason)h(that)f(ADI)f(preconditioners)i(are)f(of)g (in)o(terest)h(is)e(that)h(they)h(can)f(b)q(e)450 2581 y(sho)o(wn)c(to)g(b)q (e)h(sp)q(ectrally)g(equiv)n(alen)o(t)e(to)h(the)h(original)d(co)q(e\016cien) o(t)j(matrix.)j(Hence)d(the)g(n)o(um)o(b)q(er)450 2631 y(of)g(iterations)h (is)g(b)q(ounded)g(indep)q(enden)o(t)i(of)d(the)h(condition)g(n)o(um)o(b)q (er.)p eop %%Page: 51 63 62 bop 150 703 a Fz(Chapter)34 b(4)150 910 y FB(Related)41 b(Issues)150 1151 y Fp(4.1)70 b(Complex)21 b(Systems)150 1242 y FC(Conjugate)c(gradien)o(t)h(metho)q(ds)f(for)g(real)h(symmetric)e(systems) i(can)g(b)q(e)g(applied)f(to)h(complex)150 1292 y(Hermitian)12 b(systems)j(in)e(a)h(straigh)o(tforw)o(ard)f(manner.)18 b(F)m(or)13 b(non-Hermitian)f(complex)h(systems)150 1342 y(w)o(e)e(distinguish)g(t)o(w)o (o)g(cases.)18 b(In)12 b(general,)f(for)g(an)o(y)g(co)q(e\016cien)o(t)h (matrix)d(a)i(CGNE)g(metho)q(d)g(on)g(the)150 1391 y(normal)h(equations)j Fy(A)509 1376 y Fx(H)540 1391 y Fy(Ax)e FC(=)g Fy(A)684 1376 y Fx(H)715 1391 y Fy(b)h FC(is)h(p)q(ossible,)f(or)h(one)f(can)h(split)f(the) h(system)f(in)o(to)g(real)g(and)150 1441 y(complex)e(parts)i(and)g(use)g(a)f (metho)q(d)g(suc)o(h)h(as)g(GMRES)f(on)g(the)h(resulting)g(real)f (nonsymmetric)150 1491 y(system.)18 b(Ho)o(w)o(ev)o(er,)13 b(in)f(certain)i(practical)f(situations)f(the)i(complex)d(system)i(is)g (non-Hermitian)150 1541 y(but)h(symmetric.)212 1591 y(Complex)k(symmetric)f (systems)j(can)f(b)q(e)h(solv)o(ed)g(b)o(y)f(a)g(classical)g(conjugate)g (gradien)o(t)g(or)150 1640 y(Lanczos)13 b(metho)q(d,)e(that)i(is,)f(with)g (short)g(recurrence)q(s,)j(if)c(the)i(complex)e(inner)i(pro)q(duct)g(\()p Fy(x;)7 b(y)q FC(\))12 b(=)153 1690 y(\026)-24 b Fy(x)174 1675 y Fx(T)200 1690 y Fy(y)15 b FC(is)d(replaced)i(b)o(y)f(\()p Fy(x;)7 b(y)q FC(\))k(=)h Fy(x)670 1675 y Fx(T)696 1690 y Fy(y)q FC(.)19 b(Lik)o(e)12 b(the)i(BiConjugate)e(Gradien)o(t)g(metho)q(d,)g(this)h (metho)q(d)150 1740 y(is)i(susceptible)i(to)e(breakdo)o(wn,)h(that)f(is,)h (it)f(can)g(happ)q(en)h(that)g Fy(x)1212 1725 y Fx(T)1238 1740 y Fy(x)e FC(=)g(0)h(for)g Fy(x)f Fu(6)p FC(=)g(0.)23 b(A)15 b(lo)q(ok-)150 1790 y(ahead)e(strategy)h(can)g(remedy)f(this)h(in)e(most)h (cases)h(\(see)h(F)m(reund)f([97)o(])f(and)g(V)m(an)g(der)h(V)m(orst)f(and) 150 1840 y(Melissen)i([203)n(]\).)150 1977 y Fp(4.2)70 b(Stopping)23 b(Criteria)150 2067 y FC(An)14 b(iterativ)o(e)g(metho)q(d)f(pro)q(duces)j(a)d (sequence)j Fu(f)p Fy(x)959 2052 y Fv(\()p Fx(i)p Fv(\))999 2067 y Fu(g)d FC(of)h(v)o(ectors)h(con)o(v)o(erging)e(to)h(the)h(v)o(ector)g Fy(x)150 2117 y FC(satisfying)g(the)h Fy(n)10 b Fu(\002)h Fy(n)k FC(system)g Fy(Ax)f FC(=)h Fy(b)p FC(.)23 b(T)m(o)14 b(b)q(e)j(e\013ectiv)o (e,)f(a)g(metho)q(d)e(m)o(ust)h(decide)h(when)g(to)150 2167 y(stop.)i(A)c(go)q(o)q(d)g(stopping)f(criterion)i(should)201 2248 y(1.)20 b(iden)o(tify)13 b(when)h(the)h(error)g Fy(e)706 2233 y Fv(\()p Fx(i)p Fv(\))757 2248 y Fu(\021)d Fy(x)825 2233 y Fv(\()p Fx(i)p Fv(\))874 2248 y Fu(\000)d Fy(x)14 b FC(is)f(small)f(enough) i(to)g(stop,)201 2331 y(2.)20 b(stop)14 b(if)f(the)h(error)h(is)f(no)g (longer)f(decreasing)i(or)f(decreasing)h(to)q(o)f(slo)o(wly)m(,)e(and)201 2413 y(3.)20 b(limit)11 b(the)j(maxim)o(um)9 b(amoun)o(t)j(of)h(time)g(sp)q (en)o(t)i(iterating.)212 2494 y(F)m(or)j(the)h(user)h(wishing)e(to)g(read)h (as)f(little)g(as)g(p)q(ossible,)i(the)f(follo)o(wing)d(simple)h(stopping)150 2544 y(criterion)i(will)e(lik)o(ely)g(b)q(e)i(adequate.)33 b(The)19 b(user)g(m)o(ust)f(supply)g(the)i(quan)o(tities)e Fy(maxit)p FC(,)h Fu(k)p Fy(b)p Fu(k)p FC(,)150 2594 y Fq(stop)p 226 2594 13 2 v 15 w(tol)p FC(,)13 b(and)h(preferably)g(also)g Fu(k)p Fy(A)p Fu(k)p FC(:)212 2683 y Fu(\017)21 b FC(The)16 b(in)o(teger)h Fy(maxit)f FC(is)g(the)h(maxim)n(um)11 b(n)o(um)o(b)q(er)16 b(of)f(iterations)h(the)h(algorithm)c(will)i(b)q(e)254 2733 y(p)q(ermitted)e(to)h(p)q(erform.)904 2838 y(51)p eop %%Page: 52 64 63 bop 450 275 a FC(52)860 b Fr(CHAPTER)14 b(4.)31 b(RELA)m(TED)14 b(ISSUES)512 391 y Fu(\017)21 b FC(The)e(real)g(n)o(um)o(b)q(er)f Fu(k)p Fy(A)p Fu(k)g FC(is)g(a)h Fq(norm)j FC(of)c Fy(A)p FC(.)32 b(An)o(y)19 b(reasonable)g(\(order)h(of)e(magnitude\))554 441 y(appro)o(ximation)10 b(of)i(the)h(absolute)g(v)n(alue)f(of)g(the)i(largest)f (en)o(try)g(of)f(the)i(matrix)d Fy(A)i FC(will)e(do.)512 535 y Fu(\017)21 b FC(The)12 b(real)g(n)o(um)o(b)q(er)f Fu(k)p Fy(b)p Fu(k)g FC(is)h(a)g Fq(norm)j FC(of)c Fy(b)p FC(.)17 b(Again,)12 b(an)o(y)f(reasonable)h(appro)o(ximation)d(of)j(the)554 585 y(absolute)i(v)n(alue)f(of)g(the)i(largest)f(en)o(try)g(of)g Fy(b)f FC(will)g(do.)512 680 y Fu(\017)21 b FC(The)14 b(real)f(n)o(um)o(b)q (er)g Fq(stop)p 946 680 13 2 v 16 w(tol)g FC(measures)h(ho)o(w)f(small)f(the) i(user)h(w)o(an)o(ts)e(the)h Fq(r)n(esidual)k Fy(r)1916 664 y Fv(\()p Fx(i)p Fv(\))1967 680 y FC(=)554 729 y Fy(Ax)609 714 y Fv(\()p Fx(i)p Fv(\))659 729 y Fu(\000)12 b Fy(b)k FC(of)g(the)h (ultimate)d(solution)i Fy(x)1213 714 y Fv(\()p Fx(i)p Fv(\))1269 729 y FC(to)g(b)q(e.)27 b(One)17 b(w)o(a)o(y)f(to)g(c)o(ho)q(ose)h Fq(stop)p 1843 729 V 16 w(tol)f FC(is)g(as)554 779 y(the)g(appro)o(ximate)d (uncertain)o(t)o(y)j(in)f(the)h(en)o(tries)g(of)f Fy(A)g FC(and)g Fy(b)g FC(relativ)o(e)g(to)g Fu(k)p Fy(A)p Fu(k)g FC(and)g Fu(k)p Fy(b)p Fu(k)p FC(,)554 829 y(resp)q(ectiv)o(ely)m(.)k(F)m(or)14 b(example,)e(c)o(ho)q(osing)h Fq(stop)p 1286 829 V 16 w(tol)g FC(10)1400 814 y Fw(\000)p Fv(6)1458 829 y FC(means)g(that)h(the)h(user)g (considers)554 879 y(the)k(en)o(tries)g(of)f Fy(A)h FC(and)f Fy(b)g FC(to)h(ha)o(v)o(e)f(errors)i(in)e(the)h(range)f Fu(\006)p FC(10)1589 864 y Fw(\000)p Fv(6)1634 879 y Fu(k)p Fy(A)p Fu(k)g FC(and)g Fu(\006)p FC(10)1884 864 y Fw(\000)p Fv(6)1928 879 y Fu(k)p Fy(b)p Fu(k)p FC(,)554 929 y(resp)q(ectiv)o(ely)m(.)g(The)13 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w(tol)p FC(.)450 2336 y Fl(4.2.1)55 b(More)19 b(Details)e(ab)r(out)i(Stopping)f(Criteria)450 2417 y FC(Ideally)11 b(w)o(e)h(w)o(ould)e(lik)o(e)h(to)h(stop)g(when)g(the)g(magnitudes)e(of)h(en) o(tries)i(of)e(the)h(error)h Fy(e)1775 2402 y Fv(\()p Fx(i)p Fv(\))1827 2417 y FC(=)e Fy(x)1894 2402 y Fv(\()p Fx(i)p Fv(\))1939 2417 y Fu(\000)5 b Fy(x)450 2467 y FC(fall)15 b(b)q(elo)o(w)g(a)h (user-supplied)h(threshold.)26 b(But)16 b Fy(e)1242 2452 y Fv(\()p Fx(i)p Fv(\))1298 2467 y FC(is)g(hard)g(to)g(estimate)g(directly)m(,) g(so)g(w)o(e)g(use)450 2517 y(the)d Fq(r)n(esidual)k Fy(r)697 2502 y Fv(\()p Fx(i)p Fv(\))748 2517 y FC(=)12 b Fy(Ax)847 2502 y Fv(\()p Fx(i)p Fv(\))894 2517 y Fu(\000)7 b Fy(b)12 b FC(instead,)h(whic)o(h)g(is)g(more)e(readily)i(computed.)k(The)c(rest)h(of) e(this)450 2567 y(section)j(describ)q(es)i(ho)o(w)d(to)g(measure)h(the)g (sizes)g(of)f(v)o(ectors)i Fy(e)1444 2552 y Fv(\()p Fx(i)p Fv(\))1498 2567 y FC(and)f Fy(r)1600 2552 y Fv(\()p Fx(i)p Fv(\))1639 2567 y FC(,)f(and)h(ho)o(w)f(to)g(b)q(ound)450 2617 y Fy(e)469 2602 y Fv(\()p Fx(i)p Fv(\))523 2617 y FC(in)f(terms)h(of)f Fy(r)754 2602 y Fv(\()p Fx(i)p Fv(\))794 2617 y FC(.)p 450 2655 620 2 v 496 2681 a Fo(1)514 2693 y Fn(On)k(a)g(mac)o(hine)e(with)j(IEEE) f(Standard)e(Floating)g(P)o(oin)o(t)i(Arithmetic,)f Fh(")k Fn(=)g(2)1649 2681 y Fe(\000)p Fo(24)1725 2693 y Fg(\031)g Fn(10)1808 2681 y Fe(\000)p Fo(7)1866 2693 y Fn(in)d(single)450 2733 y(precision,)9 b(and)i Fh(")f Fn(=)g(2)757 2721 y Fe(\000)p Fo(53)823 2733 y Fg(\031)g Fn(10)896 2721 y Fe(\000)p Fo(16)964 2733 y Fn(in)h(double)f(precision.)p eop %%Page: 53 65 64 bop 150 275 a Fr(4.2.)31 b(STOPPING)14 b(CRITERIA)966 b FC(53)212 391 y(W)m(e)14 b(will)e(measure)h(errors)j(using)d Fq(ve)n(ctor)18 b FC(and)13 b Fq(matrix)i(norms)s FC(.)j(The)c(most)f(common) d(v)o(ector)150 441 y(norms)j(are:)275 514 y Fu(k)p Fy(x)p Fu(k)341 520 y Fw(1)416 514 y Fu(\021)42 b FC(max)568 520 y Fx(j)592 514 y Fu(j)p Fy(x)628 520 y Fx(j)645 514 y Fu(j)11 b Fy(;)275 563 y Fu(k)p Fy(x)p Fu(k)341 569 y Fv(1)416 563 y Fu(\021)490 532 y Ft(P)534 576 y Fx(j)558 563 y Fu(j)p Fy(x)594 569 y Fx(j)611 563 y Fu(j)g Fy(;)c FC(and)275 618 y Fu(k)p Fy(x)p Fu(k)341 624 y Fv(2)416 618 y Fu(\021)42 b FC(\()506 587 y Ft(P)550 631 y Fx(j)575 618 y Fu(j)p Fy(x)611 624 y Fx(j)627 618 y Fu(j)639 603 y Fv(2)658 618 y FC(\))674 603 y Fv(1)p Fx(=)p Fv(2)749 618 y Fy(:)150 705 y FC(F)m(or)15 b(some)f(algorithms)f(w)o (e)i(ma)o(y)e(also)h(use)i(the)g(norm)e Fu(k)p Fy(x)p Fu(k)1097 711 y Fx(B)q(;\013)1169 705 y Fu(\021)g(k)p Fy(B)r(x)p Fu(k)1314 711 y Fx(\013)1338 705 y FC(,)g(where)j Fy(B)g FC(is)e(a)g(\014xed)150 755 y(nonsingular)f(matrix)f(and)i Fy(\013)f FC(is)h(one)g(of)g Fu(1)p FC(,)f(1,)g(or)h(2.)21 b(Corresp)q(onding)15 b(to)g(these)i(v)o(ector) e(norms)150 805 y(are)f(three)h(matrix)e(norms:)275 877 y Fu(k)p Fy(A)p Fu(k)348 883 y Fw(1)424 877 y Fu(\021)42 b FC(max)575 883 y Fx(j)599 846 y Ft(P)643 890 y Fx(k)670 877 y Fu(j)p Fy(a)704 883 y Fx(j;k)748 877 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b(Henceforth)22 b Fu(k)p Fy(x)p Fu(k)e FC(and)150 1171 y Fu(k)p Fy(A)p Fu(k)f FC(will)g(refer)h(to)g(an)o(y)f(m)o(utually)e (consisten)o(t)k(pair)f(of)f(the)h(ab)q(o)o(v)o(e.)35 b(\()p Fu(k)p Fy(x)p Fu(k)1404 1177 y Fv(2)1442 1171 y FC(and)20 b Fu(k)p Fy(A)p Fu(k)1602 1177 y Fx(E)1629 1171 y FC(,)h(as)150 1221 y(w)o(ell)14 b(as)g Fu(k)p Fy(x)p Fu(k)352 1227 y Fv(2)384 1221 y FC(and)g Fu(k)p Fy(A)p Fu(k)538 1227 y Fv(2)556 1221 y FC(,)g(b)q(oth)g(form)f(m)o(utually)f(consisten)o(t)j(pairs.\))k(All)14 b(these)h(norms)e(satisfy)150 1271 y(the)k(triangle)f(inequalit)o(y)g Fu(k)p Fy(x)10 b FC(+)i Fy(y)q Fu(k)17 b(\024)f(k)p Fy(x)p Fu(k)11 b FC(+)g Fu(k)p Fy(y)q Fu(k)17 b FC(and)g Fu(k)p Fy(A)11 b FC(+)g Fy(B)r Fu(k)17 b(\024)f(k)p Fy(A)p Fu(k)11 b FC(+)g Fu(k)p Fy(B)r Fu(k)p FC(,)17 b(as)g(w)o(ell)f(as)150 1321 y Fu(k)p Fy(Ax)p Fu(k)e(\024)h(k)p Fy(A)p Fu(k)10 b(\001)g(k)p Fy(x)p Fu(k)k FC(for)i(m)o(utually)d(consisten)o(t)j(pairs.)24 b(\(F)m(or)15 b(more)g(details)g(on)h(the)g(prop)q(erties)150 1370 y(of)d(norms,)g(see)i(Golub)e(and)g(V)m(an)h(Loan)f([108)o(].\))212 1420 y(One)h(di\013erence)i(b)q(et)o(w)o(een)f(these)g(norms)d(is)h(their)h (dep)q(endence)i(on)e(dimension.)i(A)d(v)o(ector)i Fy(x)150 1470 y FC(of)e(length)h Fy(n)g FC(with)g(en)o(tries)h(uniformly)c (distributed)j(b)q(et)o(w)o(een)i(0)d(and)h(1)g(will)e(satisfy)i Fu(k)p Fy(x)p Fu(k)1577 1476 y Fw(1)1623 1470 y Fu(\024)e FC(1,)150 1520 y(but)j Fu(k)p Fy(x)p Fu(k)293 1526 y Fv(2)326 1520 y FC(will)e(gro)o(w)h(lik)o(e)583 1490 y Fu(p)p 618 1490 25 2 v 30 x Fy(n)g FC(and)h Fu(k)p Fy(x)p Fu(k)805 1526 y Fv(1)838 1520 y FC(will)e(gro)o(w)h(lik)o(e)g Fy(n)p FC(.)21 b(Therefore)16 b(a)e(stopping)h(criterion)150 1570 y(based)f(on)f Fu(k)p Fy(x)p Fu(k)388 1576 y Fv(1)419 1570 y FC(\(or)g Fu(k)p Fy(x)p Fu(k)551 1576 y Fv(2)569 1570 y FC(\))h(ma)o(y)d(ha)o(v)o(e)i(to)g(b)q(e)h(p)q (ermitted)f(to)g(gro)o(w)g(prop)q(ortional)f(to)h Fy(n)g FC(\(or)1624 1540 y Fu(p)p 1658 1540 V 1658 1570 a Fy(n)p FC(\))150 1619 y(in)g(order)i(that)f(it)g(do)q(es)g(not)g(b)q(ecome)g(m)o(uc)o(h)e(harder)j (to)f(satisfy)f(for)h(large)g Fy(n)p FC(.)212 1669 y(There)19 b(are)f(t)o(w)o(o)f(approac)o(hes)h(to)g(b)q(ounding)f(the)h(inaccuracy)g(of) f(the)h(computed)f(solution)150 1719 y(to)22 b Fy(Ax)j FC(=)h Fy(b)p FC(.)42 b(Since)23 b Fu(k)p Fy(e)576 1704 y Fv(\()p Fx(i)p Fv(\))615 1719 y Fu(k)p FC(,)h(whic)o(h)e(w)o(e)g(will)f(call)g(the)i Fq(forwar)n(d)e(err)n(or)t FC(,)i(is)f(hard)g(to)g(esti-)150 1769 y(mate)17 b(directly)m(,)i(w)o(e)g(in)o(tro)q(duce)h(the)f Fq(b)n(ackwar)n(d)g(err)n(or)t FC(,)f(whic)o(h)g(allo)o(ws)g(us)h(to)f(b)q (ound)h(the)g(for-)150 1819 y(w)o(ard)14 b(error.)21 b(The)15 b(norm)o(wise)e(bac)o(kw)o(ard)h(error)i(is)e(de\014ned)i(as)e(the)h (smallest)e(p)q(ossible)i(v)n(alue)f(of)150 1869 y(max)n Fu(fk)p Fy(\016)r(A)p Fu(k)p Fy(=)p Fu(k)p Fy(A)p Fu(k)p Fy(;)7 b Fu(k)p Fy(\016)r(b)p Fu(k)p Fy(=)p Fu(k)p Fy(b)p Fu(kg)k FC(where)k Fy(x)791 1854 y Fv(\()p Fx(i)p Fv(\))845 1869 y FC(is)f(the)h Fq(exact)j FC(solution)c(of)f(\()p Fy(A)d FC(+)g Fy(\016)r(A)p FC(\))p Fy(x)1464 1854 y Fv(\()p Fx(i)p Fv(\))1516 1869 y FC(=)i(\()p Fy(b)e FC(+)f Fy(\016)r(b)p FC(\))150 1918 y(\(here)j Fy(\016)r(A)f FC(denotes)i(a)d(general)h(matrix,)f(not)h Fy(\016)i FC(times)d Fy(A)p FC(;)h(the)h(same)e(go)q(es)h(for)g Fy(\016)r(b)p FC(\).)17 b(The)11 b(bac)o(kw)o(ard)150 1968 y(error)k(ma)o(y)d(b)q(e)i(easily)f (computed)h(from)e(the)i Fq(r)n(esidual)k Fy(r)1051 1953 y Fv(\()p Fx(i)p Fv(\))1102 1968 y FC(=)12 b Fy(Ax)1201 1953 y Fv(\()p Fx(i)p Fv(\))1249 1968 y Fu(\000)e Fy(b)p FC(;)j(w)o(e)h(sho)o(w)g (ho)o(w)f(b)q(elo)o(w.)150 2018 y(Pro)o(vided)g(one)g(has)f(some)g(b)q(ound)h (on)f(the)h(in)o(v)o(erse)h(of)e Fy(A)p FC(,)g(one)h(can)g(b)q(ound)g(the)g (forw)o(ard)f(error)i(in)150 2068 y(terms)g(of)f(the)h(bac)o(kw)o(ard)g (error)h(via)e(the)i(simple)d(equalit)o(y)254 2151 y Fy(e)273 2134 y Fv(\()p Fx(i)p Fv(\))324 2151 y FC(=)g Fy(x)392 2134 y Fv(\()p Fx(i)p Fv(\))441 2151 y Fu(\000)d Fy(x)j FC(=)g Fy(A)593 2134 y Fw(\000)p Fv(1)637 2151 y FC(\()p Fy(Ax)708 2134 y Fv(\()p Fx(i)p Fv(\))757 2151 y Fu(\000)e Fy(b)p FC(\))h(=)h Fy(A)919 2134 y Fw(\000)p Fv(1)964 2151 y Fy(r)984 2134 y Fv(\()p Fx(i)p Fv(\))1023 2151 y Fy(;)150 2234 y FC(whic)o(h)g(implies)d Fu(k)p Fy(e)445 2219 y Fv(\()p Fx(i)p Fv(\))485 2234 y Fu(k)j(\024)f(k)p Fy(A)613 2219 y Fw(\000)p Fv(1)658 2234 y Fu(k)5 b(\001)g(k)p Fy(r)742 2219 y Fv(\()p Fx(i)p Fv(\))780 2234 y Fu(k)p FC(.)18 b(Therefore,)13 b(a)e(stopping)g(criterion)i(of)e(the)h(form)e(\\stop)150 2284 y(when)k Fu(k)p Fy(r)299 2269 y Fv(\()p Fx(i)p Fv(\))339 2284 y Fu(k)d(\024)h Fy(\034)5 b FC(")14 b(also)f(yields)h(an)g(upp)q(er)h(b) q(ound)f(on)f(the)i(forw)o(ard)e(error)i Fu(k)p Fy(e)1401 2269 y Fv(\()p Fx(i)p Fv(\))1441 2284 y Fu(k)d(\024)g Fy(\034)i Fu(\001)9 b(k)p Fy(A)1623 2269 y Fw(\000)p Fv(1)1667 2284 y Fu(k)p FC(.)150 2334 y(\(Sometimes)k(w)o(e)j(ma)o(y)d(prefer)k(to)e(use)h 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Fv(\()p Fx(i)p Fv(\))150 2633 y FC(an)g(exact)h(solution)e(of)h(\()p Fy(A)t FC(+)t Fy(\016)r(A)p FC(\))p Fy(x)689 2618 y Fv(\()p Fx(i)p Fv(\))741 2633 y FC(=)h Fy(b)t FC(+)t Fy(\016)r(b)p FC(.)17 b(If)11 b(the)g(original)f(data)h Fy(A)g FC(and)g Fy(b)g FC(ha)o(v)o(e)g(errors)i(from)150 2683 y(previous)k(computations)e(or)h (measuremen)o(ts,)g(then)h(it)f(is)h(usually)e(not)i(w)o(orth)f(iterating)g (un)o(til)150 2733 y Fy(\016)r(A)g FC(and)f Fy(\016)r(b)g FC(are)h(ev)o(en)h (smaller)d(than)h(these)i(errors.)24 b(F)m(or)15 b(example,)g(if)f(the)j(mac) o(hine)d(precision)p eop %%Page: 54 66 65 bop 450 275 a FC(54)860 b Fr(CHAPTER)14 b(4.)31 b(RELA)m(TED)14 b(ISSUES)450 391 y FC(is)g Fy(")p FC(,)f(it)h(is)f(not)h(w)o(orth)g(making)d Fu(k)p Fy(\016)r(A)p Fu(k)g(\024)h Fy(")p Fu(k)p Fy(A)p Fu(k)i FC(and)f Fu(k)p Fy(\016)r(b)p Fu(k)e(\024)h Fy(")p Fu(k)p Fy(b)p Fu(k)p FC(,)h(b)q(ecause)i(just)f(rounding)g(the)450 441 y(en)o(tries)h(of)e Fy(A)h FC(and)g Fy(b)g FC(to)f(\014t)h(in)g(the)g(mac)o(hine)f(creates)j (errors)f(this)f(large.)512 491 y(Based)k(on)e(this)h(discussion,)h(w)o(e)f (discuss)g(some)f(stopping)h(criteria)g(and)f(their)h(prop)q(erties.)450 541 y(The)d(\014rst)h(one)f(w)o(e)g(discussed)i(ab)q(o)o(v)o(e)d(is)450 625 y Fs(Criterion)g(1.)21 b Fu(k)p Fy(r)756 610 y Fv(\()p Fx(i)p Fv(\))795 625 y Fu(k)12 b(\024)f Fy(S)896 631 y Fv(1)927 625 y Fu(\021)h Fq(stop)p 1049 625 13 2 v 18 w(tol)e Fu(\001)d FC(\()p Fu(k)p Fy(A)p Fu(k)g(\001)g(k)p Fy(x)1297 610 y Fv(\()p Fx(i)p Fv(\))1334 625 y Fu(k)g FC(+)g Fu(k)p Fy(b)p Fu(k)p FC(\).)17 b(This)12 b(is)h(equiv)n(alen)o(t)e(to)i(asking)554 675 y(that)f(the)h(bac)o(kw)o(ard)f(error)h Fy(\016)r(A)f FC(and)g Fy(\016)r(b)g FC(describ)q(ed)i(ab)q(o)o(v)o(e)e(satisfy)g Fu(k)p Fy(\016)r(A)p Fu(k)g(\024)f Fq(stop)p 1840 675 V 18 w(tol)f Fu(\001)c(k)p Fy(A)p Fu(k)554 724 y FC(and)13 b Fu(k)p Fy(\016)r(b)p Fu(k)e(\024)h Fq(stop)p 847 724 V 18 w(tol)h Fu(\001)c(k)p Fy(b)p Fu(k)p FC(.)17 b(This)d(criterion)h(yields)e(the)i(forw) o(ard)e(error)i(b)q(ound)658 825 y Fu(k)p Fy(e)698 808 y Fv(\()p Fx(i)p Fv(\))737 825 y Fu(k)d(\024)g(k)p Fy(A)866 808 y Fw(\000)p Fv(1)910 825 y Fu(k)d(\001)g(k)p Fy(r)1002 808 y Fv(\()p Fx(i)p Fv(\))1041 825 y Fu(k)i(\024)h Fq(stop)p 1195 825 V 18 w(tol)h Fu(\001)c(k)p Fy(A)1340 808 y Fw(\000)p Fv(1)1385 825 y Fu(k)g(\001)f FC(\()p Fu(k)p Fy(A)p Fu(k)h(\001)g(k)p Fy(x)1599 808 y Fv(\()p Fx(i)p Fv(\))1638 825 y Fu(k)g FC(+)h Fu(k)p Fy(b)p Fu(k)p FC(\))h Fy(:)450 956 y FC(The)h(second)h(stopping)f(criterion)g(w)o(e)g (discussed,)i(whic)o(h)d(do)q(es)i(not)f(require)h Fu(k)p Fy(A)p Fu(k)p FC(,)e(ma)o(y)f(b)q(e)i(m)o(uc)o(h)450 1006 y(more)h(stringen)o(t)h (than)g(Criterion)g(1:)450 1098 y Fs(Criterion)f(2.)21 b Fu(k)p Fy(r)756 1083 y Fv(\()p Fx(i)p Fv(\))795 1098 y Fu(k)h(\024)h Fy(S)918 1104 y Fv(2)959 1098 y Fu(\021)f Fq(stop)p 1091 1098 V 18 w(tol)c Fu(\001)13 b(k)p Fy(b)p Fu(k)p FC(.)36 b(This)20 b(is)g(equiv)n(alen)o(t)g(to)g(asking)f(that)i(the)554 1148 y(bac)o(kw)o(ard)c(error)g Fy(\016)r(A)g FC(and)g Fy(\016)r(b)g FC(satisfy)f Fy(\016)r(A)h FC(=)g(0)f(and)h Fu(k)p Fy(\016)r(b)p Fu(k)f(\024)h Fy(tol)12 b Fu(\001)f(k)p Fy(b)p Fu(k)p FC(.)26 b(One)18 b(di\016cult)o(y)554 1198 y(with)d(this)h(metho)q(d)e(is)i(that)f (if)g Fu(k)p Fy(A)p Fu(k)10 b(\001)g(k)p Fy(x)p Fu(k)k(\035)g(k)p Fy(b)p Fu(k)p FC(,)g(whic)o(h)i(can)g(only)e(o)q(ccur)j(if)e Fy(A)g FC(is)h(v)o(ery)554 1247 y(ill-conditioned)d(and)i Fy(x)g FC(nearly)g(lies)g(in)g(the)h(n)o(ull)e(space)i(of)f Fy(A)p FC(,)g(then)h(it)f(ma)o(y)e(b)q(e)j(di\016cult)554 1297 y(for)g(an)o(y)h (metho)q(d)f(to)h(satisfy)g(the)g(stopping)g(criterion.)28 b(T)m(o)16 b(see)i(that)f Fy(A)g FC(m)o(ust)f(b)q(e)i(v)o(ery)554 1347 y(ill-conditioned,)11 b(note)k(that)658 1472 y(1)c Fu(\034)748 1444 y(k)p Fy(A)p Fu(k)e(\001)f(k)p Fy(x)p Fu(k)p 748 1462 168 2 v 802 1501 a(k)p Fy(b)p Fu(k)932 1472 y FC(=)981 1444 y Fu(k)p Fy(A)p Fu(k)h(\001)g(k)p Fy(A)1136 1429 y Fw(\000)p Fv(1)1180 1444 y Fy(b)p Fu(k)p 981 1462 238 2 v 1070 1501 a(k)p Fy(b)p Fu(k)1235 1472 y(\024)j(k)p Fy(A)p Fu(k)d(\001)g(k)p Fy(A)1434 1455 y Fw(\000)p Fv(1)1478 1472 y Fu(k)23 b Fy(:)554 1595 y FC(This)14 b(criterion)g(yields)g(the)g(forw)o(ard)f(error)i(b)q(ound) 658 1695 y Fu(k)p Fy(e)698 1678 y Fv(\()p Fx(i)p Fv(\))737 1695 y Fu(k)d(\024)g(k)p Fy(A)866 1678 y Fw(\000)p Fv(1)910 1695 y Fu(k)d(\001)g(k)p Fy(r)1002 1678 y Fv(\()p Fx(i)p Fv(\))1041 1695 y Fu(k)i(\024)h Fq(stop)p 1195 1695 13 2 v 18 w(tol)h Fu(\001)c(k)p Fy(A)1340 1678 y Fw(\000)p Fv(1)1385 1695 y Fu(k)g(\001)f(k)p Fy(b)p Fu(k)450 1826 y FC(If)14 b(an)g(estimate)f(of)h Fu(k)p Fy(A)816 1811 y Fw(\000)p Fv(1)860 1826 y Fu(k)g FC(is)g(a)o(v)n(ailable,)d (one)j(can)h(also)e(just)i(stop)f(when)g(the)h(upp)q(er)g(b)q(ound)f(on)450 1876 y(the)e(error)g Fu(k)p Fy(A)670 1861 y Fw(\000)p Fv(1)714 1876 y Fu(k)t(\001)t(k)p Fy(r)796 1861 y Fv(\()p Fx(i)p Fv(\))834 1876 y Fu(k)f FC(falls)f(b)q(elo)o(w)g(a)h(threshold.)18 b(This)11 b(yields)g(the)g(third)h(stopping)e(criterion:)450 1968 y Fs(Criterion)j(3.) 21 b Fu(k)p Fy(r)756 1953 y Fv(\()p Fx(i)p Fv(\))795 1968 y Fu(k)14 b(\024)f Fy(S)900 1974 y Fv(3)933 1968 y Fu(\021)h Fq(stop)p 1057 1968 V 18 w(tol)g Fu(\001)9 b(k)p Fy(x)1196 1953 y Fv(\()p Fx(i)p Fv(\))1235 1968 y Fu(k)p Fy(=)p Fu(k)p Fy(A)1329 1953 y Fw(\000)p Fv(1)1373 1968 y Fu(k)p FC(.)22 b(This)15 b(stopping)f(criterion)i(guaran-)554 2018 y(tees)f(that)665 2110 y Fu(k)p Fy(e)705 2095 y Fv(\()p Fx(i)p Fv(\))745 2110 y Fu(k)p 663 2128 106 2 v 663 2167 a(k)p Fy(x)708 2155 y Fv(\()p Fx(i)p Fv(\))747 2167 y Fu(k)784 2138 y(\024)833 2110 y(k)p Fy(A)885 2095 y Fw(\000)p Fv(1)929 2110 y Fu(k)9 b(\001)g(k)p Fy(r)1021 2095 y Fv(\()p Fx(i)p Fv(\))1060 2110 y Fu(k)p 833 2128 249 2 v 905 2167 a(k)p Fy(x)950 2155 y Fv(\()p Fx(i)p Fv(\))989 2167 y Fu(k)1098 2138 y(\024)j Fq(stop)p 1220 2138 13 2 v 18 w(tol)27 b Fy(;)554 2261 y FC(p)q(ermitting)13 b(the)h(user)i(to)e (sp)q(ecify)g(the)h(desired)h(relativ)o(e)e(accuracy)h Fq(stop)p 1719 2261 V 15 w(tol)f FC(in)g(the)h(com-)554 2311 y(puted)f(solution)f Fy(x)852 2296 y Fv(\()p Fx(i)p Fv(\))892 2311 y FC(.)450 2434 y(One)j(dra)o(wbac)o(k)f(to)g(Criteria)g(1)g(and)g(2)g(is)g(that)g(they)h (usually)e(treat)i(bac)o(kw)o(ard)f(errors)i(in)e(eac)o(h)450 2483 y(comp)q(onen)o(t)d(of)g Fy(\016)r(A)h FC(and)f Fy(\016)r(b)h FC(equally)m(,)e(since)i(most)f(norms)f Fu(k)p Fy(\016)r(A)p Fu(k)h FC(and)h Fu(k)p Fy(\016)r(b)p Fu(k)f FC(measure)g(eac)o(h)i(en)o(try) 450 2533 y(of)e Fy(\016)r(A)i FC(and)e Fy(\016)r(b)h FC(equally)m(.)k(F)m(or) 12 b(example,)g(if)g Fy(A)h FC(is)g(sparse)h(and)f Fy(\016)r(A)g FC(is)g(dense,)h(this)f(loss)g(of)g(p)q(ossibly)450 2583 y(imp)q(ortan)o(t)f (structure)17 b(will)12 b(not)i(b)q(e)h(re\015ected)i(in)c Fu(k)p Fy(\016)r(A)p Fu(k)p FC(.)19 b(In)14 b(con)o(trast,)g(the)h(follo)o (wing)d(stopping)450 2633 y(criterion)20 b(giv)o(es)g(one)h(the)f(option)g (of)f(scaling)h(eac)o(h)g(comp)q(onen)o(t)g Fy(\016)r(a)1582 2639 y Fx(j;k)1645 2633 y FC(and)g Fy(\016)r(b)1770 2639 y Fx(j)1808 2633 y FC(di\013eren)o(tly)m(,)450 2683 y(including)15 b(the)i(p)q(ossibilit)o(y)e(of)h(insisting)f(that)i(some)e(en)o(tries)i(b)q (e)g(zero.)26 b(The)17 b(cost)g(is)f(an)g(extra)450 2733 y(matrix-v)o(ector)d (m)o(ultiply:)p eop %%Page: 55 67 66 bop 150 275 a Fr(4.2.)31 b(STOPPING)14 b(CRITERIA)966 b FC(55)150 391 y Fs(Criterion)13 b(4.)21 b Fy(S)440 397 y Fv(4)471 391 y Fu(\021)11 b FC(max)592 397 y Fx(j)609 391 y FC(\()p Fu(j)p Fy(r)657 376 y Fv(\()p Fx(i)p Fv(\))696 391 y Fu(j)708 397 y Fx(j)725 391 y Fy(=)p FC(\()p Fy(E)g Fu(\001)d(j)p Fy(x)860 376 y Fv(\()p Fx(i)p Fv(\))899 391 y Fu(j)h FC(+)g Fy(f)t FC(\))1001 397 y Fx(j)1019 391 y FC(\))j Fu(\024)g Fq(stop)p 1169 391 13 2 v 18 w(tol)t FC(.)18 b(Here)d Fy(E)g FC(is)f(a)f(user-de\014ned)254 441 y(matrix)g(of)h(nonnegativ)o(e)g(en)o(tries,)i Fy(f)j FC(is)c(a)f (user-de\014ned)j(v)o(ector)e(of)f(nonnegativ)o(e)h(en)o(tries,)254 491 y(and)c Fu(j)p Fy(z)r Fu(j)f FC(denotes)j(the)e(v)o(ector)h(of)f (absolute)g(v)n(alues)g(of)g(the)h(en)o(tries)g(of)e Fy(z)r FC(.)18 b(If)10 b(this)i(criterion)f(is)254 541 y(satis\014ed,)i(it)g(means)f (there)j(are)e(a)g Fy(\016)r(A)g FC(and)g(a)g Fy(\016)r(b)g FC(suc)o(h)h(that)f(\()p Fy(A)8 b FC(+)f Fy(\016)r(A)p FC(\))p Fy(x)1395 526 y Fv(\()p Fx(i)p Fv(\))1447 541 y FC(=)12 b Fy(b)7 b FC(+)h Fy(\016)r(b)p FC(,)13 b(with)254 591 y Fu(j)p Fy(\016)r(a)308 597 y Fx(j;k)351 591 y Fu(j)f(\024)h Fy(tol)e Fu(\001)e Fy(e)518 597 y Fx(j;k)562 591 y FC(,)14 b(and)g Fu(j)p Fy(\016)r(b)719 597 y Fx(j)736 591 y Fu(j)e(\024)h Fy(tol)e Fu(\001)e Fy(f)904 597 y Fx(j)936 591 y FC(for)14 b(all)f Fy(j)k FC(and)d Fy(k)q FC(.)19 b(By)c(c)o(ho)q(osing)f Fy(E)j FC(and)d Fy(f)t FC(,)h(the)254 640 y(user)f(can)g(v)n(ary)f(the)h(w)o(a)o(y)f(the)h(bac)o(kw)o(ard)g(error)g (is)f(measured)h(in)f(the)h(stopping)f(criterion.)254 690 y(F)m(or)f (example,)f(c)o(ho)q(osing)h Fy(e)686 696 y Fx(j;k)742 690 y FC(=)g Fu(k)p Fy(A)p Fu(k)859 696 y Fw(1)906 690 y FC(and)h Fy(f)1006 696 y Fx(j)1035 690 y FC(=)f Fu(k)p Fy(b)p Fu(k)1139 696 y Fw(1)1186 690 y FC(mak)o(es)f(the)j(stopping)e(criterion)254 740 y Fu(k)p Fy(r)295 725 y Fv(\()p Fx(i)p Fv(\))334 740 y Fu(k)355 746 y Fw(1)390 740 y Fy(=)p FC(\()p Fy(n)p Fu(k)p Fy(A)p Fu(k)525 746 y Fw(1)560 740 y Fu(k)p Fy(x)605 725 y Fv(\()p Fx(i)p Fv(\))644 740 y Fu(k)665 746 y Fw(1)710 740 y FC(+)f Fu(k)p Fy(b)p Fu(k)813 746 y Fw(1)847 740 y FC(\),)16 b(whic)o(h)g(is)f(essen)o(tially)h(the)g(same)f(as)g(Criterion)h(1.)254 790 y(Cho)q(osing)g Fy(e)457 796 y Fx(j;k)518 790 y FC(=)g Fu(j)p Fy(a)600 796 y Fx(j;k)644 790 y Fu(j)g FC(and)h Fy(f)776 796 y Fx(j)810 790 y FC(=)g Fu(j)p Fy(b)889 796 y Fx(j)906 790 y Fu(j)f FC(mak)o(es)g(the)h(stopping)g(criterion)g(measure)g(the)254 840 y Fq(c)n(omp)n(onentwise)f(r)n(elative)f(b)n(ackwar)n(d)h(err)n(or)t FC(,)f Fq(i.e.)p FC(,)f(the)i(smallest)e(relativ)o(e)h(p)q(erturbations)254 889 y(in)e(an)o(y)g(comp)q(onen)o(t)h(of)f Fy(A)h FC(and)f Fy(b)h FC(whic)o(h)g(is)f(necessary)j(to)e(mak)o(e)e Fy(x)1324 874 y Fv(\()p Fx(i)p Fv(\))1377 889 y FC(an)i(exact)h(solution.)254 939 y(This)h(tigh)o(ter)g(stopping)g(criterion)h(requires,)g(among)d(other)j (things,)f(that)g Fy(\016)r(A)h FC(ha)o(v)o(e)f(the)254 989 y(same)h(sparsit)o(y)g(pattern)i(as)f Fy(A)p FC(.)29 b(Other)19 b(c)o(hoices)g(of)e Fy(E)i FC(and)f Fy(f)k FC(can)c(b)q(e)g(used)h(to)e (re\015ect)254 1039 y(other)12 b(structured)j(uncertain)o(ties)e(in)e Fy(A)h FC(and)g Fy(b)p FC(.)17 b(This)12 b(criterion)g(yields)g(the)h(forw)o (ard)e(error)254 1089 y(b)q(ound)358 1187 y Fu(k)p Fy(e)398 1170 y Fv(\()p Fx(i)p Fv(\))437 1187 y Fu(k)458 1193 y Fw(1)505 1187 y Fu(\024)h(k)7 b(j)p Fy(A)620 1170 y Fw(\000)p Fv(1)664 1187 y Fu(j)h(\001)h(j)p Fy(r)737 1170 y Fv(\()p Fx(i)p Fv(\))776 1187 y Fu(j)e(k)k(\024)h Fy(S)896 1193 y Fv(4)924 1187 y Fu(\001)d(k)e(j)p Fy(A)1016 1170 y Fw(\000)p Fv(1)1060 1187 y Fu(j)p FC(\()p Fy(E)r Fu(j)p Fy(x)1157 1170 y Fv(\()p Fx(i)p Fv(\))1196 1187 y Fu(j)h FC(+)i Fy(f)t FC(\))p Fu(k)1319 1193 y Fw(1)254 1285 y FC(where)15 b Fu(j)p Fy(A)417 1270 y Fw(\000)p Fv(1)461 1285 y Fu(j)e FC(is)h(the)h(matrix)d(of)h(absolute)h(v)n(alues)g(of)f(en)o(tries)i (of)e Fy(A)1328 1270 y Fw(\000)p Fv(1)1373 1285 y FC(.)150 1404 y(Finally)m(,)e(w)o(e)j(men)o(tion)f(one)h(more)f(criterion,)h(not)f(b)q (ecause)j(w)o(e)e(recommend)f(it,)g(but)h(b)q(ecause)i(it)150 1454 y(is)e(widely)f(used.)19 b(W)m(e)13 b(men)o(tion)g(it)g(in)h(order)g(to) g(explain)f(its)h(p)q(oten)o(tial)f(dra)o(wbac)o(ks:)150 1544 y Fs(Dubious)g(Criterion)g(5.)21 b Fu(k)p Fy(r)644 1529 y Fv(\()p Fx(i)p Fv(\))683 1544 y Fu(k)12 b(\024)f Fy(S)784 1550 y Fv(5)815 1544 y Fu(\021)h Fq(stop)p 937 1544 V 18 w(tol)f Fu(\001)6 b(k)p Fy(r)1066 1529 y Fv(\(0\))1111 1544 y Fu(k)p FC(.)17 b(This)c(commonl)o(y)d(used)k(criterion)254 1594 y(has)j(the)g(disadv)n(an)o (tage)f(of)g(dep)q(ending)i(to)q(o)e(strongly)h(on)f(the)i(initial)d (solution)h Fy(x)1589 1578 y Fv(\(0\))1633 1594 y FC(.)27 b(If)254 1643 y Fy(x)278 1628 y Fv(\(0\))337 1643 y FC(=)16 b(0,)g(a)g(common)d(c)o (hoice,)k(then)g Fy(r)894 1628 y Fv(\(0\))954 1643 y FC(=)f Fy(b)p FC(.)24 b(Then)17 b(this)f(criterion)h(is)f(equiv)n(alen)o(t)f(to)254 1693 y(Criterion)g(2)g(ab)q(o)o(v)o(e,)h(whic)o(h)f(ma)o(y)f(b)q(e)i (di\016cult)f(to)g(satisfy)g(for)h(an)o(y)f(algorithm)d(if)j Fu(k)p Fy(b)p Fu(k)f(\034)254 1743 y(k)p Fy(A)p Fu(k)s(\001)s(k)p Fy(x)p Fu(k)p FC(.)g(On)d(the)g(other)g(hand,)g(if)f Fy(x)848 1728 y Fv(\(0\))903 1743 y FC(is)g(v)o(ery)h(large)f(and)h(v)o(ery)f (inaccurate,)i(then)f Fu(k)p Fy(r)1634 1728 y Fv(\(0\))1678 1743 y Fu(k)254 1793 y FC(will)i(b)q(e)i(v)o(ery)g(large)f(and)h Fy(S)691 1799 y Fv(5)724 1793 y FC(will)e(b)q(e)i(arti\014cially)e(large;)h (this)h(means)f(the)h(iteration)f(ma)o(y)254 1843 y(stop)e(to)q(o)h(so)q(on.) k(This)c(criterion)f(yields)h(the)g(forw)o(ard)f(error)h(b)q(ound)f Fu(k)p Fy(e)1386 1828 y Fv(\()p Fx(i)p Fv(\))1426 1843 y Fu(k)g(\024)f Fy(S)1527 1849 y Fv(5)1553 1843 y Fu(\001)6 b(k)p Fy(A)1623 1828 y Fw(\000)p Fv(1)1667 1843 y Fu(k)p FC(.)150 1959 y Fl(4.2.2)55 b(When)19 b Fd(r)509 1940 y Fn(\()p Fh(i)p Fn(\))569 1959 y Fl(or)g Fc(k)p Fd(r)687 1940 y Fn(\()p Fh(i)p Fn(\))729 1959 y Fc(k)f Fl(is)g(not)h(readily)e(a)n(v)m(ailable)150 2035 y FC(It)c(is)f(p)q(ossible)h(to)f(design)h(an)f(iterativ)o(e)h(algorithm)d(for) i(whic)o(h)g Fy(r)1170 2020 y Fv(\()p Fx(i)p Fv(\))1222 2035 y FC(=)f Fy(Ax)1320 2020 y Fv(\()p Fx(i)p Fv(\))1367 2035 y Fu(\000)c Fy(b)12 b FC(or)g Fu(k)p Fy(r)1526 2020 y Fv(\()p Fx(i)p Fv(\))1566 2035 y Fu(k)g FC(is)g(not)150 2085 y(directly)k(a)o(v)n (ailable,)d(although)i(this)h(is)f Fq(not)20 b FC(the)c(case)h(for)e(an)o(y)g (algorithms)f(in)h(this)h(b)q(o)q(ok.)23 b(F)m(or)150 2135 y(completeness,)14 b(ho)o(w)o(ev)o(er,)g(w)o(e)g(discuss)h(stopping)f 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Fy(r)1948 684 y Fv(\()p Fx(i)p Fv(\))1988 699 y FC(.)450 749 y(There)g(is)f(a)g(large)f(n)o(um)o(b)q(er)g (of)h(problem)e(dep)q(enden)o(t)k(w)o(a)o(ys)e(to)f(estimate)h Fy(A)1680 734 y Fw(\000)p Fv(1)1725 749 y FC(;)g(w)o(e)g(men)o(tion)e(a)450 799 y(few)g(here.)512 850 y(When)g(a)g(splitting)f Fy(A)f FC(=)f Fy(M)j Fu(\000)c Fy(N)19 b FC(is)13 b(used)i(to)f(get)g(an)g(iteration)554 938 y Fy(x)578 921 y Fv(\()p Fx(i)p Fv(\))629 938 y FC(=)e Fy(M)718 921 y Fw(\000)p Fv(1)762 938 y Fy(N)5 b(x)824 921 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))915 938 y FC(+)10 b Fy(M)1002 921 y Fw(\000)p Fv(1)1046 938 y Fy(b)h FC(=)h Fy(Gx)1176 921 y Fv(\()p Fx(i)p Fw(\000)p Fv(1\))1267 938 y FC(+)d Fy(c;)450 1025 y FC(then)j(the)g(matrix)e(whose)i(in)o(v)o(erse)g(norm)f(w)o(e)g(need)i (is)e Fy(I)e Fu(\000)c Fy(G)p FC(.)16 b(Often,)c(w)o(e)g(kno)o(w)f(ho)o(w)g (to)h(estimate)450 1075 y Fu(k)p Fy(G)p Fu(k)j FC(if)g(the)h(splitting)f(is)h (a)g(standard)g(one)g(suc)o(h)g(as)g(Jacobi)g(or)g(SOR,)f(and)h(the)g(matrix) e Fy(A)i FC(has)450 1125 y(sp)q(ecial)k(c)o(haracteristics)h(suc)o(h)f(as)g (Prop)q(ert)o(y)h(A.)e(Then)h(w)o(e)g(ma)o(y)d(estimate)i Fu(k)p FC(\()p Fy(I)e Fu(\000)c Fy(G)p FC(\))1881 1110 y Fw(\000)p Fv(1)1925 1125 y Fu(k)21 b(\024)450 1175 y FC(1)p Fy(=)p FC(\(1)9 b Fu(\000)g(k)p Fy(G)p Fu(k)p FC(\).)512 1226 y(When)h Fy(A)f FC(is)g(symmetric)f(p)q(ositiv)o(e)h(de\014nite,)i(and)e(Cheb)o(yshev)h (acceleration)g(with)f(adaptation)450 1276 y(of)j(parameters)h(is)g(b)q(eing) f(used,)i(then)f(at)g(eac)o(h)g(step)h(the)f(algorithm)d(estimates)j(the)g (largest)g(and)450 1325 y(smallest)e(eigen)o(v)n(alues)h Fy(\025)845 1331 y Fv(max)908 1325 y FC(\()p Fy(A)p FC(\))h(and)f Fy(\025)1087 1331 y Fv(min)1144 1325 y FC(\()p Fy(A)p FC(\))h(of)e Fy(A)h FC(an)o(yw)o(a)o(y)m(.)k(Since)d Fy(A)f FC(is)g(symmetric)e(p)q(ositiv)o(e) 450 1375 y(de\014nite,)k Fu(k)p Fy(A)661 1360 y Fw(\000)p Fv(1)706 1375 y Fu(k)727 1381 y Fv(2)756 1375 y FC(=)e Fy(\025)824 1357 y Fw(\000)p Fv(1)824 1387 y(min)882 1375 y FC(\()p Fy(A)p FC(\).)512 1426 y(This)f(adaptiv)o(e)g(estimation)e(is)i(often)g(done)g(using)g(the)g Fq(L)n(anczos)i(algorithm)s FC(,)e(whic)o(h)g(can)g(usu-)450 1476 y(ally)k(pro)o(vide)h(go)q(o)q(d)g(estimates)g(of)f(the)i(largest)f (\(righ)o(tmost\))f(and)h(smallest)f(\(leftmost\))g(eigen-)450 1526 y(v)n(alues)d(of)f(a)g(symmetric)f(matrix)g(at)i(the)g(cost)h(of)e(a)g (few)h(matrix-v)o(ector)f(m)o(ultiplies.)k(F)m(or)c(general)450 1576 y(nonsymmetric)f Fy(A)p FC(,)i(w)o(e)g(ma)o(y)d(apply)i(the)i(Lanczos)f (metho)q(d)f(to)h Fy(AA)1515 1561 y Fx(T)1553 1576 y FC(or)g Fy(A)1633 1561 y Fx(T)1659 1576 y Fy(A)p FC(,)g(and)f(use)i(the)f(fact)450 1631 y(that)i Fu(k)p Fy(A)592 1616 y Fw(\000)p Fv(1)636 1631 y Fu(k)657 1637 y Fv(2)687 1631 y FC(=)e(1)p Fy(=\025)797 1610 y Fv(1)p Fx(=)p Fv(2)797 1643 y(min)854 1631 y FC(\()p Fy(AA)932 1616 y Fx(T)958 1631 y FC(\))g(=)g(1)p Fy(=\025)1096 1610 y Fv(1)p Fx(=)p Fv(2)1096 1643 y(min)1153 1631 y FC(\()p Fy(A)1200 1616 y Fx(T)1226 1631 y Fy(A)p FC(\).)512 1682 y(It)f(is)g(also)f(p)q (ossible)h(to)f(estimate)h Fu(k)p Fy(A)1089 1667 y Fw(\000)p Fv(1)1133 1682 y Fu(k)1154 1688 y Fw(1)1200 1682 y FC(pro)o(vided)f(one)h(is) g(willing)e(to)h(solv)o(e)h(a)f(few)h(systems)450 1732 y(of)f(linear)f (equations)i(with)e Fy(A)i FC(and)f Fy(A)1028 1717 y Fx(T)1064 1732 y FC(as)h(co)q(e\016cien)o(t)g(matrices.)16 b(This)10 b(is)g(often)g(done)h(with)f(dense)450 1782 y(linear)j(system)g(solv)o(ers,)h (b)q(ecause)h(the)f(extra)g(cost)g(of)f(these)h(systems)g(is)f Fy(O)q FC(\()p Fy(n)1685 1767 y Fv(2)1704 1782 y FC(\),)g(whic)o(h)g(is)g (small)450 1832 y(compared)h(to)h(the)g(cost)g Fy(O)q FC(\()p Fy(n)924 1817 y Fv(3)943 1832 y FC(\))f(of)g(the)i(LU)e(decomp)q(osition)g (\(see)i(Hager)f([120)o(],)e(Higham)g([123)o(])450 1882 y(and)i(Anderson,)h Fq(et)g(al.)e FC([3]\).)21 b(This)15 b(is)g(not)f(the)i(case)g(for)f (iterativ)o(e)g(solv)o(ers,)g(where)h(the)g(cost)g(of)450 1931 y(these)g(solv)o(es)g(ma)o(y)d(w)o(ell)h(b)q(e)i(sev)o(eral)f(times)g(as)g(m) o(uc)o(h)f(as)h(the)g(original)e(linear)i(system.)21 b(Still,)14 b(if)450 1981 y(man)o(y)h(linear)i(systems)h(with)f(the)h(same)e(co)q (e\016cien)o(t)i(matrix)e(and)h(di\013ering)g(righ)o(t-hand-sides)450 2031 y(are)d(to)g(b)q(e)h(solv)o(ed,)e(it)g(is)h(a)g(viable)f(metho)q(d.)512 2082 y(The)g(approac)o(h)g(in)f(the)h(last)g(paragraph)f(also)g(lets)h(us)g (estimate)f(the)h(alternate)h(error)f(b)q(ound)450 2132 y Fu(k)p Fy(e)490 2117 y Fv(\()p Fx(i)p Fv(\))530 2132 y Fu(k)551 2138 y Fw(1)602 2132 y Fu(\024)j(k)7 b(j)p Fy(A)721 2117 y Fw(\000)p Fv(1)765 2132 y Fu(j)k(\001)f(j)p Fy(r)842 2117 y Fv(\()p Fx(i)p Fv(\))881 2132 y Fu(j)d(k)921 2138 y Fw(1)956 2132 y FC(.)26 b(This)16 b(ma)o(y)f(b)q(e)i(m)o(uc)o(h)e(smaller)g(than)i(the)g(simpler)e Fu(k)p Fy(A)1876 2117 y Fw(\000)p Fv(1)1921 2132 y Fu(k)1942 2138 y Fw(1)1988 2132 y Fu(\001)450 2182 y(k)p Fy(r)491 2167 y Fv(\()p Fx(i)p Fv(\))530 2182 y Fu(k)551 2188 y 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1238 y(The)j(in\014nit)o(y)e(norm)g Fu(k)p Fy(x)p Fu(k)619 1244 y Fw(1)667 1238 y FC(=)g(max)790 1244 y Fx(j)815 1238 y Fu(j)p Fy(x)851 1244 y Fx(j)867 1238 y Fu(j)h FC(requires)h(the)g(few)o(est)g(\015oating)e(p)q(oin)o(t)h(op)q (erations)150 1288 y(to)i(compute,)f(and)h(cannot)g(o)o(v)o(er\015o)o(w)g(or) g(cause)h(other)f(exceptions)h(if)e(the)i Fy(x)1399 1294 y Fx(j)1433 1288 y FC(are)f(themselv)o(es)150 1337 y(\014nite)242 1322 y Fv(2)261 1337 y FC(.)23 b(On)15 b(the)i(other)f(hand,)f(computing)f Fu(k)p Fy(x)p Fu(k)940 1343 y Fv(2)972 1337 y FC(=)g(\()1034 1306 y Ft(P)1078 1350 y Fx(j)1103 1337 y Fu(j)p Fy(x)1139 1343 y Fx(j)1155 1337 y Fu(j)1167 1322 y Fv(2)1186 1337 y FC(\))1202 1322 y Fv(1)p Fx(=)p Fv(2)1269 1337 y FC(in)h(the)i(most)d(straigh)o(tfor-) 150 1387 y(w)o(ard)k(manner)f(can)h(easily)f(o)o(v)o(er\015o)o(w)h(or)f(lose) h(accuracy)h(to)f(under\015o)o(w)g(ev)o(en)h(when)f(the)g(true)150 1437 y(result)12 b(is)g(far)f(from)f(either)j(the)f(o)o(v)o(er\015o)o(w)f(or) 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Fv(\()p Fx(i)p Fv(\))1467 1886 y FC(=)e Fy(A)1545 1871 y Fw(\000)p Fv(1)1590 1886 y Fy(r)1610 1871 y Fv(\()p Fx(i)p Fv(\))1666 1886 y FC(of)150 1936 y(at)k(most)f Fy(O)q FC(\()p Fy(")p FC(\))p Fu(k)p Fy(A)449 1921 y Fw(\000)p Fv(1)493 1936 y Fu(k)13 b(\001)f FC(\()p Fu(k)p Fy(A)p Fu(k)g(\001)g(k)p Fy(x)721 1921 y Fv(\()p Fx(i)p Fv(\))761 1936 y Fu(k)g FC(+)h Fu(k)p Fy(b)p Fu(k)p FC(\).)33 b(A)19 b(more)f(re\014ned)i(b)q(ound)f(is)g(that)g(the)g(error)150 1986 y(\()p Fy(\016)r(r)206 1971 y Fv(\()p Fx(i)p Fv(\))246 1986 y FC(\))262 1992 y Fx(j)298 1986 y FC(in)e(the)i Fy(j)r FC(th)g(comp)q(onen)o(t)e(of)h Fy(r)791 1971 y Fv(\()p Fx(i)p Fv(\))849 1986 y FC(is)g(b)q(ounded)h(b)o(y)f Fy(O)q FC(\()p Fy(")p FC(\))g(times)f(the)i Fy(j)r FC(th)g(comp)q(onen)o(t)150 2036 y(of)f Fu(j)p Fy(A)p Fu(j)11 b(\001)g(j)p Fy(x)327 2020 y Fv(\()p Fx(i)p Fv(\))366 2036 y Fu(j)h FC(+)g Fu(j)p Fy(b)p Fu(j)p FC(,)18 b(or)g(more)f(tersely)i Fu(j)p Fy(\016)r(r)858 2020 y Fv(\()p Fx(i)p Fv(\))897 2036 y 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b(the)g(co)q(e\016cien)o(t)h(matrix)d Fy(A)i FC(is)g(sparse,)i(large-scale)e(linear)g(systems)g(of)f(the)i(form)d Fy(Ax)j FC(=)f Fy(b)450 935 y FC(can)c(b)q(e)g(most)f(e\016cien)o(tly)h(solv) o(ed)f(if)g(the)i(zero)f(elemen)o(ts)g(of)f Fy(A)h FC(are)g(not)g(stored.)19 b(Sparse)c(storage)450 985 y(sc)o(hemes)d(allo)q(cate)e(con)o(tiguous)h (storage)h(in)f(memory)d(for)j(the)h(nonzero)g(elemen)o(ts)f(of)g(the)h (matrix,)450 1035 y(and)g(p)q(erhaps)i(a)e(limited)e(n)o(um)o(b)q(er)h(of)h (zeros.)19 b(This,)12 b(of)g(course,)h(requires)h(a)e(sc)o(heme)g(for)g(kno)o (wing)450 1084 y(where)j(the)g(elemen)o(ts)e(\014t)h(in)o(to)f(the)i(full)e (matrix.)512 1135 y(There)21 b(are)f(man)o(y)d(metho)q(ds)i(for)h(storing)f (the)h(data)f(\(see)i(for)e(instance)i(Saad)e([182)n(])g(and)450 1184 y(Eijkhout)f([84)o(]\).)32 b(Here)20 b(w)o(e)f(will)f(discuss)h (Compressed)h(Ro)o(w)d(and)i(Column)e(Storage,)i(Blo)q(c)o(k)450 1234 y(Compressed)f(Ro)o(w)f(Storage,)i(Diagonal)c(Storage,)k(Jagged)f 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y(sparse)d(matrix)d Fy(A)p FC(,)i(w)o(e)h(create)g(3)f(v)o(ectors:)23 b(one)16 b(for)g(\015oating-p)q(oin)o(t)e(n)o(um)o(b)q(ers)i(\()p FA(val)p FC(\),)g(and)g(the)450 1869 y(other)i(t)o(w)o(o)f(for)h(in)o(tegers) g(\()p FA(col)p 954 1869 14 2 v 15 w(ind)p FC(,)g FA(row)p 1131 1869 V 15 w(ptr)p FC(\).)29 b(The)18 b FA(val)f FC(v)o(ector)h(stores)h (the)f(v)n(alues)g(of)f(the)450 1919 y(nonzero)g(elemen)o(ts)f(of)g(the)h (matrix)d Fy(A)p FC(,)j(as)f(they)h(are)g(tra)o(v)o(ersed)g(in)f(a)g(ro)o (w-wise)g(fashion.)25 b(The)450 1969 y FA(col)p 519 1969 V 15 w(ind)11 b FC(v)o(ector)i(stores)g(the)g(column)d(indexes)j(of)e(the)i (elemen)o(ts)f(in)f(the)i FA(val)e FC(v)o(ector.)18 b(That)12 b(is,)g(if)450 2019 y FA(val)p FC(\()p FA(k)p FC(\))f(=)h Fy(a)647 2025 y Fx(i;j)698 2019 y FC(then)g FA(col)p 859 2019 V 15 w(ind)p FC(\()p FA(k)p FC(\))f(=)h Fy(j)r FC(.)18 b(The)12 b FA(row)p 1247 2019 V 16 w(ptr)f FC(v)o(ector)i(stores)g(the)f(lo)q(cations)g(in)f(the) i FA(val)450 2069 y FC(v)o(ector)e(that)f(start)h(a)f(ro)o(w,)g(that)h(is,)f (if)g FA(val)p FC(\()p FA(k)p FC(\))h(=)g Fy(a)1240 2075 y Fx(i;j)1290 2069 y FC(then)g FA(row)p 1450 2069 V 15 w(ptr)p FC(\()p FA(i)p FC(\))g Fu(\024)g Fy(k)i(<)f FA(row)p 1784 2069 V 15 w(ptr)p FC(\()p FA(i)c FC(+)i FA(1)p FC(\).)450 2118 y(By)20 b(con)o(v)o(en)o(tion,)g(w)o(e)g(de\014ne)g FA(row)p 1008 2118 V 15 w(ptr)p FC(\()p FA(n)9 b FC(+)g FA(1)p FC(\))21 b(=)g Fy(nnz)15 b FC(+)e(1,)21 b(where)f Fy(nnz)i FC(is)d(the)h(n)o(um)o(b)q(er)f (of)450 2168 y(nonzeros)13 b(in)f(the)g(matrix)f Fy(A)p FC(.)17 b(The)c(storage)f(sa)o(vings)g(for)f(this)i(approac)o(h)f(is)f(signi\014can)o (t.)17 b(Instead)450 2218 y(of)c(storing)h Fy(n)661 2203 y Fv(2)693 2218 y FC(elemen)o(ts,)g(w)o(e)g(need)h(only)e(2)p Fy(nnz)e FC(+)e Fy(n)g FC(+)h(1)j(storage)i(lo)q(cations.)512 2268 y(As)g(an)e(example,)f(consider)j(the)g(nonsymmetric)d(matrix)g Fy(A)i FC(de\014ned)h(b)o(y)554 2474 y Fy(A)c FC(=)640 2316 y Ft(0)640 2389 y(B)640 2414 y(B)640 2439 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401 V 50 w(8)p 1009 401 V 50 w(7)p 1079 401 V 49 w(3)14 b Fu(\001)7 b(\001)g(\001)12 b FC(9)p 1247 401 V 50 w(13)p 1338 401 V 49 w(4)p 1409 401 V 49 w(2)p 1479 401 V 50 w(-1)p 1564 401 V 285 403 1280 2 v 284 453 2 50 v 310 438 a FA(col)p 379 438 14 2 v 15 w(ind)p 480 453 2 50 v 70 w FC(1)p 572 453 V 63 w(5)p 656 453 V 50 w(1)p 727 453 V 49 w(2)p 797 453 V 50 w(6)p 868 453 V 49 w(2)p 938 453 V 50 w(3)p 1009 453 V 50 w(4)p 1079 453 V 49 w(1)i Fu(\001)7 b(\001)g(\001)12 b FC(5)p 1247 453 V 70 w(6)p 1338 453 V 50 w(2)p 1409 453 V 49 w(5)p 1479 453 V 64 w(6)p 1564 453 V 285 454 1280 2 v 529 466 753 2 v 528 516 2 50 v 554 501 a FA(row)p 623 501 14 2 v 15 w(ptr)p 724 516 2 50 v 49 w FC(1)p 795 516 V 50 w(3)p 865 516 V 49 w(6)p 936 516 V 50 w(9)p 1006 516 V 49 w(13)p 1098 516 V 50 w(17)p 1189 516 V 49 w(20)p 1280 516 V 529 518 753 2 v 1309 502 a(.)150 592 y(If)e(the)g(matrix)e Fy(A)i FC(is)g(symmetric,)e(w)o(e)j(need)f(only)g 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15 w(ind)p FC(,)h FA(col)p 1461 1078 V 15 w(ptr)p Fu(g)p FC(,)f(where)150 1128 y FA(row)p 219 1128 V 15 w(ind)15 b FC(stores)j(the)e(ro)o(w)g(indices)g(of)f(eac)o(h)i(nonzero,)g (and)e FA(col)p 1191 1128 V 15 w(ptr)h FC(stores)h(the)f(index)g(of)g(the)150 1178 y(elemen)o(ts)c(in)h FA(val)f FC(whic)o(h)g(start)h(a)g(column)e(of)h Fy(A)p FC(.)17 b(The)c(CCS)g(format)e(for)h(the)h(matrix)e Fy(A)i FC(in)f(\(4.1\))150 1228 y(is)i(giv)o(en)f(b)o(y)p 292 1276 1266 2 v 291 1326 2 50 v 398 1311 a FA(val)p 487 1326 V 49 w FC(10)p 578 1326 V 49 w(3)p 649 1326 V 50 w(3)p 720 1326 V 49 w(9)p 790 1326 V 50 w(7)p 861 1326 V 49 w(8)p 931 1326 V 50 w(4)p 1002 1326 V 50 w(8)p 1072 1326 V 49 w(8)h Fu(\001)7 b(\001)g(\001)12 b FC(9)p 1240 1326 V 50 w(2)p 1310 1326 V 49 w(3)p 1381 1326 V 50 w(13)p 1472 1326 V 49 w(-1)p 1557 1326 V 292 1328 1266 2 v 291 1378 2 50 v 317 1363 a FA(row)p 386 1363 14 2 v 15 w(ind)p 487 1378 2 50 v 70 w FC(1)p 578 1378 V 49 w(2)p 649 1378 V 50 w(4)p 720 1378 V 49 w(2)p 790 1378 V 50 w(3)p 861 1378 V 49 w(5)p 931 1378 V 50 w(6)p 1002 1378 V 50 w(3)p 1072 1378 V 49 w(4)i Fu(\001)7 b(\001)g(\001)12 b FC(5)p 1240 1378 V 50 w(6)p 1310 1378 V 49 w(2)p 1381 1378 V 70 w(5)p 1472 1378 V 64 w(6)p 1557 1378 V 292 1379 1266 2 v 518 1391 774 2 v 517 1441 2 50 v 543 1426 a FA(col)p 612 1426 14 2 v 15 w(ptr)p 714 1441 2 50 v 50 w FC(1)p 784 1441 V 49 w(4)p 855 1441 V 50 w(8)p 926 1441 V 49 w(10)p 1017 1441 V 50 w(13)p 1108 1441 V 49 w(17)p 1200 1441 V 49 w(20)p 1291 1441 V 518 1443 774 2 v 52 w(.)150 1570 y Fs(Blo)q(c)o(k)j(Compressed)g(Ro)o (w)g(Storage)g(\(BCRS\))150 1648 y FC(If)d(the)h(sparse)h(matrix)c Fy(A)j FC(is)f(comprised)g(of)g(square)h(dense)h(blo)q(c)o(ks)e(of)g (nonzeros)i(in)e(some)f(regular)150 1698 y(pattern,)h(w)o(e)e(can)h(mo)q (dify)d(the)j(CRS)f(\(or)h(CCS\))g(format)d(to)j(exploit)f(suc)o(h)h(blo)q(c) o(k)f(patterns.)18 b(Blo)q(c)o(k)150 1747 y(matrices)10 b(t)o(ypically)f (arise)i(from)d(the)k(discretization)e(of)g(partial)g(di\013eren)o(tial)g (equations)g(in)g(whic)o(h)150 1797 y(there)17 b(are)g(sev)o(eral)g Fq(de)n(gr)n(e)n(es)f(of)h(fr)n(e)n(e)n(dom)f FC(asso)q(ciated)h(with)f(a)f (grid)h(p)q(oin)o(t.)24 b(W)m(e)16 b(then)h(partition)150 1847 y(the)d(matrix)e(in)g(small)g(blo)q(c)o(ks)h(with)g(a)g(size)h(equal)f(to)h (the)g(n)o(um)o(b)q(er)e(of)h(degrees)i(of)e(freedom,)f(and)150 1897 y(treat)j(eac)o(h)f(blo)q(c)o(k)g(as)f(a)h(dense)h(matrix,)d(ev)o(en)i (though)g(it)g(ma)o(y)e(ha)o(v)o(e)h(some)g(zeros.)212 1947 y(If)i Fy(n)280 1953 y Fx(b)312 1947 y FC(is)g(the)g(dimension)f(of)h(eac)o (h)g(blo)q(c)o(k)g(and)g Fy(nnz)r(b)g FC(is)g(the)h(n)o(um)o(b)q(er)e(of)h (nonzero)g(blo)q(c)o(ks)h(in)150 1997 y(the)i Fy(n)11 b Fu(\002)g Fy(n)17 b FC(matrix)e Fy(A)p FC(,)j(then)f(the)h(total)e(storage)i(needed)g (is)f Fy(nnz)i FC(=)e Fy(nnz)r(b)11 b Fu(\002)g Fy(n)1457 1982 y Fv(2)1457 2009 y Fx(b)1476 1997 y FC(.)27 b(The)18 b(blo)q(c)o(k)150 2047 y(dimension)12 b Fy(n)371 2053 y Fx(d)404 2047 y FC(of)i Fy(A)g FC(is)f(then)i(de\014ned)g(b)o(y)f Fy(n)859 2053 y Fx(d)889 2047 y FC(=)e Fy(n=n)1004 2053 y Fx(b)1020 2047 y FC(.)212 2097 y(Similar)d(to)i(the)g(CRS)g(format,)f(w)o(e)h(require)h(3)f(arra)o(ys)g (for)g(the)h(BCRS)f(format:)k(a)c(rectangular)150 2147 y(arra)o(y)20 b(for)f(\015oating-p)q(oin)o(t)f(n)o(um)o(b)q(ers)i(\()f FA(val\()p FC(1)i(:)g Fy(nnz)r(b)p FA(,)p FC(1)f(:)h Fy(n)1180 2153 y Fx(b)1197 2147 y FA(,)p FC(1)g(:)g Fy(n)1319 2153 y Fx(b)1335 2147 y FA(\))p FC(\))f(whic)o(h)g(stores)h(the)150 2197 y(nonzero)11 b(blo)q(c)o(ks)g(in)f(\(blo)q(c)o(k\))g(ro)o(w-wise)g(fashion,)g(an)g(in)o (teger)h(arra)o(y)f(\()p FA(col)p 1302 2197 14 2 v 16 w(ind\()p FC(1)g(:)h Fy(nnz)r(b)p FA(\))p FC(\))f(whic)o(h)150 2246 y(stores)15 b(the)g(actual)f(column)f(indices)h(in)g(the)h(original)d(matrix)g Fy(A)i FC(of)g(the)h(\(1)p Fy(;)7 b FC(1\))13 b(elemen)o(ts)h(of)g(the)150 2296 y(nonzero)h(blo)q(c)o(ks,)g(and)f(a)g(p)q(oin)o(ter)h(arra)o(y)f(\()p FA(row)p 898 2296 V 16 w(blk\()p FC(1)d(:)h Fy(n)1080 2302 y Fx(d)1109 2296 y FC(+)e(1)p FA(\))p FC(\))k(whose)i(en)o(tries)f(p)q(oin)o (t)f(to)h(the)150 2346 y(b)q(eginning)f(of)h(eac)o(h)g(blo)q(c)o(k)f(ro)o(w)h (in)g FA(val\(:,:,:\))j FC(and)d FA(col)p 1114 2346 V 15 w(ind\(:\))p FC(.)20 b(The)15 b(sa)o(vings)f(in)h(storage)150 2396 y(lo)q(cations)f(and)h (reduction)h(in)e(indirect)i(addressing)f(for)g(BCRS)g(o)o(v)o(er)g(CRS)f (can)h(b)q(e)h(signi\014can)o(t)150 2446 y(for)e(matrices)f(with)g(a)h(large) g Fy(n)636 2452 y Fx(b)652 2446 y FC(.)150 2556 y Fs(Compressed)g(Diagonal)g (Storage)g(\(CDS\))150 2633 y FC(If)j(the)h(matrix)e Fy(A)h FC(is)h(banded)f(with)h(bandwidth)f(that)g(is)h(fairly)e(constan)o(t)i(from)d (ro)o(w)j(to)f(ro)o(w,)150 2683 y(then)i(it)g(is)f(w)o(orth)o(while)g(to)h (tak)o(e)g(adv)n(an)o(tage)e(of)h(this)h(structure)i(in)d(the)h(storage)h(sc) o(heme)e(b)o(y)150 2733 y(storing)12 b(sub)q(diagonals)f(of)g(the)h(matrix)e (in)h(consecutiv)o(e)j(lo)q(cations.)i(Not)c(only)f(can)h(w)o(e)g(eliminate)p eop 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y(B)640 1244 y(@)697 1053 y FC(10)41 b Fu(\000)p FC(3)h(0)f(0)g(0)74 b(0)718 1103 y(3)g(9)41 b(6)g(0)g(0)74 b(0)718 1152 y(0)g(7)41 b(8)g(7)g(0)74 b(0)718 1202 y(0)g(0)41 b(8)g(7)g(5)74 b(0)718 1252 y(0)g(0)41 b(0)g(9)g(9)53 b(13)718 1302 y(0)74 b(0)41 b(0)g(0)g(2)h Fu(\000)p FC(1)1136 1020 y Ft(1)1136 1093 y(C)1136 1118 y(C)1136 1142 y(C)1136 1167 y(C)1136 1192 y(C)1136 1217 y(C)1136 1244 y(A)1193 1178 y Fy(:)709 b FC(\(4.2\))450 1366 y(Using)18 b(the)h(CDS)e(format,)g(w)o(e)h(store)h(this)f(matrix)f Fy(A)h FC(in)f(an)h(arra)o(y)g(of)f(dimension)g FA(\(6,-1:1\))450 1416 y FC(using)d(the)g(mapping)554 1483 y FA(val)p FC(\()p FA(i)p Fy(;)7 b FA(j)p FC(\))j(=)i Fy(a)791 1489 y Fx(i;i)p Fv(+)p Fx(j)867 1483 y Fy(:)1035 b FC(\(4.3\))450 1549 y(Hence,)15 b(the)f(ro)o(ws)g(of)g(the)g FA(val\(:,:\))j FC(arra)o(y)d(are)p 843 1580 725 2 v 842 1629 2 50 v 868 1615 a FA(val\(:,-1\))p 1088 1629 V 68 w FC(0)p 1179 1629 V 50 w(3)p 1250 1629 V 49 w(7)p 1320 1629 V 50 w(8)p 1391 1629 V 70 w(9)p 1482 1629 V 64 w(2)p 1567 1629 V 843 1631 725 2 v 842 1681 2 50 v 868 1666 a FA(val\(:,)20 b(0\))p 1088 1681 V 50 w FC(10)p 1179 1681 V 49 w(9)p 1250 1681 V 49 w(8)p 1320 1681 V 50 w(7)p 1391 1681 V 70 w(9)p 1482 1681 V 50 w(-1)p 1567 1681 V 843 1683 725 2 v 842 1732 2 50 v 868 1717 a FA(val\(:,+1\))p 1088 1732 V 54 w FC(-3)p 1179 1732 V 50 w(6)p 1250 1732 V 49 w(7)p 1320 1732 V 50 w(5)p 1391 1732 V 50 w(13)p 1482 1732 V 63 w(0)p 1567 1732 V 843 1734 725 2 v 1595 1666 a(.)450 1784 y(Notice)14 b(the)h(t)o(w)o(o)e(zeros)i(corresp)q(onding)g(to)f(non-existing)f(matrix)f (elemen)o(ts.)512 1834 y(A)i(generalization)e(of)h(the)h(CDS)f(format)f(more) g(suitable)h(for)g(manipulating)d(general)k(sparse)450 1883 y(matrices)i(on)f(v)o(ector)i(sup)q(ercomputers)h(is)e(discussed)h(b)o(y)f (Melhem)f(in)h([151)o(].)24 b(This)16 b(v)n(arian)o(t)f(of)450 1933 y(CDS)c(uses)i(a)e Fq(strip)n(e)f FC(data)h(structure)j(to)d(store)h (the)g(matrix)e Fy(A)p FC(.)17 b(This)12 b(structure)h(is)e(more)g(e\016cien) o(t)450 1983 y(in)k(storage)h(in)f(the)h(case)g(of)f(v)n(arying)f(bandwidth,) h(but)h(it)f(mak)o(es)f(the)i(matrix-v)o(ector)e(pro)q(duct)450 2033 y(sligh)o(tly)f(more)f(exp)q(ensiv)o(e,)j(as)f(it)f(in)o(v)o(olv)o(es)g (a)h(gather)g(op)q(eration.)512 2083 y(As)21 b(de\014ned)g(in)f([151)o(],)h (a)f(strip)q(e)h(in)f(the)h Fy(n)13 b Fu(\002)h Fy(n)20 b FC(matrix)f Fy(A)h FC(is)g(a)g(set)h(of)f(p)q(ositions)g Fy(S)25 b FC(=)450 2132 y Fu(f)p FC(\()p Fy(i;)7 b(\033)q FC(\()p Fy(i)p FC(\)\);)22 b Fy(i)14 b Fu(2)g Fy(I)j Fu(\022)e Fy(I)811 2138 y Fx(n)833 2132 y Fu(g)p FC(,)g(where)i Fy(I)1021 2138 y Fx(n)1057 2132 y FC(=)e Fu(f)p FC(1)p Fy(;)7 b(:)g(:)g(:)t(;)g(n)p Fu(g)14 b FC(and)h Fy(\033)i FC(is)e(a)g(strictly)h(increasing)f(function.)450 2182 y(Sp)q(eci\014cally)m(,)e(if)g(\()p Fy(i;)7 b(\033)q FC(\()p Fy(i)p FC(\)\))15 b(and)e(\()p Fy(j;)7 b(\033)q FC(\()p Fy(j)r FC(\)\))15 b(are)f(in)g Fy(S)r FC(,)g(then)554 2249 y Fy(i)e(<)f(j)j Fu(!)d Fy(\033)q FC(\()p Fy(i)p FC(\))i Fy(<)f(\033)q 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y(0)h(9)41 b(6)g(0)g Fu(\000)p FC(2)74 b(0)718 2585 y(3)g(0)41 b(8)g(7)74 b(0)f(0)718 2635 y(0)h(6)41 b(0)g(7)74 b(5)f(4)718 2684 y(0)h(0)41 b(0)g(0)74 b(9)53 b(13)718 2734 y(0)74 b(0)41 b(0)g(0)74 b(5)41 b Fu(\000)p FC(1)1168 2452 y Ft(1)1168 2525 y(C)1168 2550 y(C)1168 2575 y(C)1168 2600 y(C)1168 2625 y(C)1168 2650 y(C)1168 2676 y(A)1225 2610 y Fy(;)677 b FC(\(4.4\))p eop %%Page: 61 73 72 bop 150 275 a Fr(4.3.)31 b(D)o(A)m(T)m(A)13 b(STR)o(UCTURES)994 b FC(61)150 391 y(the)14 b(4)g(strip)q(es)h(of)f(the)g(matrix)e Fy(A)i FC(stored)h(in)e(the)i(ro)o(ws)f(of)f(the)i FA(val\(:,:\))h FC(arra)o(y)e(w)o(ould)f(b)q(e)p 536 443 739 2 v 535 492 2 50 v 561 477 a FA(val\(:,-1\))p 781 492 V 68 w FC(0)p 872 492 V 64 w(0)p 957 492 V 63 w(3)p 1041 492 V 50 w(6)p 1112 492 V 49 w(0)p 1182 492 V 71 w(5)p 1273 492 V 536 494 739 2 v 535 544 2 50 v 561 529 a FA(val\(:,)20 b(0\))p 781 544 V 50 w FC(10)p 872 544 V 63 w(9)p 957 544 V 63 w(8)p 1041 544 V 50 w(7)p 1112 544 V 49 w(9)p 1182 544 V 57 w(-1)p 1273 544 V 536 546 739 2 v 535 595 2 50 v 561 580 a FA(val\(:,+1\))p 781 595 V 68 w FC(0)p 872 595 V 50 w(-3)p 957 595 V 63 w(6)p 1041 595 V 50 w(7)p 1112 595 V 49 w(5)p 1182 595 V 50 w(13)p 1273 595 V 536 597 739 2 v 535 647 2 50 v 561 632 a FA(val\(:,+2\))p 781 647 V 68 w FC(0)p 872 647 V 64 w(1)p 957 647 V 49 w(-2)p 1041 647 V 50 w(0)p 1112 647 V 49 w(4)p 1182 647 V 71 w(0)p 1273 647 V 536 648 739 2 v 1302 555 a(.)150 783 y Fs(Jagged)15 b(Diagonal)f(Storage)g(\(JDS\))150 861 y FC(The)f(Jagged)g(Diagonal)d (Storage)j(format)e(can)i(b)q(e)g(useful)g(for)f(the)h(implemen)o(tation)c (of)j(iterativ)o(e)150 911 y(metho)q(ds)19 b(on)h(parallel)f(and)g(v)o(ector) i(pro)q(cessors)h(\(see)f(Saad)f([181)n(]\).)36 b(Lik)o(e)19 b(the)i(Compressed)150 961 y(Diagonal)15 b(format,)h(it)h(giv)o(es)g(a)g(v)o (ector)h(length)g(essen)o(tially)f(of)g(the)h(size)g(of)e(the)i(matrix.)27 b(It)17 b(is)150 1011 y(more)c(space-e\016cien)o(t)i(than)f(CDS)g(at)f(the)i 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b(The)15 b(compressed)h(and)f(p)q(erm)o(uted)g(diagonals)f (are)h(then)h(stored)g(in)e(a)h(linear)f(arra)o(y)m(.)21 b(The)150 2433 y(new)14 b(data)g(structure)i(is)e(called)f Fq(jagge)n(d)i(diagonals)p FC(.)212 2483 y(The)d(n)o(um)o(b)q(er)f(of)g(jagged)g(diagonals)f(is)h(equal) h(to)f(the)h(n)o(um)o(b)q(er)f(of)g(nonzeros)h(in)f(the)i(\014rst)f(ro)o(w,) 150 2533 y Fq(i.e.)p FC(,)h(the)h(largest)h(n)o(um)o(b)q(er)e(of)g(nonzeros)j (in)d(an)o(y)h(ro)o(w)g(of)f Fy(A)p FC(.)19 b(The)14 b(data)g(structure)i(to) e(represen)o(t)150 2583 y(the)21 b Fy(n)14 b Fu(\002)f Fy(n)21 b FC(matrix)d Fy(A)j FC(therefore)h(consists)f(of)f(a)g(p)q(erm)o(utation)f (arra)o(y)i(\()p FA(perm\(1:n\))p FC(\))d(whic)o(h)150 2633 y(reorders)f(the)e(ro)o(ws,)g(a)g(\015oating-p)q(oin)o(t)e(arra)o(y)i(\()p FA(jdiag\(:\))p FC(\))k(con)o(taining)14 b(the)h(jagged)g(diagonals)150 2683 y(in)k(succession,)j(an)d(in)o(teger)g(arra)o(y)g(\()p FA(col)p 823 2683 14 2 v 16 w(ind\(:\))p FC(\))28 b(con)o(taining)18 b(the)i(corresp)q(onding)g(column)150 2733 y(indices)c(indices,)g(and)f (\014nally)g(a)g(p)q(oin)o(ter)h(arra)o(y)f(\()p FA(jd)p 1002 2733 V 16 w(ptr\(:\))p FC(\))20 b(whose)d(elemen)o(ts)e(p)q(oin)o(t)g(to)h (the)p eop %%Page: 62 74 73 bop 450 275 a FC(62)860 b Fr(CHAPTER)14 b(4.)31 b(RELA)m(TED)14 b(ISSUES)450 391 y FC(b)q(eginning)g(of)f(eac)o(h)i(jagged)e(diagonal.)k(The) e(adv)n(an)o(tages)f(of)f(JDS)h(for)g(matrix)e(m)o(ultiplications)450 441 y(are)i(discussed)i(b)o(y)e(Saad)f(in)g([181)o(].)512 491 y(The)j(JDS)g(format)d(for)i(the)h(ab)q(o)o(v)o(e)f(matrix)f Fy(A)h FC(in)g(using)g(the)h(linear)f(arra)o(ys)h Fu(f)p FA(perm)p FC(,)e FA(jdiag)p FC(,)450 541 y FA(col)p 519 541 14 2 v 15 w(ind)p FC(,)f FA(jd)p 669 541 V 15 w(ptr)p Fu(g)g FC(is)h(giv)o(en)f(b)q (elo)o(w)h(\(jagged)f(diagonals)g(are)h(separated)h(b)o(y)f(semicolons\))p 539 581 1372 2 v 538 631 2 50 v 602 616 a FA(jdiag)p 735 631 V 48 w FC(1)p 805 631 V 50 w(3)p 876 631 V 49 w(7)p 946 631 V 50 w(8)p 1017 631 V 50 w(10)p 1108 631 V 49 w(2;)p 1190 631 V 49 w(9)p 1261 631 V 50 w(9)p 1331 631 V 49 w(8)g Fu(\001)7 b(\001)g(\001)12 b FC(-1;)p 1524 631 V 49 w(9)p 1595 631 V 49 w(6)p 1665 631 V 50 w(7)p 1736 631 V 50 w(5;)p 1818 631 V 49 w(13)p 1909 631 V 539 632 1372 2 v 538 682 2 50 v 564 667 a FA(col)p 633 667 14 2 v 15 w(ind)p 735 682 2 50 v 49 w FC(1)p 805 682 V 50 w(1)p 876 682 V 49 w(2)p 946 682 V 50 w(3)p 1017 682 V 70 w(1)p 1108 682 V 50 w(5;)p 1190 682 V 49 w(4)p 1261 682 V 50 w(2)p 1331 682 V 63 w(3)i Fu(\001)7 b(\001)g(\001)12 b FC(6;)p 1524 682 V 49 w(5)p 1595 682 V 49 w(3)p 1665 682 V 50 w(4)p 1736 682 V 50 w(5;)p 1818 682 V 69 w(6)p 1909 682 V 539 684 1372 2 v 669 696 561 2 v 668 746 2 50 v 694 731 a FA(perm)p 805 746 V 49 w FC(5)p 876 746 V 49 w(2)p 946 746 V 50 w(3)p 1017 746 V 49 w(4)p 1087 746 V 50 w(1)p 1158 746 V 50 w(6)p 1228 746 V 669 747 561 2 v 1243 696 499 2 v 1242 746 2 50 v 63 w FA(jd)p 1315 731 14 2 v 15 w(ptr)p 1417 746 2 50 v 49 w FC(1)p 1487 746 V 50 w(7)p 1558 746 V 50 w(13)p 1649 746 V 49 w(17)p 1740 746 V 1243 747 499 2 v 52 w(.)450 867 y Fs(Skyline)i(Storage)g(\(SKS\))450 944 y FC(The)19 b(\014nal)g(storage) g(sc)o(heme)g(w)o(e)g(consider)h(is)f(for)f(skyline)h(matrices,)g(whic)o(h)g (are)g(also)f(called)450 993 y(v)n(ariable)e(band)h(or)f(pro\014le)h (matrices)g(\(see)h(Du\013,)f(Erisman)f(and)h(Reid)f([77)o(]\).)27 b(It)17 b(is)g(mostly)e(of)450 1043 y(imp)q(ortance)g(in)g(direct)i(solution) e(metho)q(ds,)h(but)g(it)f(can)h(b)q(e)h(used)f(for)g(handling)f(the)h (diagonal)450 1093 y(blo)q(c)o(ks)i(in)g(blo)q(c)o(k)g(matrix)e (factorization)h(metho)q(ds.)30 b(A)19 b(ma)r(jor)d(adv)n(an)o(tage)h(of)h (solving)f(linear)450 1143 y(systems)d(ha)o(ving)e(skyline)h(co)q(e\016cien)o (t)i(matrices)e(is)g(that)h(when)g(piv)o(oting)e(is)h(not)h(necessary)m(,)h (the)450 1193 y(skyline)c(structure)j(is)d(preserv)o(ed)i(during)e(Gaussian)g (elimination.)j(If)d(the)h(matrix)d(is)i(symmetric,)450 1243 y(w)o(e)19 b(only)f(store)i(its)e(lo)o(w)o(er)h(triangular)e(part.)33 b(A)19 b(straigh)o(tforw)o(ard)f(approac)o(h)h(in)f(storing)h(the)450 1292 y(elemen)o(ts)e(of)f(a)h(skyline)g(matrix)e(is)i(to)g(place)g(all)f(the) i(ro)o(ws)f(\(in)g(order\))h(in)o(to)e(a)h(\015oating-p)q(oin)o(t)450 1342 y(arra)o(y)12 b(\()p FA(val\(:\))p FC(\),)f(and)g(then)i(k)o(eep)g(an)f (in)o(teger)g(arra)o(y)g(\()p FA(row)p 1391 1342 14 2 v 15 w(ptr\(:\))p FC(\))k(whose)d(elemen)o(ts)f(p)q(oin)o(t)f(to)450 1392 y(the)k(b)q(eginning)g(of)f(eac)o(h)h(ro)o(w.)21 b(The)15 b(column)f(indices)h(of)f(the)h(nonzeros)h(stored)g(in)f FA(val\(:\))k FC(are)450 1442 y(easily)13 b(deriv)o(ed)i(and)f(are)g(not)g(stored.)512 1492 y(F)m(or)9 b(a)g(nonsymmetric)f(skyline)h(matrix)e(suc)o(h)j(as)g(the)g (one)f(illustrated)h(in)e(Figure)i(4.1,)f(w)o(e)g(store)450 1541 y(the)14 b(lo)o(w)o(er)f(triangular)g(elemen)o(ts)g(in)g(SKS)h(format,)d (and)j(store)g(the)g(upp)q(er)h(triangular)d(elemen)o(ts)450 1591 y(in)j(a)g(column-orien)o(ted)g(SKS)h(format)d(\(transp)q(ose)k(stored)g (in)e(ro)o(w-wise)h(SKS)g(format\).)21 b(These)450 1641 y(t)o(w)o(o)9 b(separated)i Fq(substructur)n(es)f FC(can)g(b)q(e)g(link)o(ed)f(in)h(a)f(v)n (ariet)o(y)g(of)h(w)o(a)o(ys.)16 b(One)10 b(approac)o(h,)h(discussed)450 1691 y(b)o(y)17 b(Saad)f(in)g([182)o(],)g(is)h(to)g(store)g(eac)o(h)h(ro)o(w) e(of)g(the)i(lo)o(w)o(er)e(triangular)g(part)h(and)g(eac)o(h)g(column)450 1741 y(of)e(the)g(upp)q(er)i(triangular)d(part)h(con)o(tiguously)g(in)o(to)f (the)i(\015oating-p)q(oin)o(t)e(arra)o(y)h(\()p FA(val\(:\))p FC(\).)21 b(An)450 1790 y(additional)16 b(p)q(oin)o(ter)j(is)e(then)i(needed) h(to)e(determine)g(where)h(the)g(diagonal)d(elemen)o(ts,)i(whic)o(h)450 1840 y(separate)12 b(the)f(lo)o(w)o(er)g(triangular)f(elemen)o(ts)g(from)f (the)j(upp)q(er)g(triangular)d(elemen)o(ts,)i(are)g(lo)q(cated.)450 2004 y Fl(4.3.2)55 b(Matrix)18 b(v)n(ector)g(pro)r(ducts)450 2081 y FC(In)c(man)o(y)f(of)g(the)i(iterativ)o(e)g(metho)q(ds)e(discussed)j (earlier,)f(b)q(oth)f(the)h(pro)q(duct)g(of)f(a)g(matrix)e(and)450 2131 y(that)17 b(of)f(its)h(transp)q(ose)h(times)e(a)h(v)o(ector)g(are)h (needed,)g(that)f(is,)g(giv)o(en)g(an)f(input)h(v)o(ector)g Fy(x)g FC(w)o(e)450 2181 y(w)o(an)o(t)c(to)h(compute)g(pro)q(ducts)554 2254 y Fy(y)f FC(=)f Fy(Ax)83 b FC(and)g Fy(y)13 b FC(=)f Fy(A)1027 2237 y Fx(T)1053 2254 y Fy(x:)450 2328 y FC(W)m(e)j(will)f(presen)o(t)j (these)g(algorithms)c(for)i(t)o(w)o(o)g(of)g(the)h(storage)g(formats)e(from)g Fu(x)p FC(4.3:)20 b(CRS)15 b(and)450 2378 y(CDS.)450 2484 y Fs(CRS)h(Matrix-V)l(ector)e(Pro)q(duct)450 2561 y FC(The)i(matrix)e(v)o (ector)i(pro)q(duct)h Fy(y)f FC(=)f Fy(Ax)g FC(using)g(CRS)g(format)f(can)i (b)q(e)g(expressed)i(in)d(the)h(usual)450 2611 y(w)o(a)o(y:)554 2690 y Fy(y)574 2696 y Fx(i)599 2690 y FC(=)643 2651 y Ft(X)666 2739 y Fx(j)710 2690 y Fy(a)732 2696 y Fx(i;j)771 2690 y Fy(x)795 2696 y Fx(j)812 2690 y Fy(;)p eop 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a(+)56 b(x)k(x)p 1545 1099 V 149 1148 V 1165 1134 a(x)g(x)g(x)55 b(+)h(x)p 1545 1148 V 149 1198 V 1489 1183 a(+)p 1545 1198 V 150 1200 1396 2 v 285 1274 a(Figure)14 b(4.1:)j(Pro\014le)d(of)g(a)f(nonsymmetric)f(skyline)i(or)g (v)n(ariable-band)e(matrix.)150 1372 y(since)g(this)g(tra)o(v)o(erses)h(the)f (ro)o(ws)g(of)f(the)h(matrix)d Fy(A)p FC(.)18 b(F)m(or)11 b(an)g Fy(n)t Fu(\002)t Fy(n)i FC(matrix)c(A,)i(the)h(matrix-v)o(ector)150 1421 y(m)o(ultiplicatio)o(n)f(is)j(giv)o(en)f(b)o(y)150 1502 y FA(for)21 b(i)h(=)f(1,)h(n)237 1552 y(y\(i\))43 b(=)22 b(0)237 1602 y(for)f(j)h(=)g(row_ptr\(i\),)d(row_ptr\(i+1\))g(-)j(1)324 1652 y(y\(i\))f(=)h(y\(i\))f(+)g(val\(j\))g(*)h(x\(col_ind\(j\)\))237 1702 y(end;)150 1751 y(end;)150 1832 y FC(Since)10 b(this)g(metho)q(d)f(only) g(m)o(ultiplies)f(nonzero)i(matrix)e(en)o(tries,)k(the)e(op)q(eration)g(coun) o(t)g(is)g(2)f(times)150 1882 y(the)18 b(n)o(um)o(b)q(er)f(of)g(nonzero)i (elemen)o(ts)e(in)h Fy(A)p FC(,)g(whic)o(h)f(is)h(a)f(signi\014can)o(t)h(sa)o (vings)f(o)o(v)o(er)h(the)g(dense)150 1932 y(op)q(eration)c(requiremen)o(t)g (of)f(2)p Fy(n)660 1917 y Fv(2)678 1932 y FC(.)212 1982 y(F)m(or)h(the)g (transp)q(ose)h(pro)q(duct)g Fy(y)f FC(=)d Fy(A)808 1967 y Fx(T)835 1982 y Fy(x)i FC(w)o(e)h(cannot)g(use)h(the)g(equation)254 2068 y Fy(y)274 2074 y Fx(i)299 2068 y FC(=)343 2029 y Ft(X)366 2117 y Fx(j)403 2068 y FC(\()p Fy(A)450 2051 y Fx(T)477 2068 y FC(\))493 2074 y Fx(i;j)532 2068 y Fy(x)556 2074 y Fx(j)585 2068 y FC(=)628 2029 y Ft(X)651 2117 y Fx(j)695 2068 y Fy(a)717 2074 y Fx(j;i)755 2068 y Fy(x)779 2074 y Fx(j)796 2068 y Fy(;)150 2191 y FC(since)f(this)g(implies)e(tra)o(v)o(ersing)h(columns)g(of)g(the)h (matrix,)d(an)i(extremely)h(ine\016cien)o(t)f(op)q(eration)150 2241 y(for)h(matrices)f(stored)i(in)e(CRS)h(format.)i(Hence,)f(w)o(e)f(switc) o(h)g(indices)g(to)254 2322 y(for)f(all)g Fy(j)r FC(,)h(do)f(for)h(all)f Fy(i)p FC(:)83 b Fy(y)728 2328 y Fx(i)753 2322 y Fu( )11 b Fy(y)826 2328 y Fx(i)850 2322 y FC(+)e Fy(a)913 2328 y Fx(j;i)950 2322 y Fy(x)974 2328 y Fx(j)992 2322 y Fy(:)150 2403 y FC(The)14 b(matrix-v)o(ector)f(m)o(ultiplicatio)o(n)e(in)o(v)o(olving)h Fy(A)970 2388 y Fx(T)1010 2403 y FC(is)i(then)g(giv)o(en)g(b)o(y)150 2483 y FA(for)21 b(i)h(=)f(1,)h(n)237 2533 y(y\(i\))f(=)44 b(0)150 2583 y(end;)150 2633 y(for)21 b(j)h(=)f(1,)h(n)237 2683 y(for)f(i)h(=)g(row_ptr\(j\),)d(row_ptr\(j+1\)-1)324 2733 y(y\(col_ind\(i\)\))g(=)j(y\(col_ind\(i\)\))d(+)j(val\(i\))e(*)i(x\(j\))p eop %%Page: 64 76 75 bop 450 275 a FC(64)864 b Fr(CHAPTER)15 b(4.)26 b(RELA)m(TED)14 b(ISSUES)537 391 y FA(end;)450 441 y(end;)512 528 y FC(Both)h(matrix-v)o (ector)d(pro)q(ducts)k(ab)q(o)o(v)o(e)e(ha)o(v)o(e)f(largely)h(the)g(same)g (structure,)h(and)f(b)q(oth)g(use)450 578 y(indirect)h(addressing.)k(Hence,)c (their)f(v)o(ectorizabilit)o(y)f(prop)q(erties)j(are)e(the)h(same)e(on)h(an)o (y)g(giv)o(en)450 628 y(computer.)21 b(Ho)o(w)o(ev)o(er,)15 b(the)h(\014rst)g(pro)q(duct)g(\()p Fy(y)f FC(=)f Fy(Ax)p FC(\))h(has)g(a)g (more)f(fa)o(v)o(orable)g(memory)e(access)450 678 y(pattern)17 b(in)g(that)f(\(p)q(er)i(iteration)e(of)g(the)h(outer)g(lo)q(op\))f(it)g (reads)i(t)o(w)o(o)e(v)o(ectors)i(of)e(data)g(\(a)g(ro)o(w)450 728 y(of)h(matrix)e Fy(A)i FC(and)g(the)h(input)f(v)o(ector)h Fy(x)p FC(\))f(and)g(writes)h(one)g(scalar.)28 b(The)18 b(transp)q(ose)g(pro) q(duct)450 778 y(\()p Fy(y)24 b FC(=)e Fy(A)595 762 y Fx(T)622 778 y Fy(x)p FC(\))e(on)f(the)i(other)g(hand)f(reads)h(one)f(elemen)o(t)g(of) g(the)g(input)g(v)o(ector,)i(one)f(ro)o(w)f(of)450 827 y(matrix)10 b Fy(A)p FC(,)i(and)g(b)q(oth)h(reads)g(and)f(writes)h(the)f(result)h(v)o (ector)g Fy(y)q FC(.)19 b(Unless)13 b(the)f(mac)o(hine)f(on)h(whic)o(h)450 877 y(these)j(metho)q(ds)e(are)h(implemen)o(ted)e(has)h(three)i(separate)g (memory)c(paths)j(\()p Fq(e.g.)p FC(,)f(Cra)o(y)g(Y-MP\),)450 927 y(the)f(memory)c(tra\016c)k(will)d(then)j(limit)d(the)j(p)q(erformance.) 17 b(This)11 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y Fy(x)1293 1416 y Fx(i)p Fv(+)p Fx(j)1361 1410 y Fy(:)450 1497 y FC(With)15 b(the)h(index)f Fy(i)h FC(in)f(the)h(inner)f(lo) q(op)g(w)o(e)g(see)i(that)e(the)h(expression)h Fy(a)1617 1503 y Fx(i;i)p Fv(+)p Fx(j)1709 1497 y FC(accesses)h(the)e Fy(j)r FC(th)450 1547 y(diagonal)c(of)h(the)i(matrix)d(\(where)j(the)g(main)c (diagonal)h(has)i(n)o(um)o(b)q(er)g(0\).)512 1598 y(The)e(algorithm)d(will)g (no)o(w)i(ha)o(v)o(e)g(a)g(doubly-nested)h(lo)q(op)f(with)g(the)h(outer)g(lo) q(op)e(en)o(umerating)450 1648 y(the)15 b(diagonals)d FA(diag=-p,q)g FC(with)i Fy(p)g FC(and)g Fy(q)h FC(the)f(\(nonnegativ)o(e\))g(n)o(um)o(b)q (ers)g(of)f(diagonals)g(to)h(the)450 1698 y(left)j(and)g(righ)o(t)f(of)h(the) g(main)e(diagonal.)26 b(The)17 b(b)q(ounds)h(for)e(the)i(inner)f(lo)q(op)f (follo)o(w)f(from)h(the)450 1748 y(requiremen)o(t)e(that)554 1836 y(1)d Fu(\024)h FA(i)p Fy(;)7 b FA(i)h FC(+)i FA(j)h Fu(\024)h Fy(n:)512 1924 y FC(The)j(algorithm)c(b)q(ecomes)450 2013 y FA(for)21 b(i)h(=)f(1,)h(n)537 2062 y(y\(i\))f(=)h(0)450 2112 y(end;)450 2162 y(for)f(diag)g(=)h(-diag_left,)d(diag_right)537 2212 y(for)i(loc)h(=)f(max\(1,1-diag\),)e(min\(n,n-diag\))624 2262 y(y\(loc\))i(=)g(y\(loc\))g(+)h(val\(loc,diag\))d(*)i(x\(loc+diag\))537 2312 y(end;)450 2361 y(end;)512 2449 y FC(The)11 b(transp)q(ose)h(matrix-v)o (ector)d(pro)q(duct)j Fy(y)h FC(=)f Fy(A)1294 2434 y Fx(T)1320 2449 y Fy(x)f FC(is)f(a)g(minor)f(v)n(ariation)g(of)h(the)h(algorithm)450 2498 y(ab)q(o)o(v)o(e.)18 b(Using)13 b(the)i(up)q(date)f(form)o(ula)554 2584 y Fy(y)574 2590 y Fx(i)629 2584 y Fu( )41 b Fy(y)732 2590 y Fx(i)756 2584 y FC(+)9 b Fy(a)819 2590 y Fx(i)p Fv(+)p Fx(j;i)894 2584 y Fy(x)918 2590 y Fx(j)634 2647 y FC(=)46 b Fy(y)732 2653 y Fx(i)756 2647 y FC(+)9 b Fy(a)819 2653 y Fx(i)p Fv(+)p Fx(j;i)p Fv(+)p Fx(j)r Fw(\000)p Fx(j)976 2647 y Fy(x)1000 2653 y Fx(i)p Fv(+)p Fx(j)450 2733 y FC(w)o(e)14 b(obtain)p eop %%Page: 65 77 76 bop 150 275 a Fr(4.3.)31 b(D)o(A)m(T)m(A)13 b(STR)o(UCTURES)994 b FC(65)150 391 y FA(for)21 b(i)h(=)f(1,)h(n)237 441 y(y\(i\))f(=)h(0)150 491 y(end;)150 541 y(for)f(diag)g(=)h(-diag_right,)d(diag_left)237 591 y(for)i(loc)h(=)f(max\(1,1-diag\),)e(min\(n,n-diag\))324 640 y(y\(loc\))i(=)g(y\(loc\))g(+)h(val\(loc+diag,)d(-diag\))h(*)i (x\(loc+diag\))237 690 y(end;)150 740 y(end;)150 819 y FC(The)13 b(memory)d(access)15 b(for)d(the)i(CDS-based)f(matrix-v)o(ector)e(pro)q(duct) j Fy(y)f FC(=)f Fy(Ax)g FC(\(or)h Fy(y)h FC(=)e Fy(A)1593 804 y Fx(T)1619 819 y Fy(x)p FC(\))g(is)150 868 y(three)k(v)o(ectors)g(p)q(er)g (inner)f(iteration.)21 b(On)15 b(the)g(other)h(hand,)e(there)i(is)f(no)g (indirect)g(addressing,)150 918 y(and)20 b(the)g(algorithm)d(is)j(v)o (ectorizable)h(with)e(v)o(ector)i(lengths)f(of)f(essen)o(tially)h(the)g (matrix)e(or-)150 968 y(der)d Fy(n)p FC(.)k(Because)d(of)e(the)h(regular)f (data)g(access,)h(most)e(mac)o(hines)h(can)g(p)q(erform)g(this)g(algorithm) 150 1018 y(e\016cien)o(tly)g(b)o(y)f(k)o(eeping)h(three)i(base)e(registers)h 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FC(in)k(diagonal,)e(lo)o(w)o(er)h(and)h(upp)q(er)h(triangular)e(part,)h (then)h(the)g(factorization)e(will)f(b)q(e)j(of)e(the)i(form)150 1509 y(\()p Fy(D)200 1515 y Fx(A)238 1509 y FC(+)f Fy(L)310 1515 y Fx(A)337 1509 y FC(\))p Fy(D)388 1491 y Fw(\000)p Fv(1)387 1521 y Fx(A)433 1509 y FC(\()p Fy(D)483 1515 y Fx(A)522 1509 y FC(+)f Fy(U)593 1515 y Fx(A)620 1509 y FC(\).)26 b(In)16 b(this)g(w)o(a)o(y)m(,)g(w)o(e)g(only)g(need)h(to)f(store)i(a)e(diagonal)e (matrix)h Fy(D)150 1558 y FC(con)o(taining)g(the)i(piv)o(ots)f(of)f(the)i (factorization,)f(and)g(a)g(preconditioner)h(solv)o(e)f(co)q(de)h(is)f (needed)150 1608 y(for)e(b)q(oth)254 1687 y Fy(LU)5 b(y)13 b FC(=)f Fy(x)83 b FC(and)g(\()p Fy(LU)5 b FC(\))742 1670 y Fx(T)768 1687 y Fy(y)14 b FC(=)d Fy(x:)150 1765 y FC(The)18 b(transp)q(ose)h(preconditioner)f(solv)o(e)f(is)g(for)g(an)h(iterativ)o(e)f (metho)q(d)g(in)o(v)o(olving)e(a)i(transp)q(ose)150 1815 y(matrix-v)o(ector)c (pro)q(duct.)150 1922 y 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FC(\)\))e(=)h Fy(a)691 2304 y Fx(i;i)726 2298 y FC(.)212 2347 y(The)j(factorization)e(b)q(egins)h(b)o(y)g(cop)o(ying)f (the)h(matrix)e(diagonal)150 2426 y FA(for)21 b(i)h(=)f(1,)h(n)237 2476 y(pivots\(i\))e(=)i(val\(diag_ptr\(i\)\))150 2526 y(end;)150 2604 y FC(Eac)o(h)14 b(elimination)d(step)k(starts)g(b)o(y)e(in)o(v)o(erting) h(the)g(piv)o(ot)150 2683 y FA(for)21 b(i)h(=)f(1,)h(n)237 2733 y(pivots\(i\))e(=)i(1)f(/)h(pivots\(i\))p eop %%Page: 66 78 77 bop 450 275 a FC(66)864 b Fr(CHAPTER)15 b(4.)26 b(RELA)m(TED)14 b(ISSUES)450 391 y FC(F)m(or)d(all)e(nonzero)j(elemen)o(ts)f Fy(a)917 397 y Fx(i;j)967 391 y FC(with)g Fy(j)j(>)e(i)p FC(,)f(w)o(e)g(next) h(c)o(hec)o(k)g(whether)g Fy(a)1607 397 y Fx(j;i)1656 391 y FC(is)f(a)f(nonzero)i(matrix)450 441 y(elemen)o(t,)h(since)i(this)f(is)f(the) i(only)e(elemen)o(t)g(that)h(can)g(cause)h(\014ll)e(with)h Fy(a)1609 447 y Fx(i;j)1648 441 y FC(.)537 525 y FA(for)21 b(j)h(=)g(diag_ptr\(i\)+1,)c(row_ptr\(i+1\)-1)624 575 y(found)j(=)h(FALSE)624 625 y(for)g(k)f(=)h(row_ptr\(col_ind\()o(j\)\),)c(diag_ptr\(col_ind\(j\))o (\)-1)712 675 y(if\(col_ind\(k\))h(=)i(i\))h(then)799 725 y(found)e(=)i(TRUE) 799 774 y(element)e(=)i(val\(k\))712 824 y(endif)624 874 y(end;)450 958 y FC(If)14 b(so,)f(w)o(e)h(up)q(date)h Fy(a)777 964 y Fx(j;j)818 958 y FC(.)624 1042 y FA(if)22 b(\(found)e(=)i(TRUE\))712 1092 y(val\(diag_ptr\(co)o(l_ind)o(\(j\)\)\))c(=)k(val\(diag_ptr\(col)o(_ind\()o (j\)\)\))1213 1142 y(-)f(element)g(*)g(pivots\(i\))f(*)i(val\(j\))537 1192 y(end;)450 1242 y(end;)450 1351 y Fs(CRS-based)15 b(F)l(actorization)e (Solv)o(e)450 1428 y FC(The)i(system)f Fy(LU)5 b(y)15 b FC(=)e Fy(x)h FC(can)h(b)q(e)g(solv)o(ed)g(in)f(the)h(usual)f(manner)g(b)o(y)g(in)o (tro)q(ducing)h(a)f(temp)q(orary)450 1478 y(v)o(ector)h Fy(z)r FC(:)554 1562 y Fy(Lz)f FC(=)e Fy(x;)89 b(U)5 b(y)13 b FC(=)f Fy(z)r(:)450 1646 y FC(W)m(e)h(ha)o(v)o(e)h(a)g(c)o(hoice)g(b)q(et)o(w)o(een) h(sev)o(eral)g(equiv)n(alen)o(t)e(w)o(a)o(ys)h(of)f(solving)g(the)h(system:) 554 1730 y Fy(LU)46 b FC(=)c(\()p Fy(D)11 b FC(+)f Fy(L)861 1736 y Fx(A)888 1730 y FC(\))p Fy(D)939 1713 y Fw(\000)p Fv(1)984 1730 y FC(\()p Fy(D)h FC(+)f Fy(U)1115 1736 y Fx(A)1142 1730 y FC(\))656 1792 y(=)42 b(\()p Fy(I)13 b FC(+)d Fy(L)847 1798 y Fx(A)874 1792 y Fy(D)909 1775 y Fw(\000)p Fv(1)954 1792 y FC(\)\()p Fy(D)h FC(+)f Fy(U)1101 1798 y Fx(A)1128 1792 y FC(\))656 1854 y(=)42 b(\()p Fy(D)11 b FC(+)f Fy(L)861 1860 y Fx(A)888 1854 y FC(\)\()p Fy(I)j FC(+)c Fy(D)1027 1837 y Fw(\000)p Fv(1)1073 1854 y Fy(U)1101 1860 y Fx(A)1128 1854 y FC(\))656 1917 y(=)42 b(\()p Fy(I)13 b FC(+)d Fy(L)847 1923 y Fx(A)874 1917 y Fy(D)909 1900 y Fw(\000)p Fv(1)954 1917 y FC(\))p Fy(D)q FC(\()p Fy(I)k FC(+)9 b Fy(D)1129 1900 y Fw(\000)p Fv(1)1174 1917 y Fy(U)1202 1923 y Fx(A)1229 1917 y FC(\))450 2000 y(The)16 b(\014rst)g(and)f(fourth)g (form)o(ulae)f(are)h(not)h(suitable)f(since)h(they)g(require)g(b)q(oth)g(m)o (ultiplicati)o(on)450 2050 y(and)f(division)f(with)h Fy(D)q FC(;)h(the)g(di\013erence)h(b)q(et)o(w)o(een)g(the)f(second)g(and)f(third)g (is)h(only)e(one)i(of)e(ease)450 2100 y(of)f(co)q(ding.)k(In)c(this)h (section)f(w)o(e)h(use)g(the)g(third)f(form)o(ula;)d(in)j(the)h(next)g (section)f(w)o(e)h(will)e(use)i(the)450 2150 y(second)h(for)f(the)g(transp)q (ose)h(system)f(solution.)512 2200 y(Both)j(halv)o(es)f(of)g(the)g(solution)g (ha)o(v)o(e)g(largely)f(the)i(same)e(structure)k(as)d(the)h(matrix)d(v)o (ector)450 2250 y(m)o(ultiplicatio)o(n.)450 2334 y FA(for)21 b(i)h(=)f(1,)h(n)537 2384 y(sum)f(=)44 b(0)537 2434 y(for)21 b(j)h(=)g(row_ptr\(i\),)d(diag_ptr\(i\)-1)624 2483 y(sum)j(=)f(sum)g(+)h (val\(j\))e(*)i(z\(col_ind\(j\)\))537 2533 y(end;)537 2583 y(z\(i\))f(=)h(pivots\(i\))e(*)h(\(x\(i\)-sum\))450 2633 y(end;)450 2683 y(for)g(i)h(=)f(n,)h(1,)f(\(step)g(-1\))537 2733 y(sum)g(=)h(0)p eop %%Page: 67 79 78 bop 150 275 a Fr(4.3.)31 b(D)o(A)m(T)m(A)13 b(STR)o(UCTURES)994 b FC(67)237 391 y FA(for)21 b(j)h(=)g(diag\(i\)+1,)d(row_ptr\(i+1\)-1)324 441 y(sum)j(=)f(sum)g(+)h(val\(j\))e(*)i(y\(col_ind\(j\)\))324 491 y(y\(i\))f(=)h(z\(i\))f(-)g(pivots\(i\))f(*)i(sum)237 541 y(end;)150 591 y(end;)150 699 y FC(The)14 b(temp)q(orary)e(v)o(ector)i FA(z)f FC(can)h(b)q(e)f(eliminated)f(b)o(y)h(reusing)g(the)h(space)g(for)f FA(y)p FC(;)g(algorithmicall)o(y)m(,)150 749 y FA(z)h FC(can)g(ev)o(en)g(o)o (v)o(erwrite)h FA(x)p FC(,)e(but)h(o)o(v)o(erwriting)f(input)h(data)f(is)h (in)f(general)h(not)g(recommended.)150 902 y Fs(CRS-based)h(F)l(actorization) e(T)l(ransp)q(ose)i(Solv)o(e)150 995 y FC(Solving)j(the)h(transp)q(ose)i (system)e(\()p Fy(LU)5 b FC(\))806 980 y Fx(T)832 995 y Fy(y)22 b FC(=)f Fy(x)e FC(is)f(sligh)o(tly)g(more)g(in)o(v)o(olv)o(ed.)33 b(In)19 b(the)g(usual)150 1045 y(form)o(ulation)14 b(w)o(e)j(tra)o(v)o(erse)h (ro)o(ws)f(when)g(solving)f(a)h(factored)g(system,)g(but)g(here)h(w)o(e)f (can)g(only)150 1095 y(access)h(columns)e(of)g(the)h(matrices)f Fy(L)761 1080 y Fx(T)804 1095 y FC(and)g Fy(U)920 1080 y Fx(T)963 1095 y FC(\(at)h(less)g(than)f(prohibitiv)o(e)g(cost\).)27 b(The)17 b(k)o(ey)150 1144 y(idea)11 b(is)f(to)h(distribute)h(eac)o(h)f (newly)g(computed)f(comp)q(onen)o(t)g(of)g(a)h(triangular)f(solv)o(e)h(imm)o (ediately)150 1194 y(o)o(v)o(er)j(the)g(remaining)e(righ)o(t-hand-side.)212 1253 y(F)m(or)j(instance,)h(if)e(w)o(e)i(write)f(a)g(lo)o(w)o(er)g (triangular)f(matrix)g(as)h Fy(L)f FC(=)g(\()p Fy(l)1318 1259 y Fw(\003)p Fv(1)1354 1253 y Fy(;)7 b(l)1385 1259 y Fw(\003)p Fv(2)1421 1253 y Fy(;)g(:)g(:)g(:)13 b FC(\),)i(then)h(the)150 1302 y(system)f Fy(Ly)h FC(=)e Fy(x)h FC(can)g(b)q(e)h(written)g(as)f Fy(x)f FC(=)g Fy(l)868 1308 y Fw(\003)p Fv(1)904 1302 y Fy(y)924 1308 y Fv(1)953 1302 y FC(+)d Fy(l)1008 1308 y Fw(\003)p Fv(2)1044 1302 y Fy(y)1064 1308 y Fv(2)1093 1302 y FC(+)f Fu(\001)d(\001)g(\001)n FC(.)23 b(Hence,)16 b(after)g(computing)d Fy(y)1680 1308 y Fv(1)150 1352 y FC(w)o(e)20 b(mo)q(dify)d Fy(x)22 b Fu( )e Fy(x)13 b Fu(\000)h Fy(l)565 1358 y Fw(\003)p Fv(1)601 1352 y Fy(y)621 1358 y Fv(1)640 1352 y FC(,)20 b(and)g(so)g(on.)35 b(Upp)q(er)21 b(triangular)e(systems)h(are)g(treated)h(in)e(a)150 1402 y(similar)9 b(manner.)16 b(With)10 b(this)h(algorithm)e(w)o(e)i(only)f (access)j(columns)d(of)h(the)g(triangular)g(systems.)150 1452 y(Solving)f(a)i(transp)q(ose)h(system)e(with)h(a)f(matrix)f(stored)j(in)e (CRS)h(format)e(essen)o(tially)h(means)g(that)150 1502 y(w)o(e)j(access)i(ro) o(ws)e(of)f Fy(L)h FC(and)g Fy(U)5 b FC(.)212 1560 y(The)15 b(algorithm)c(no)o(w)i(b)q(ecomes)150 1677 y FA(for)21 b(i)h(=)f(1,)h(n)237 1727 y(x_tmp\(i\))e(=)i(x\(i\))150 1777 y(end;)150 1827 y(for)f(i)h(=)f(1,)h (n)237 1877 y(z\(i\))f(=)h(x_tmp\(i\))237 1926 y(tmp)f(=)h(pivots\(i\))e(*)h (z\(i\))237 1976 y(for)g(j)h(=)g(diag_ptr\(i\)+1,)c(row_ptr\(i+1\)-1)324 2026 y(x_tmp\(col_ind\(j\)\))h(=)i(x_tmp\(col_ind\(j\)\))d(-)k(tmp)f(*)h (val\(j\))237 2076 y(end;)150 2126 y(end;)150 2225 y(for)f(i)h(=)f(n,)h(1)f (\(step)g(-1\))237 2275 y(y\(i\))g(=)h(pivots\(i\))e(*)h(z\(i\))237 2325 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Fv(\()p Fx(i)p Fv(\))765 2733 y FC(,)g(after)h(whic)o(h)f (the)h(inner)g(pro)q(duct)g Fy(r)1346 2717 y Fv(\()p Fx(i)p Fv(\))1384 2705 y Fj(T)1409 2733 y Fy(M)1454 2717 y Fw(\000)p Fv(1)1498 2733 y Fy(r)1518 2717 y Fv(\()p Fx(i)p Fv(\))1569 2733 y FC(can)g(b)q(e)g(computed)f(as)g Fy(s)1953 2717 y Fx(T)1980 2733 y Fy(s)p eop %%Page: 69 81 80 bop 150 275 a Fr(4.4.)31 b(P)m(ARALLELISM)1111 b FC(69)150 391 y(where)12 b Fy(s)g FC(=)g Fy(L)370 376 y Fw(\000)p Fv(1)415 391 y Fy(r)435 376 y Fv(\()p Fx(i)p Fv(\))474 391 y FC(.)17 b(The)12 b(computation)d(of)h Fy(L)897 376 y Fw(\000)p Fx(T)950 391 y Fy(s)h FC(will)f(then)h(mask)f(the)h(comm)o(unication)d(stage)150 441 y(of)13 b(the)i(inner)f(pro)q(duct.)p 325 547 1200 2 v 325 1675 2 1129 v 572 638 a Fy(x)596 622 y Fv(\()p Fw(\000)p Fv(1\))678 638 y FC(=)e Fy(x)746 622 y Fv(\(0\))790 638 y FC(=)i(initial)e (guess;)j Fy(r)1097 622 y Fv(\(0\))1153 638 y FC(=)d Fy(b)d Fu(\000)g Fy(Ax)1320 622 y Fv(\(0\))1364 638 y FC(;)572 689 y Fy(p)593 674 y 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y(previous)g(iteration)f(step\).)201 2241 y(2.)20 b(The)10 b(reduction)h(of)f(the)h(inner)f(pro)q(duct)h(for)f Fy(\032)963 2247 y Fx(i)p Fv(+1)1029 2241 y FC(can)h(b)q(e)f(o)o(v)o(erlapp)q(ed)h(with)e (the)i(remaining)254 2291 y(part)j(of)f(the)i(preconditioning)e(op)q(eration) h(at)f(the)i(b)q(eginning)e(of)g(the)i(next)f(iteration.)201 2383 y(3.)20 b(The)15 b(computation)d(of)i(a)h(segmen)o(t)f(of)g Fy(p)896 2368 y Fv(\()p Fx(i)p Fv(\))950 2383 y FC(can)g(b)q(e)h(follo)o(w)o (ed)e(immediately)e(b)o(y)k(the)g(com-)254 2433 y(putation)h(of)g(a)h(segmen) o(t)g(of)f Fy(q)747 2418 y Fv(\()p Fx(i)p Fv(\))787 2433 y FC(,)h(and)g(this)g(can)g(b)q(e)g(follo)o(w)o(ed)e(b)o(y)i(the)h(computation) d(of)254 2483 y(a)g(part)h(of)f(the)h(inner)f(pro)q(duct.)24 b(This)16 b(sa)o(v)o(es)g(on)f(load)f(op)q(erations)i(for)f(segmen)o(ts)h(of) f Fy(p)1660 2468 y Fv(\()p Fx(i)p Fv(\))254 2533 y FC(and)e Fy(q)354 2517 y Fv(\()p Fx(i)p Fv(\))394 2533 y FC(.)150 2633 y(F)m(or)g(a)g(more)g(detailed)g(discussion)h(see)h(Demmel,)10 b(Heath)k(and)f(V)m(an)g(der)h(V)m(orst)g([64)o(].)j(This)d(algo-)150 2683 y(rithm)f(can)h(b)q(e)g(extended)i(trivially)c(to)i(preconditioners)h (of)e Fy(LD)q(L)1212 2668 y Fx(T)1253 2683 y FC(form,)f(and)i(nonsymmetric) 150 2733 y(preconditioners)h(in)e(the)i(Biconjugate)f(Gradien)o(t)f(Metho)q (d.)p eop %%Page: 70 82 81 bop 450 275 a FC(70)860 b Fr(CHAPTER)14 b(4.)31 b(RELA)m(TED)14 b(ISSUES)450 391 y Fs(F)l(ew)o(er)i(sync)o(hronizati)o(on)c(p)q(oin)o(ts)450 473 y FC(Sev)o(eral)k(authors)f(ha)o(v)o(e)h(found)f(w)o(a)o(ys)g(to)g (eliminate)f(some)g(of)h(the)h(sync)o(hronization)f(p)q(oin)o(ts)h(in-)450 523 y(duced)d(b)o(y)f(the)h(inner)g(pro)q(ducts)g(in)f(metho)q(ds)g(suc)o(h)h (as)f(CG.)f(One)i(strategy)g(has)g(b)q(een)g(to)f(replace)450 573 y(one)19 b(of)f(the)i(t)o(w)o(o)e(inner)h(pro)q(ducts)i(t)o(ypically)c (presen)o(t)k(in)d(conjugate)h(gradien)o(t-lik)o(e)f(metho)q(ds)450 623 y(b)o(y)e(one)g(or)g(t)o(w)o(o)f(others)i(in)f(suc)o(h)h(a)e(w)o(a)o(y)h (that)g(all)e(inner)j(pro)q(ducts)g(can)f(b)q(e)h(p)q(erformed)e(sim)o(ul-) 450 672 y(taneously)m(.)32 b(The)19 b(global)e(comm)o(unicatio)o(n)f(can)j (then)g(b)q(e)h(pac)o(k)n(aged.)31 b(A)19 b(\014rst)h(suc)o(h)f(metho)q(d)450 722 y(w)o(as)14 b(prop)q(osed)i(b)o(y)e(Saad)g([177)o(])g(with)g(a)g(mo)q (di\014cation)e(to)i(impro)o(v)o(e)f(its)h(stabilit)o(y)g(suggested)h(b)o(y) 450 772 y(Meuran)o(t)h([153)o(].)23 b(Recen)o(tly)m(,)15 b(related)h(metho)q (ds)f(ha)o(v)o(e)h(b)q(een)g(prop)q(osed)h(b)o(y)e(Chronop)q(oulos)g(and)450 822 y(Gear)j([53)o(],)f(D'Azev)o(edo)h(and)g(Romine)d([59)o(],)j(and)g (Eijkhout)f([85)o(].)29 b(These)19 b(sc)o(hemes)f(can)g(also)450 872 y(b)q(e)d(applied)e(to)h(nonsymmetric)f(metho)q(ds)g(suc)o(h)i(as)g (BiCG.)e(The)h(stabilit)o(y)f(of)h(suc)o(h)h(metho)q(ds)e(is)450 922 y(discussed)j(b)o(y)d(D'Azev)o(edo,)h(Eijkhout)f(and)h(Romine)e([58)o(].) 512 974 y(Another)h(approac)o(h)f(is)h(to)f(generate)h(a)f(n)o(um)o(b)q(er)g (of)f(successiv)o(e)k(Krylo)o(v)c(v)o(ectors)j(\(see)f Fu(x)q FC(2.3.4\))450 1024 y(and)j(orthogonalize)f(these)i(as)f(a)f(blo)q(c)o(k)g (\(see)j(V)m(an)d(Rosendale)g([205)o(],)h(and)f(Chronop)q(oulos)g(and)450 1074 y(Gear)f([53)o(]\).)450 1205 y Fl(4.4.2)55 b(V)-5 b(ector)19 b(up)r(dates)450 1287 y FC(V)m(ector)c(up)q(dates)f(are)h(trivially)d (parallelizable:)k(eac)o(h)f(pro)q(cessor)g(up)q(dates)g(its)f(o)o(wn)g (segmen)o(t.)450 1418 y Fl(4.4.3)55 b(Matrix-v)n(ector)17 b(pro)r(ducts)450 1500 y FC(The)e(matrix{v)o(ector)d(pro)q(ducts)k(are)e(often)h(easily)e (parallelized)h(on)g(shared-memory)e(mac)o(hines)450 1550 y(b)o(y)g (splitting)f(the)i(matrix)e(in)g(strips)i(corresp)q(onding)g(to)g(the)f(v)o (ector)h(segmen)o(ts.)18 b(Eac)o(h)12 b(pro)q(cessor)450 1600 y(then)k(computes)g(the)g(matrix{v)o(ector)f(pro)q(duct)i(of)e(one)h(strip.) 24 b(F)m(or)15 b(distributed-memory)f(ma-)450 1649 y(c)o(hines,)e(there)h(ma) o(y)c(b)q(e)j(a)f(problem)f(if)g(eac)o(h)i(pro)q(cessor)h(has)e(only)g(a)g (segmen)o(t)g(of)f(the)i(v)o(ector)g(in)f(its)450 1699 y(memory)m(.)j(Dep)q (ending)f(on)e(the)i(bandwidth)f(of)f(the)i(matrix,)d(w)o(e)i(ma)o(y)e(need)j (comm)o(unication)c(for)450 1749 y(other)k(elemen)o(ts)e(of)h(the)g(v)o (ector,)h(whic)o(h)f(ma)o(y)e(lead)h(to)h(comm)o(unication)d(b)q(ottlenec)o (ks.)18 b(Ho)o(w)o(ev)o(er,)450 1799 y(man)o(y)d(sparse)k(matrix)c(problems)h (arise)i(from)d(a)i(net)o(w)o(ork)h(in)f(whic)o(h)g(only)f(nearb)o(y)i(no)q (des)g(are)450 1849 y(connected.)32 b(F)m(or)17 b(example,)g(matrices)h (stemming)d(from)h(\014nite)i(di\013erence)i(or)e(\014nite)g(elemen)o(t)450 1898 y(problems)13 b(t)o(ypically)f(in)o(v)o(olv)o(e)g(only)h(lo)q(cal)f (connections:)19 b(matrix)12 b(elemen)o(t)h Fy(a)1674 1904 y Fx(i;j)1726 1898 y FC(is)h(nonzero)g(only)450 1948 y(if)g(v)n(ariables)f Fy(i)i FC(and)f Fy(j)j FC(are)e(ph)o(ysically)f(close.)20 b(In)14 b(suc)o(h)i(a)e(case,)h(it)f(seems)h(natural)f(to)g(sub)q(divide)450 1998 y(the)20 b(net)o(w)o(ork,)h(or)f(grid,)h(in)o(to)e(suitable)g(blo)q(c)o (ks)h(and)g(to)f(distribute)i(them)e(o)o(v)o(er)h(the)g(pro)q(ces-)450 2048 y(sors.)f(When)c(computing)d Fy(Ap)929 2054 y Fx(i)943 2048 y FC(,)i(eac)o(h)g(pro)q(cessor)i(requires)g(the)e(v)n(alues)g(of)g Fy(p)1665 2054 y Fx(i)1693 2048 y FC(at)f(some)h(no)q(des)h(in)450 2098 y(neigh)o(b)q(oring)e(blo)q(c)o(ks.)18 b(If)c(the)g(n)o(um)o(b)q(er)f (of)h(connections)h(to)e(these)j(neigh)o(b)q(oring)d(blo)q(c)o(ks)h(is)f (small)450 2148 y(compared)f(to)h(the)g(n)o(um)o(b)q(er)f(of)g(in)o(ternal)h (no)q(des,)g(then)h(the)f(comm)o(unication)c(time)j(can)h(b)q(e)g(o)o(v)o (er-)450 2197 y(lapp)q(ed)18 b(with)f(computational)e(w)o(ork.)29 b(F)m(or)18 b(more)e(detailed)i(discussions)g(on)g(implemen)o(tati)o(on)450 2247 y(asp)q(ects)e(for)d(distributed)i(memory)c(systems,)j(see)h(de)f (Sturler)h([60)o(])e(and)h(P)o(ommerell)d([170)o(].)450 2378 y Fl(4.4.4)55 b(Preconditioning)450 2460 y FC(Preconditioning)10 b(is)g(often)g(the)h(most)e(problematic)g(part)h(of)g(parallelizing)e(an)i (iterativ)o(e)h(metho)q(d.)450 2510 y(W)m(e)i(will)g(men)o(tion)f(a)i(n)o(um) o(b)q(er)f(of)g(approac)o(hes)i(to)f(obtaining)e(parallelism)f(in)j (preconditioning.)450 2633 y Fs(Disco)o(v)o(ering)f(parallelism)g(in)i (sequen)o(tial)f(preconditi)o(on)o(ers.)39 b FC(Certain)14 b(precondition-)450 2683 y(ers)g(w)o(ere)f(not)g(dev)o(elop)q(ed)g(with)g (parallelism)d(in)i(mind,)f(but)i(they)g(can)g(b)q(e)g(executed)i(in)d (parallel.)450 2733 y(Some)j(examples)g(are)i(domain)d(decomp)q(osition)h (metho)q(ds)g(\(see)j Fu(x)p FC(5.4\),)e(whic)o(h)g(pro)o(vide)g(a)g(high)p eop %%Page: 71 83 82 bop 150 275 a Fr(4.4.)31 b(P)m(ARALLELISM)1111 b FC(71)150 391 y(degree)14 b(of)f(coarse)h(grained)e(parallelism,)e(and)j(p)q(olynomial) d(preconditioners)k(\(see)g Fu(x)q FC(3.5\),)d(whic)o(h)150 441 y(ha)o(v)o(e)j(the)g(same)f(parallelism)e(as)j(the)h(matrix-v)o(ector)e (pro)q(duct.)212 492 y(Incomplete)19 b(factorization)f(preconditioners)j(are) f(usually)e(m)o(uc)o(h)g(harder)i(to)f(parallelize:)150 542 y(using)h(w)o(a)o(v)o(efron)o(ts)g(of)g(indep)q(enden)o(t)i(computations)d (\(see)i(for)g(instance)g(P)o(aolini)d(and)j(Radi-)150 592 y(cati)14 b(di)f(Brozolo)g([166)o(]\))g(a)h(mo)q(dest)f(amoun)o(t)f(of)h (parallelism)e(can)j(b)q(e)g(attained,)f(but)h(the)g(imple-)150 642 y(men)o(tation)g(is)h(complicated.)21 b(F)m(or)15 b(instance,)h(a)f(cen)o (tral)g(di\013erence)i(discretization)f(on)f(regular)150 692 y(grids)f(giv)o(es)g(w)o(a)o(v)o(efron)o(ts)f(that)h(are)g(h)o(yp)q(erplanes) h(\(see)g(V)m(an)f(der)g(V)m(orst)g([198)o(,)g(200)o(]\).)150 807 y Fs(More)i(parallel)d(v)m(arian)o(ts)i(of)h(sequen)o(tial)d(preconditi)o (one)o(rs.)39 b FC(V)m(arian)o(ts)13 b(of)h(existing)g(se-)150 857 y(quen)o(tial)h(incomplete)g(factorization)g(preconditioners)j(with)d(a)h 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(um)o(b)q(er)g(of)f(w)o(a)o(v)o(efron)o(ts.)212 2633 y(Next,)22 b(w)o(e)e(observ)o(e)h(that)f(the)h(unkno)o(wns)f(in)f(a)h(w)o(a)o(v)o(efron) o(t)f(can)h(b)q(e)h(computed)e(as)h(so)q(on)150 2683 y(as)d(all)g(w)o(a)o(v)o (efron)o(ts)g(con)o(taining)f(its)h(predecessors)k(ha)o(v)o(e)c(b)q(een)i (computed.)27 b(Th)o(us)18 b(w)o(e)g(can,)g(in)150 2733 y(the)g(absence)g(of) e(tests)j(for)d(con)o(v)o(ergence,)j(ha)o(v)o(e)e(comp)q(onen)o(ts)g(from)e (sev)o(eral)i(iterations)g(b)q(eing)p eop %%Page: 72 84 83 bop 450 275 a FC(72)860 b Fr(CHAPTER)14 b(4.)31 b(RELA)m(TED)14 b(ISSUES)450 391 y FC(computed)f(sim)o(ultaneously)m(.)i(Adams)d(and)h (Jordan)g([2])f(observ)o(e)i(that)g(in)e(this)h(w)o(a)o(y)g(the)h(natural)450 441 y(ordering)h(of)f(unkno)o(wns)h(giv)o(es)g(an)f(iterativ)o(e)h(metho)q(d) f(that)h(is)f(mathematically)d(equiv)n(alen)o(t)j(to)450 491 y(a)g(m)o(ulti-color)d(ordering.)512 541 y(In)16 b(the)h(m)o(ulti-color)c 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y(the)f(preconditioned)f(GMRES)g(metho)q(d)f(\(see)j Fu(x)p FC(2.3.4\))d(has)h(a)g(k)o(ernel)h(where)g(one)f(new)h(v)o(ector,)450 1431 y Fy(M)495 1416 y Fw(\000)p Fv(1)539 1431 y Fy(Av)591 1416 y Fv(\()p Fx(j)r Fv(\))636 1431 y FC(,)11 b(is)g(orthogonalized)g (against)f(the)i(previously)f(built)g(orthogonal)f(set)i Fu(f)p Fy(v)1786 1416 y Fv(\(1\))1831 1431 y FC(,)g Fy(v)1876 1416 y Fv(\(2\))1921 1431 y FC(,)p Fy(:)7 b(:)g(:)e FC(,)450 1481 y Fy(v)471 1466 y Fv(\()p Fx(j)r Fv(\))515 1481 y Fu(g)p FC(.)18 b(In)13 b(our)g(v)o(ersion,)g(this)g(is)g(done)h(using)f(Lev)o(el)g(1)g (BLAS,)g(whic)o(h)g(ma)o(y)f(b)q(e)h(quite)h(ine\016cien)o(t.)450 1531 y(T)m(o)i(incorp)q(orate)h(Lev)o(el)f(2)g(BLAS)h(w)o(e)g(can)g(apply)e (either)j(Householder)f(orthogonalization)d(or)450 1580 y(classical)f (Gram-Sc)o(hmidt)d(t)o(wice)k(\(whic)o(h)f(mitigates)e(classical)i(Gram-Sc)o (hmidt's)d(p)q(oten)o(tial)j(in-)450 1630 y(stabilit)o(y;)k(see)i(Saad)e ([179)n(]\).)28 b(Both)18 b(approac)o(hes)g(signi\014can)o(tly)e(increase)j (the)f(computational)450 1680 y(w)o(ork,)12 b(but)g(using)g(classical)g (Gram-Sc)o(hmidt)d(has)k(the)g(adv)n(an)o(tage)e(that)i(all)e(inner)h(pro)q (ducts)i(can)450 1730 y(b)q(e)e(p)q(erformed)e(sim)o(ultaneously;)g(that)h (is,)g(their)g(comm)o(unication)d(can)j(b)q(e)g(pac)o(k)n(aged.)17 b(This)11 b(ma)o(y)450 1780 y(increase)k(the)g(e\016ciency)f(of)g(the)g (computation)e(signi\014can)o(tly)m(.)512 1829 y(Another)21 b(w)o(a)o(y)d(to)i(obtain)f(more)f(parallelism)f(and)i(data)h(lo)q(calit)o(y) e(is)h(to)h(generate)g(a)g(ba-)450 1879 y(sis)d Fu(f)p Fy(v)553 1864 y Fv(\(1\))598 1879 y FC(,)f Fy(Av)678 1864 y Fv(\(1\))723 1879 y FC(,)h(...,)e Fy(A)846 1864 y Fx(m)878 1879 y Fy(v)899 1864 y Fv(\(1\))944 1879 y Fu(g)h FC(for)g(the)h(Krylo)o(v)f(subspace)i (\014rst,)f(and)f(to)h(orthogonalize)e(this)450 1929 y(set)i(afterw)o(ards;)g (this)e(is)h(called)g Fy(m)p FC(-step)h(GMRES\()p Fy(m)p FC(\))f(\(see)h(Kim) d(and)i(Chronop)q(oulos)f([137)o(]\).)450 1979 y(\(Compare)10 b(this)i(to)f(the)i(GMRES)d(metho)q(d)h(in)g Fu(x)q FC(2.3.4,)e(where)k(eac)o (h)f(new)g(v)o(ector)g(is)g(imm)o(ediately)450 2029 y(orthogonalized)j(to)g (all)f(previous)i(v)o(ectors.\))24 b(This)15 b(approac)o(h)g(do)q(es)h(not)g (increase)g(the)g(compu-)450 2079 y(tational)g(w)o(ork)g(and,)i(in)e(con)o (trast)i(to)f(CG,)f(the)i(n)o(umerical)d(instabilit)o(y)h(due)h(to)g (generating)g(a)450 2128 y(p)q(ossibly)d(near-dep)q(enden)o(t)i(set)e(is)g (not)g(necessarily)h(a)e(dra)o(wbac)o(k.)p eop %%Page: 73 85 84 bop 150 707 a Fz(Chapter)34 b(5)150 919 y FB(Remaining)42 b(topics)150 1164 y Fp(5.1)70 b(The)22 b(Lanczos)h(Connection)150 1262 y FC(As)12 b(discussed)i(b)o(y)e(P)o(aige)g(and)f(Saunders)j(in)d([164)o (])g(and)h(b)o(y)g(Golub)f(and)h(V)m(an)f(Loan)h(in)f([108)o(],)h(it)f(is)150 1312 y(straigh)o(tforw)o(ard)f(to)g(deriv)o(e)h(the)g(conjugate)g(gradien)o (t)f(metho)q(d)f(for)i(solving)e(symmetric)f(p)q(ositiv)o(e)150 1362 y(de\014nite)k(linear)g(systems)f(from)f(the)j(Lanczos)f(algorithm)d (for)i(solving)g(symmetric)f(eigensystems)150 1412 y(and)i(vice)g(v)o(ersa.) 18 b(As)12 b(an)g(example,)f(let)h(us)g(consider)h(ho)o(w)e(one)i(can)f (deriv)o(e)g(the)h(Lanczos)f(pro)q(cess)150 1462 y(for)i(symmetric)e (eigensystems)i(from)e(the)j(\(unpreconditioned\))f(conjugate)g(gradien)o(t)g (metho)q(d.)212 1516 y(Supp)q(ose)h(w)o(e)f(de\014ne)h(the)g Fy(n)9 b Fu(\002)g Fy(k)15 b FC(matrix)d Fy(R)907 1501 y Fv(\()p Fx(k)q Fv(\))967 1516 y FC(b)o(y)254 1615 y Fy(R)286 1621 y Fx(k)317 1615 y FC(=)g([)p Fy(r)393 1598 y Fv(\(0\))437 1615 y Fy(;)20 b(r)489 1598 y Fv(\(1\))534 1615 y Fy(;)g(:)7 b(:)g(:)e(;)21 b(r)674 1598 y Fv(\()p Fx(k)q Fw(\000)p Fv(1\))762 1615 y FC(])p Fy(;)150 1711 y FC(and)14 b(the)g Fy(k)c Fu(\002)g Fy(k)15 b FC(upp)q(er)f(bidiagonal)e(matrix)g Fy(B)898 1717 y Fx(k)933 1711 y FC(b)o(y)254 1950 y Fy(B)285 1956 y Fx(k)317 1950 y FC(=)361 1779 y Ft(2)361 1852 y(6)361 1877 y(6)361 1902 y(6)361 1927 y(6)361 1952 y(6)361 1977 y(6)361 2001 y(6)361 2028 y(4)409 1808 y FC(1)42 b Fu(\000)p Fy(\014)527 1814 y Fv(2)704 1808 y Fu(\001)7 b(\001)g(\001)67 b FC(0)499 1886 y(1)h Fu(\000)p Fy(\014)643 1892 y Fv(3)826 1852 y FC(.)826 1869 y(.)826 1886 y(.)487 1934 y(.)503 1946 y(.)519 1959 y(.)603 1934 y(.)619 1946 y(.)635 1959 y(.)706 1934 y(.)722 1946 y(.)738 1959 y(.)414 2007 y(.)414 2024 y(.)414 2040 y(.)603 2011 y(.)619 2024 y(.)635 2036 y(.)706 2011 y(.)722 2024 y(.)738 2036 y(.)793 2040 y Fu(\000)p Fy(\014)848 2046 y Fx(k)409 2090 y FC(0)55 b Fu(\001)7 b(\001)g(\001)286 b FC(1)890 1779 y Ft(3)890 1852 y(7)890 1877 y(7)890 1902 y(7)890 1927 y(7)890 1952 y(7)890 1977 y(7)890 2001 y(7)890 2028 y(5)925 1950 y Fy(;)150 2192 y FC(where)16 b(the)g(sequences)i Fu(f)p Fy(r)575 2177 y Fv(\()p Fx(k)q Fv(\))621 2192 y Fu(g)d FC(and)g Fu(f)p Fy(\014)783 2198 y Fx(k)804 2192 y Fu(g)g FC(are)g(de\014ned)i(b)o(y)e(the)g(standard)h(conjugate)f(gradien)o (t)150 2242 y(algorithm)c(discussed)16 b(in)d 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FC(-conjugate,)e(it)h(follo)o(ws)e(that)262 2722 y(^)254 2733 y Fy(T)278 2739 y Fx(k)310 2733 y FC(=)g Fy(R)386 2715 y Fx(T)386 2743 y(k)412 2733 y Fy(AR)475 2739 y Fx(k)506 2733 y FC(=)g Fy(B)583 2715 y Fx(T)581 2743 y(k)614 2722 y FC(^)610 2733 y(\003)639 2739 y Fx(k)659 2733 y Fy(B)690 2739 y Fx(k)904 2838 y FC(73)p eop %%Page: 74 86 85 bop 450 275 a FC(74)794 b Fr(CHAPTER)14 b(5.)32 b(REMAINING)14 b(TOPICS)450 391 y FC(is)g(a)f(tridiagonal)f(matrix)g(since)558 606 y(^)554 616 y(\003)583 622 y Fx(k)615 616 y FC(=)658 433 y Ft(2)658 506 y(6)658 531 y(6)658 556 y(6)658 581 y(6)658 606 y(6)658 631 y(6)658 656 y(6)658 681 y(6)658 707 y(4)707 475 y Fy(p)728 460 y Fv(\(1\))771 447 y Fj(T)795 475 y Fy(Ap)847 460 y Fv(\(1\))1015 475 y FC(0)214 b Fu(\001)7 b(\001)g(\001)124 b FC(0)789 552 y(0)f Fy(p)954 537 y Fv(\(2\))997 525 y Fj(T)1022 552 y Fy(Ap)1074 537 y Fv(\(2\))1428 519 y FC(.)1428 536 y(.)1428 552 y(.)1004 601 y(.)1020 613 y(.)1036 626 y(.)1162 601 y(.)1178 613 y(.)1194 626 y(.)1252 601 y(.)1268 613 y(.)1284 626 y(.)794 674 y(.)794 690 y(.)794 707 y(.)1162 678 y(.)1178 690 y(.)1194 703 y(.)1252 678 y(.)1268 690 y(.)1284 703 y(.)1424 707 y(0)789 764 y(0)192 b Fu(\001)7 b(\001)g(\001)212 b FC(0)55 b Fy(p)1361 749 y Fv(\()p Fx(k)q Fv(\))1406 736 y Fj(T)1430 764 y Fy(Ap)1482 749 y Fv(\()p Fx(k)q Fv(\))1549 433 y Ft(3)1549 506 y(7)1549 531 y(7)1549 556 y(7)1549 581 y(7)1549 606 y(7)1549 631 y(7)1549 656 y(7)1549 681 y(7)1549 707 y(5)1584 616 y Fy(:)512 849 y FC(Since)15 b(span)p Fu(f)p Fy(p)746 834 y Fv(\(1\))790 849 y Fy(;)21 b(p)844 834 y Fv(\(2\))889 849 y Fy(;)f(:)7 b(:)g(:)f(;)20 b(p)1030 834 y Fv(\()p Fx(j)r Fv(\))1074 849 y Fu(g)14 b FC(=)g(span)p Fu(f)p Fy(r)1279 834 y Fv(\(0\))1323 849 y Fy(;)21 b(r)1376 834 y Fv(\(1\))1421 849 y Fy(;)f(:)7 b(:)g(:)f(;)20 b(r)1561 834 y Fv(\()p Fx(j)r Fw(\000)p Fv(1\))1647 849 y Fu(g)14 b FC(and)g(since)h(the)g(ele-)450 899 y(men)o(ts)10 b(of)h Fu(f)p Fy(r)654 884 y Fv(\()p Fx(j)r Fv(\))697 899 y Fu(g)g FC(are)g(m)o(utually)e 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FC(\003)822 1182 y Fx(k)842 1176 y Fy(B)873 1182 y Fx(k)894 1176 y FC(\001)929 1159 y Fw(\000)p Fv(1)987 1176 y Fy(;)915 b FC(\(5.1\))450 1260 y(where)14 b(\003)598 1266 y Fx(k)630 1260 y FC(is)f(the)g(diagonal)e(matrix)f(whose)j Fy(i)p FC(th)g(diagonal)e (elemen)o(t)h(is)g Fy(p)1604 1245 y Fv(\()p Fx(i)p Fv(\))1642 1233 y Fj(T)1667 1260 y Fy(p)1688 1245 y Fv(\()p Fx(i)p Fv(\))1728 1260 y FC(.)17 b(The)c(extremal)450 1310 y(eigen)o(v)n(alues)j(of)g Fy(T)741 1316 y Fx(k)778 1310 y FC(appro)o(ximate)e(those)j(of)e(the)i (matrix)e Fy(A)p FC(.)25 b(Hence,)18 b(the)e(diagonal)f(and)h(sub-)450 1360 y(diagonal)e(elemen)o(ts)i(of)g Fy(T)864 1366 y Fx(k)901 1360 y FC(in)g(\(5.1\),)f(whic)o(h)i(are)f(readily)g(a)o(v)n(ailable)e (during)i(iterations)g(of)g(the)450 1410 y(conjugate)g(gradien)o(t)f (algorithm)e(\()p Fu(x)p FC(2.3.1\),)h(can)i(b)q(e)g(used)g(to)f(construct)i Fy(T)1650 1416 y Fx(k)1686 1410 y FC(after)f Fy(k)g FC(CG)f(itera-)450 1459 y(tions.)j(This)13 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b(Blo)r(c)n(k)22 b(Iterativ)n(e)f (Metho)r(ds)450 2035 y FC(The)12 b(metho)q(ds)g(discussed)h(so)f(far)g(are)g (all)f(subspace)i(metho)q(ds,)f(that)f(is,)h(in)g(ev)o(ery)g(iteration)g (they)450 2085 y(extend)i(the)g(dimension)e(of)g(the)i(subspace)h(generated.) k(In)13 b(fact,)g(they)h(generate)g(an)f(orthogonal)450 2135 y(basis)g(for)g(this)g(subspace,)h(b)o(y)f(orthogonalizing)e(the)i(newly)g (generated)i(v)o(ector)e(with)g(resp)q(ect)i(to)450 2185 y(the)f(previous)h (basis)f(v)o(ectors.)512 2234 y(Ho)o(w)o(ev)o(er,)21 b(in)e(the)i(case)f(of)f (nonsymmetric)f(co)q(e\016cien)o(t)i(matrices)f(the)h(newly)g(generated)450 2284 y(v)o(ector)g(ma)o(y)d(b)q(e)i(almost)e(linearly)h(dep)q(enden)o(t)i(on) f(the)h(existing)e(basis.)33 b(T)m(o)18 b(prev)o(en)o(t)i(break-)450 2334 y(do)o(wn)c(or)g(sev)o(ere)i(n)o(umerical)d(error)i(in)f(suc)o(h)g (instances,)i(metho)q(ds)e(ha)o(v)o(e)g(b)q(een)h(prop)q(osed)g(that)450 2384 y(p)q(erform)c(a)g(lo)q(ok-ahead)f(step)i(\(see)h(F)m(reund,)f(Gutknec)o 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b(the)g(blo)q(c)o(k)f(size)h(constan)o(t)f (throughout)h(the)f(iteration)g(leads)g(to)g(the)h(Blo)q(c)o(k)f(Lanczos)150 441 y(algorithm)g(and)i(the)h(Blo)q(c)o(k)f(Conjugate)g(Gradien)o(t)h(metho)q (d)e(\(see)j(O'Leary)e([159)o(]\).)17 b(In)c(fact,)f(one)150 491 y(can)h(sho)o(w)g(that)g(the)g(sp)q(ectrum)h(of)e(the)h(matrix)e(is)i (e\013ectiv)o(ely)h(reduced)h(b)o(y)d(the)i Fy(n)1457 497 y Fx(b)1480 491 y Fu(\000)8 b FC(1)13 b(smallest)150 541 y(eigen)o(v)n(alues,)g (where)i Fy(n)521 547 y Fx(b)552 541 y FC(is)e(the)i(blo)q(c)o(k)e(size.)150 682 y Fp(5.3)70 b(Reduced)21 b(System)h(Preconditioning)150 775 y FC(As)14 b(w)o(e)f(ha)o(v)o(e)g(seen)h(earlier,)f(a)g(suitable)g (preconditioner)h(for)e(CG)h(is)g(a)g(matrix)e Fy(M)18 b FC(suc)o(h)13 b(that)h(the)150 824 y(system)254 911 y Fy(M)299 894 y Fw(\000)p Fv(1)343 911 y Fy(Ax)e FC(=)f Fy(M)498 894 y Fw(\000)p Fv(1)543 911 y Fy(f)150 996 y FC(requires)17 b(few)o(er)g(iterations)f(to)f(solv)o(e)h (than)g Fy(Ax)f FC(=)g Fy(f)21 b FC(do)q(es,)16 b(and)g(for)f(whic)o(h)h (systems)g Fy(M)5 b(z)17 b FC(=)e Fy(r)150 1046 y FC(can)e(b)q(e)h(solv)o(ed) g(e\016cien)o(tly)m(.)j(The)c(\014rst)i(prop)q(ert)o(y)f(is)f(indep)q(enden)o (t)h(of)f(the)h(mac)o(hine)e(used,)i(while)150 1096 y(the)k(second)g(is)f (highly)f(mac)o(hine)g(dep)q(enden)o(t.)29 b(Cho)q(osing)16 b(the)i(b)q(est)g(preconditioner)g(in)o(v)o(olv)o(es)150 1146 y(balancing)e(those)j(t)o(w)o(o)d(criteria)i(in)f(a)g(w)o(a)o(y)g(that)h (minim)o(i)o(zes)e(the)i(o)o(v)o(erall)e(computation)g(time.)150 1195 y(One)e(balancing)f(approac)o(h)g(used)h(for)f(matrices)g Fy(A)h FC(arising)e(from)g(5-p)q(oin)o(t)h(\014nite)g(di\013erence)i(dis-)150 1245 y(cretization)e(of)f(second)i(order)f(elliptic)f(partial)g(di\013eren)o (tial)g(equations)h(\(PDEs\))g(with)g(Diric)o(hlet)150 1295 y(b)q(oundary)g(conditions)h(in)o(v)o(olv)o(es)e(solving)g(a)h Fq(r)n(e)n(duc)n(e)n(d)i(system)p FC(.)i(Sp)q(eci\014cally)m(,)c(for)g(an)g Fy(n)8 b Fu(\002)h Fy(n)k FC(grid,)150 1345 y(w)o(e)h(can)g(use)h(a)f(p)q (oin)o(t)f(red-blac)o(k)h(ordering)g(of)f(the)i(no)q(des)f(to)g(get)254 1454 y Fy(Ax)d FC(=)364 1395 y Ft(\024)407 1428 y Fy(D)441 1434 y Fx(R)525 1428 y Fy(C)408 1478 y(C)441 1463 y Fx(T)510 1478 y Fy(D)544 1484 y Fx(B)593 1395 y Ft(\025)c(\024)666 1428 y Fy(x)690 1434 y Fx(R)665 1478 y Fy(x)689 1484 y Fx(B)738 1395 y Ft(\025)771 1454 y FC(=)815 1395 y Ft(\024)859 1428 y Fy(f)879 1434 y Fx(R)858 1478 y Fy(f)878 1484 y Fx(B)928 1395 y Ft(\025)970 1454 y Fy(;)632 b FC(\(5.2\))150 1562 y(where)24 b Fy(D)313 1568 y Fx(R)364 1562 y FC(and)e Fy(D)487 1568 y Fx(B)539 1562 y FC(are)h(diagonal,)g(and)g Fy(C)j FC(is)c(a)h(w)o (ell-structured)h(sparse)g(matrix)e(with)150 1611 y(5)17 b(nonzero)g (diagonals)f(if)g Fy(n)g FC(is)h(ev)o(en)h(and)e(4)h(nonzero)g(diagonals)f (if)g Fy(n)h FC(is)f(o)q(dd.)27 b(Applying)16 b(one)150 1661 y(step)h(of)f(blo)q(c)o(k)g(Gaussian)g(elimination)d(\(or)k(computing)e(the)i (Sc)o(h)o(ur)g(complemen)o(t;)e(see)i(Golub)150 1711 y(and)d(V)m(an)f(Loan)g ([108)o(]\))h(w)o(e)g(ha)o(v)o(e)254 1764 y Ft(\024)371 1797 y Fy(I)120 b(O)296 1848 y Fu(\000)p Fy(C)361 1833 y Fx(T)388 1848 y Fy(D)423 1831 y Fw(\000)p Fv(1)422 1860 y Fx(R)515 1848 y Fy(I)563 1764 y Ft(\025)7 b(\024)634 1798 y Fy(D)668 1804 y Fx(R)752 1798 y Fy(C)636 1848 y(C)669 1832 y Fx(T)737 1848 y Fy(D)771 1854 y Fx(B)821 1764 y Ft(\025)g(\024)893 1798 y Fy(x)917 1804 y Fx(R)892 1847 y Fy(x)916 1853 y Fx(B)966 1764 y Ft(\025)999 1823 y FC(=)1043 1764 y Ft(\024)1160 1797 y Fy(I)120 b(O)1085 1848 y Fu(\000)p Fy(C)1150 1833 y Fx(T)1177 1848 y Fy(D)1212 1831 y Fw(\000)p Fv(1)1211 1860 y Fx(R)1304 1848 y Fy(I)1352 1764 y Ft(\025)7 b(\024)1424 1798 y Fy(f)1444 1804 y Fx(R)1423 1847 y Fy(f)1443 1853 y Fx(B)1493 1764 y Ft(\025)150 1932 y FC(whic)o(h)14 b(reduces)i(to)254 1975 y Ft(\024)296 2007 y Fy(D)330 2013 y Fx(R)539 2007 y Fy(C)311 2059 y(O)57 b(D)434 2065 y Fx(B)472 2059 y Fu(\000)9 b Fy(C)546 2044 y Fx(T)572 2059 y Fy(D)606 2065 y Fx(R)634 2041 y Fw(\000)p Fv(1)678 2059 y Fy(C)732 1975 y Ft(\025)d(\024)804 2008 y Fy(x)828 2014 y Fx(R)803 2058 y Fy(x)827 2064 y Fx(B)876 1975 y Ft(\025)910 2034 y FC(=)953 1975 y Ft(\024)1128 2007 y Fy(f)1148 2013 y Fx(R)996 2059 y Fy(f)1016 2065 y Fx(B)1054 2059 y Fu(\000)k Fy(C)1129 2044 y Fx(T)1154 2059 y Fy(D)1188 2065 y Fx(R)1216 2041 y Fw(\000)p Fv(1)1261 2059 y Fy(f)1281 2065 y Fx(R)1329 1975 y Ft(\025)1358 2034 y Fy(:)150 2154 y FC(With)17 b(prop)q(er)i(scaling)f (\(left)g(and)f(righ)o(t)h(m)o(ultiplicatio)o(n)d(b)o(y)j Fy(D)1187 2160 y Fx(B)1216 2135 y Fw(\000)p Fv(1)p Fx(=)p Fv(2)1294 2154 y FC(\),)h(w)o(e)f(obtain)f(from)g(the)150 2203 y(second)e(blo)q(c)o(k)f (equation)f(the)i(reduced)g(system)254 2290 y(\()p Fy(I)e Fu(\000)c Fy(H)380 2272 y Fx(T)406 2290 y Fy(H)s FC(\))p Fy(y)14 b FC(=)d Fy(g)16 b(;)1029 b FC(\(5.3\))150 2383 y(where)15 b Fy(H)g FC(=)d Fy(D)399 2361 y Fw(\000)p Fv(1)p Fx(=)p Fv(2)398 2395 y Fx(R)477 2383 y Fy(C)s(D)545 2361 y Fw(\000)p Fv(1)p Fx(=)p Fv(2)544 2395 y Fx(B)624 2383 y FC(,)h Fy(y)h FC(=)e Fy(D)762 2361 y Fv(1)p Fx(=)p Fv(2)761 2395 y Fx(B)814 2383 y Fy(x)838 2389 y Fx(B)867 2383 y FC(,)h(and)h Fy(g)f FC(=)f Fy(D)1085 2361 y Fw(\000)p Fv(1)p Fx(=)p Fv(2)1084 2395 y Fx(B)1163 2383 y FC(\()p Fy(f)1199 2389 y Fx(B)1238 2383 y Fu(\000)d Fy(C)1312 2368 y Fx(T)1338 2383 y Fy(D)1373 2365 y Fw(\000)p Fv(1)1372 2395 y Fx(R)1418 2383 y Fy(f)1438 2389 y Fx(R)1466 2383 y FC(\).)18 b(The)d(linear)150 2433 y(system)f(\(5.3\))g(is)g(of)g(order)h Fy(n)612 2418 y Fv(2)630 2433 y Fy(=)p FC(2)f(for)g(ev)o(en)h Fy(n)f FC(and)h(of)e(order)i(\()p Fy(n)1163 2418 y Fv(2)1192 2433 y Fu(\000)9 b FC(1\))p Fy(=)p FC(2)14 b(for)g(o)q(dd)g Fy(n)p FC(.)20 b(Once)15 b Fy(y)h FC(is)150 2483 y(determined,)f(the)g (solution)g Fy(x)f FC(is)h(easily)g(retriev)o(ed)h(from)d Fy(y)q FC(.)22 b(The)16 b(v)n(alues)e(on)h(the)h(blac)o(k)e(p)q(oin)o(ts)150 2532 y(are)i(those)h(that)e(w)o(ould)g(b)q(e)i(obtained)e(from)f(a)i 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(Jacobi)g(preconditioning)g(for)g(3-dimensional)450 889 y(problems.)450 1023 y Fp(5.4)70 b(Domain)22 b(Decomp)r(osition)e(Metho)r(ds)450 1114 y FC(In)14 b(recen)o(t)h(y)o(ears,)f(m)o(uc)o(h)e(atten)o(tion)i(has)f (b)q(een)i(giv)o(en)e(to)h(domain)d(decomp)q(osition)h(metho)q(ds)i(for)450 1164 y(linear)f(elliptic)g(problems)f(that)i(are)f(based)i(on)e(a)g (partitioning)f(of)h(the)h(domain)d(of)i(the)h(ph)o(ysical)450 1214 y(problem.)i(Since)e(the)f(sub)q(domains)f(can)h(b)q(e)h(handled)e (indep)q(enden)o(tly)m(,)h(suc)o(h)h(metho)q(ds)f(are)g(v)o(ery)450 1264 y(attractiv)o(e)e(for)f(coarse-grain)h(parallel)f(computers.)17 b(On)11 b(the)g(other)h(hand,)f(it)f(should)g(b)q(e)i(stressed)450 1313 y(that)i(they)g(can)g(b)q(e)h(v)o(ery)f(e\013ectiv)o(e)h(ev)o(en)g(on)f (sequen)o(tial)g(computers.)512 1363 y(In)j(this)f(brief)g(surv)o(ey)m(,)h(w) o(e)f(shall)g(restrict)h(ourselv)o(es)g(to)g(the)f(standard)h(second)g(order) g(self-)450 1413 y(adjoin)o(t)c(scalar)h(elliptic)f(problems)g(in)g(t)o(w)o (o)g(dimensions)g(of)g(the)i(form:)554 1480 y Fu(\000r)9 b(\001)g FC(\()p Fy(a)p FC(\()p Fy(x;)e(y)q FC(\))p Fu(r)p Fy(u)p FC(\))k(=)h Fy(f)t FC(\()p Fy(x;)7 b(y)q FC(\))879 b(\(5.4\))450 1547 y(where)15 b Fy(a)p FC(\()p Fy(x;)7 b(y)q FC(\))13 b(is)h(a)f(p)q(ositiv)o(e)g(function) g(on)g(the)h(domain)d(\012,)i(on)h(whose)g(b)q(oundary)f(the)h(v)n(alue)f(of) 450 1597 y Fy(u)g FC(is)g(prescrib)q(ed)i(\(the)f(Diric)o(hlet)f(problem\).)k (F)m(or)12 b(more)h(general)g(problems,)f(and)h(a)g(go)q(o)q(d)g(set)h(of)450 1647 y(references,)h(the)e(reader)g(is)f(referred)i(to)e(the)h(series)g(of)f (pro)q(ceedings)h([46)o(,)f(47)o(,)g(48,)f(106)o(,)h(135)o(,)g(172)o(].)512 1696 y(F)m(or)g(the)g(discretization)h(of)e(\(5.4\),)g(w)o(e)h(shall)g (assume)f(for)h(simplicit)o(y)d(that)j(\012)g(is)g(triangulated)450 1746 y(b)o(y)j(a)g(set)i Fy(T)636 1752 y Fx(H)683 1746 y FC(of)e(nono)o(v)o (erlapping)f(coarse)i(triangles)g(\(sub)q(domains\))e(\012)1611 1752 y Fx(i)1625 1746 y Fy(;)7 b(i)14 b FC(=)g(1)p Fy(;)7 b(:::;)g(p)13 b FC(with)i Fy(n)1968 1752 y Fx(H)450 1796 y FC(in)o(ternal)h(v)o(ertices.)28 b(The)17 b(\012)899 1802 y Fx(i)913 1796 y FC('s)f(are)h(in)f(turn)h(further) h(re\014ned)g(in)o(to)e(a)g(set)h(of)f(smaller)f(triangles)450 1846 y Fy(T)474 1852 y Fx(h)512 1846 y FC(with)h Fy(n)g FC(in)o(ternal)f(v)o (ertices)j(in)e(total.)24 b(Here)17 b Fy(H)q(;)7 b(h)15 b FC(denote)i(the)g (coarse)g(and)f(\014ne)h(mesh)e(size)450 1896 y(resp)q(ectiv)o(ely)m(.)30 b(By)18 b(a)g(standard)g(Ritz-Galerkin)e(metho)q(d)h(using)g(piecewise)i (linear)e(triangular)450 1945 y(basis)d(elemen)o(ts)g(on)g(\(5.4\),)f(w)o(e)i (obtain)e(an)h Fy(n)c Fu(\002)f Fy(n)14 b FC(symmetric)f(p)q(ositiv)o(e)h (de\014nite)h(linear)e(system)450 1995 y Fy(Au)e FC(=)h Fy(f)t FC(.)512 2045 y(Generally)m(,)e(there)i(are)f(t)o(w)o(o)f(kinds)h(of)e (approac)o(hes)j(dep)q(ending)f(on)f(whether)i(the)f(sub)q(domains)450 2095 y(o)o(v)o(erlap)17 b(with)g(one)h(another)g(\(Sc)o(h)o(w)o(arz)g(metho)q (ds\))f(or)h(are)g(separated)h(from)d(one)h(another)h(b)o(y)450 2145 y(in)o(terfaces)d(\(Sc)o(h)o(ur)f(Complemen)o(t)e(metho)q(ds,)h (iterativ)o(e)h(substructuring\).)512 2194 y(W)m(e)g(shall)g(presen)o(t)i (domain)d(decomp)q(osition)g(metho)q(ds)h(as)h(preconditioners)g(for)f(the)i (linear)450 2244 y(system)j Fy(Au)h FC(=)g Fy(f)j Fq(or)g FC(to)c(a)g (reduced)i(\(Sc)o(h)o(ur)e(Complemen)o(t\))e(system)h Fy(S)1649 2250 y Fx(B)1678 2244 y Fy(u)1702 2250 y Fx(B)1751 2244 y FC(=)i Fy(g)1823 2250 y Fx(B)1870 2244 y FC(de\014ned)450 2294 y(on)c(the)h(in)o (terfaces)g(in)f(the)h(non-o)o(v)o(erlapping)d(form)o(ulation.)22 b(When)17 b(used)g(with)f(the)h(standard)450 2344 y(Krylo)o(v)e(subspace)i (metho)q(ds)f(discussed)h(elsewhere)h(in)d(this)h(b)q(o)q(ok,)f(the)i(user)f (has)g(to)g(supply)f(a)450 2394 y(pro)q(cedure)i(for)e(computing)f Fy(Av)j FC(or)f Fy(S)r(w)h FC(giv)o(en)e Fy(v)i FC(or)e Fy(w)i FC(and)e(the)h(algorithms)d(to)i(b)q(e)h(describ)q(ed)450 2444 y(herein)21 b(computes)g Fy(K)811 2428 y Fw(\000)p Fv(1)856 2444 y Fy(v)q FC(.)38 b(The)21 b(computation)e(of)h Fy(Av)j FC(is)d(a)g(simple)g(sparse)h(matrix-v)o(ector)450 2493 y(m)o(ultiply)m(,)10 b(but)k Fy(S)r(w)i FC(ma)o(y)c(require)i(sub)q(domain)f(solv)o(es,)g(as)h (will)f(b)q(e)h(describ)q(ed)i(later.)450 2606 y Fl(5.4.1)55 b(Ov)n(erlapping)18 b(Sub)r(domain)g(Metho)r(ds)450 2683 y FC(In)f(this)g(approac)o(h,)g(eac)o(h)g(substructure)j(\012)1153 2689 y Fx(i)1183 2683 y FC(is)d(extended)h(to)f(a)f(larger)h(substructure)j (\012)1893 2668 y Fw(0)1893 2694 y Fx(i)1923 2683 y FC(con-)450 2733 y(taining)12 b Fy(n)615 2717 y Fw(0)615 2743 y Fx(i)642 2733 y FC(in)o(ternal)h(v)o(ertices)i(and)e(all)f(the)i(triangles)f Fy(T)1351 2717 y Fw(0)1345 2743 y Fx(i)1375 2733 y Fu(\032)e Fy(T)1442 2739 y Fx(h)1464 2733 y FC(,)i(within)g(a)g(distance)h Fy(\016)h FC(from)d(\012)1974 2739 y Fx(i)1988 2733 y FC(,)p eop %%Page: 77 89 88 bop 150 275 a Fr(5.4.)31 b(DOMAIN)14 b(DECOMPOSITION)h(METHODS)623 b FC(77)150 391 y(where)15 b Fy(\016)h FC(refers)f(to)f(the)g(amoun)o(t)e(of) i(o)o(v)o(erlap.)212 450 y(Let)h Fy(A)318 434 y Fw(0)318 460 y Fx(i)332 450 y Fy(;)7 b(A)382 456 y Fx(H)427 450 y FC(denote)15 b(the)f(the)h(discretizations)g(of)e(\(5.4\))g(on)h(the)h(sub)q(domain)d (triangulation)150 499 y Fy(T)180 484 y Fw(0)174 510 y Fx(i)206 499 y FC(and)h(the)i(coarse)g(triangulation)d Fy(T)756 505 y Fx(H)801 499 y FC(resp)q(ectiv)o(ely)m(.)212 557 y(Let)18 b Fy(R)322 542 y Fx(T)322 568 y(i)365 557 y FC(denote)g(the)f(extension)h(op) q(erator)f(whic)o(h)g(extends)i(\(b)o(y)d(zero\))i(a)f(function)g(on)g Fy(T)1688 542 y Fw(0)1682 568 y Fx(i)150 607 y FC(to)j Fy(T)231 613 y Fx(h)274 607 y FC(and)g Fy(R)393 613 y Fx(i)427 607 y FC(the)h(corresp)q(onding)g(p)q(oin)o(t)o(wise)f(restriction)i(op)q(erator.) 38 b(Similarly)l(,)19 b(let)h Fy(R)1668 592 y Fx(T)1668 619 y(H)150 657 y FC(denote)c(the)g(in)o(terp)q(olation)e(op)q(erator)i(whic)o(h) f(maps)f(a)g(function)h(on)g(the)h(coarse)g(grid)f Fy(T)1573 663 y Fx(H)1620 657 y FC(on)o(to)150 707 y(the)f(\014ne)h(grid)e Fy(T)409 713 y Fx(h)445 707 y FC(b)o(y)g(piecewise)i(linear)f(in)o(terp)q (olation)f(and)g Fy(R)1157 713 y Fx(H)1202 707 y FC(the)h(corresp)q(onding)h (w)o(eigh)o(ted)150 757 y(restriction)g(op)q(erator.)212 815 y(With)k(these)i(notations,)f(the)g Fq(A)n(dditive)g(Schwarz)g(Pr)n(e)n(c)n (onditioner)k Fy(K)1374 821 y Fx(as)1429 815 y FC(for)19 b(the)h(system)150 865 y Fy(Au)11 b FC(=)h Fy(f)19 b FC(can)14 b(b)q(e)g(compactly)f(describ)q (ed)j(as:)562 1006 y Fy(K)600 989 y Fw(\000)p Fv(1)597 1016 y Fx(as)645 1006 y Fy(v)d FC(=)f Fy(R)754 989 y Fx(T)754 1016 y(H)785 1006 y Fy(A)816 988 y Fw(\000)p Fv(1)816 1018 y Fx(H)861 1006 y Fy(R)893 1012 y Fx(H)924 1006 y Fy(v)f FC(+)1018 952 y Fx(p)996 966 y Ft(X)999 1055 y Fx(i)p Fv(=1)1063 1006 y Fy(R)1095 989 y Fx(T)1095 1016 y(i)1121 1006 y Fy(A)1152 989 y Fw(0)1164 983 y(\000)p Fv(1)1164 1016 y Fx(i)1208 1006 y Fy(R)1240 1012 y Fx(i)1254 1006 y Fy(v)q(:)212 1158 y FC(Note)i(that)f(the)h(righ)o(t)e (hand)h(side)h(can)f(b)q(e)h(computed)e(using)h Fy(p)g FC(sub)q(domain)e (solv)o(es)j(using)f(the)150 1208 y Fy(A)181 1193 y Fw(0)181 1219 y Fx(i)195 1208 y FC('s,)g(plus)g(a)f(coarse)i(grid)f(solv)o(e)f(using)h Fy(A)809 1214 y Fx(H)841 1208 y FC(,)f(all)g(of)g(whic)o(h)h(can)g(b)q(e)h (computed)e(in)h(parallel.)k(Eac)o(h)150 1263 y(term)g Fy(R)284 1248 y Fx(T)284 1274 y(i)310 1263 y Fy(A)341 1248 y Fw(0)353 1242 y(\000)p Fv(1)353 1273 y Fx(i)397 1263 y Fy(R)429 1269 y Fx(i)443 1263 y Fy(v)i FC(should)e(b)q(e)h(ev)n(aluated)g(in)f(three)i (steps:)24 b(\(1\))17 b(Restriction:)23 b Fy(v)1472 1269 y Fx(i)1502 1263 y FC(=)17 b Fy(R)1583 1269 y Fx(i)1596 1263 y Fy(v)q FC(,)g(\(2\))150 1313 y(Sub)q(domain)d(solv)o(es)i(for)f Fy(w)585 1319 y Fx(i)598 1313 y FC(:)22 b Fy(A)663 1298 y Fw(0)663 1323 y Fx(i)677 1313 y Fy(w)707 1319 y Fx(i)735 1313 y FC(=)15 b Fy(v)802 1319 y Fx(i)816 1313 y FC(,)g(\(3\))h(In)o(terp)q(olation:)21 b Fy(y)1202 1319 y Fx(i)1230 1313 y FC(=)15 b Fy(R)1309 1298 y Fx(T)1309 1323 y(i)1335 1313 y Fy(w)1365 1319 y Fx(i)1379 1313 y FC(.)23 b(The)16 b(coarse)h(grid)150 1362 y(solv)o(e)d(is)f(handled)h (in)g(the)g(same)f(manner.)212 1421 y(The)f(theory)f(of)f(Dryja)h(and)f (Widlund)g([73)o(])h(sho)o(ws)g(that)g(the)h(condition)e(n)o(um)o(b)q(er)g (of)h Fy(K)1585 1406 y Fw(\000)p Fv(1)1582 1431 y Fx(as)1629 1421 y Fy(A)g FC(is)150 1470 y(b)q(ounded)g(indep)q(enden)o(tly)h(of)e(b)q (oth)h(the)g(coarse)h(grid)e(size)h Fy(H)j FC(and)c(the)i(\014ne)f(grid)f (size)i Fy(h)p FC(,)e(pro)o(vided)150 1520 y(there)i(is)g(\\su\016cien)o(t")f (o)o(v)o(erlap)f(b)q(et)o(w)o(een)j(\012)835 1526 y Fx(i)860 1520 y FC(and)e(\012)968 1505 y Fw(0)968 1531 y Fx(i)993 1520 y FC(\(essen)o(tially)g(it)g(means)f(that)h(the)h(ratio)f Fy(\016)r(=H)150 1570 y FC(of)g(the)h(distance)h Fy(\016)g FC(of)e(the)h(b)q(oundary)g Fy(@)r FC(\012)808 1555 y Fw(0)808 1581 y Fx(i)834 1570 y FC(to)f Fy(@)r FC(\012)936 1576 y Fx(i)962 1570 y FC(should)g(b)q(e)h(uniformly)d(b)q (ounded)k(from)c(b)q(elo)o(w)150 1620 y(as)15 b Fy(h)e Fu(!)f FC(0.\))20 b(If)15 b(the)g(coarse)h(grid)e(solv)o(e)g(term)g(is)h(left)f (out,)h(then)g(the)g(condition)f(n)o(um)o(b)q(er)g(gro)o(ws)150 1670 y(as)g Fy(O)q FC(\(1)p Fy(=H)330 1655 y Fv(2)348 1670 y FC(\),)f(re\015ecting)i(the)g(lac)o(k)e(of)h(global)e(coupling)h(pro)o (vided)h(b)o(y)f(the)i(coarse)g(grid.)212 1728 y(F)m(or)i(the)h(purp)q(ose)g (of)f(implemen)o(tatio)o(ns,)f(it)g(is)i(often)f(useful)g(to)g(in)o(terpret)i (the)f(de\014nition)150 1778 y(of)g Fy(K)237 1784 y Fx(as)292 1778 y FC(in)h(matrix)e(notation.)32 b(Th)o(us)19 b(the)h(decomp)q(osition)d (of)i(\012)g(in)o(to)f(\012)1369 1763 y Fw(0)1369 1789 y Fx(i)1383 1778 y FC('s)g(corresp)q(onds)j(to)150 1828 y(partitioning)d(of)h(the)h(comp) q(onen)o(ts)f(of)g(the)h(v)o(ector)g Fy(u)f FC(in)o(to)f Fy(p)i FC(o)o(v)o(erlapping)e(groups)h(of)g(index)150 1877 y(sets)j Fy(I)256 1883 y Fx(i)270 1877 y FC('s,)f(eac)o(h)g(with)f Fy(n)557 1862 y Fw(0)557 1888 y Fx(i)591 1877 y FC(comp)q(onen)o(ts.)37 b(The)21 b Fy(n)972 1862 y Fw(0)972 1888 y Fx(i)1000 1877 y Fu(\002)13 b Fy(n)1070 1862 y Fw(0)1070 1888 y Fx(i)1104 1877 y FC(matrix)19 b Fy(A)1277 1862 y Fw(0)1277 1888 y Fx(i)1311 1877 y FC(is)i(simply)d(a)i(principal)150 1927 y(submatrix)e(of)g Fy(A)i FC(corresp)q(onding)g(to)f(the)g(index)g(set)i Fy(I)1066 1933 y Fx(i)1080 1927 y FC(.)33 b Fy(R)1157 1912 y Fx(T)1157 1938 y(i)1202 1927 y FC(is)19 b(a)g Fy(n)13 b Fu(\002)g Fy(n)1397 1912 y Fw(0)1397 1938 y Fx(i)1430 1927 y FC(matrix)k(de\014ned)150 1977 y(b)o(y)g(its)f(action)h(on)g(a)f(v)o(ector)i Fy(u)649 1983 y Fx(i)679 1977 y FC(de\014ned)g(on)f Fy(T)916 1962 y Fw(0)910 1988 y Fx(i)944 1977 y FC(as:)24 b(\()p Fy(R)1065 1962 y Fx(T)1065 1988 y(i)1091 1977 y Fy(u)1115 1983 y Fx(i)1129 1977 y FC(\))1145 1983 y Fx(j)1179 1977 y FC(=)17 b(\()p Fy(u)1268 1983 y Fx(i)1281 1977 y FC(\))1297 1983 y Fx(j)1332 1977 y FC(if)f Fy(j)j Fu(2)d Fy(I)1471 1983 y Fx(i)1502 1977 y FC(but)h(is)f(zero) 150 2027 y(otherwise.)31 b(Similarly)l(,)16 b(the)i(action)g(of)f(its)h (transp)q(ose)h Fy(R)1092 2033 y Fx(i)1105 2027 y Fy(u)f FC(forms)f(an)g Fy(n)1352 2012 y Fw(0)1352 2038 y Fx(i)1366 2027 y FC(-v)o(ector)h(b)o(y)g (pic)o(king)150 2077 y(o\013)e(the)h(comp)q(onen)o(ts)f(of)g Fy(u)f FC(corresp)q(onding)j(to)e Fy(I)946 2083 y Fx(i)960 2077 y FC(.)25 b(Analogously)m(,)14 b Fy(R)1275 2062 y Fx(T)1275 2088 y(H)1323 2077 y FC(is)i(an)g Fy(n)10 b Fu(\002)h Fy(n)1530 2083 y Fx(H)1578 2077 y FC(matrix)150 2126 y(with)k(en)o(tries)i(corresp)q (onding)f(to)g(piecewise)g(linear)g(in)o(terp)q(olation)e(and)h(its)h(transp) q(ose)h(can)f(b)q(e)150 2176 y(in)o(terpreted)k(as)f(a)g(w)o(eigh)o(ted)f (restriction)i(matrix.)31 b(If)18 b Fy(T)1081 2182 y Fx(h)1122 2176 y FC(is)g(obtained)h(from)e Fy(T)1471 2182 y Fx(H)1521 2176 y FC(b)o(y)i(nested)150 2226 y(re\014nemen)o(t,)10 b(the)h(action)e(of)g Fy(R)625 2232 y Fx(H)656 2226 y Fy(;)e(R)707 2211 y Fx(T)707 2237 y(H)747 2226 y FC(can)j(b)q(e)h(e\016cien)o(tly)e(computed)g(as)h(in)f (a)h(standard)g(m)o(ultigrid)150 2276 y(algorithm.)25 b(Note)17 b(that)g(the)h(matrices)e Fy(R)837 2261 y Fx(T)837 2287 y(i)863 2276 y Fy(;)7 b(R)914 2282 y Fx(i)927 2276 y Fy(;)g(R)978 2261 y Fx(T)978 2287 y(H)1008 2276 y Fy(;)g(R)1059 2282 y Fx(H)1107 2276 y FC(are)17 b(de\014ned)h(b)o(y)f(their)g(actions)g(and)150 2326 y(need)e(not)f(b)q(e)g(stored)h(explicitly)m(.)212 2384 y(W)m(e)k(also)f(note)h(that)g(in)f(this)h(algebraic)f(form)o(ulation,)f(the) i(preconditioner)h Fy(K)1521 2390 y Fx(as)1576 2384 y FC(can)f(b)q(e)150 2434 y(extended)11 b(to)f(an)o(y)g(matrix)e Fy(A)p FC(,)i(not)g(necessarily)h (one)f(arising)f(from)f(a)i(discretization)g(of)f(an)h(elliptic)150 2483 y(problem.)16 b(Once)11 b(w)o(e)g(ha)o(v)o(e)g(the)g(partitioning)e (index)h(sets)i Fy(I)1077 2489 y Fx(i)1091 2483 y FC('s,)f(the)g(matrices)f Fy(R)1405 2489 y Fx(i)1419 2483 y Fy(;)d(A)1469 2468 y Fw(0)1469 2494 y Fx(i)1493 2483 y FC(are)k(de\014ned.)150 2533 y(F)m(urthermore,)16 b(if)f Fy(A)h FC(is)g(p)q(ositiv)o(e)g(de\014nite,)g(then)h Fy(A)981 2518 y Fw(0)981 2544 y Fx(i)1011 2533 y FC(is)e(guaran)o(teed)i(to)f (b)q(e)g(nonsingular.)24 b(The)150 2583 y(di\016cult)o(y)9 b(is)g(in)h(de\014ning)f(the)i(\\coarse)f(grid")f(matrices)g Fy(A)1057 2589 y Fx(H)1089 2583 y Fy(;)e(R)1140 2589 y Fx(H)1170 2583 y FC(,)j(whic)o(h)g(inheren)o(tly)g(dep)q(ends)h(on)150 2633 y(kno)o(wledge)h(of)f(the)i(grid)e(structure.)20 b(This)12 b(part)g(of)f(the)i(preconditioner)f(can)h(b)q(e)f(left)g(out,)g(at)g(the)150 2683 y(exp)q(ense)k(of)d(a)h(deteriorating)g(con)o(v)o(ergence)i(rate)f(as)f Fy(p)g FC(increases.)20 b(Radicati)13 b(and)h(Rob)q(ert)g([173)o(])150 2733 y(ha)o(v)o(e)g(exp)q(erimen)o(ted)g(with)g(suc)o(h)g(an)g(algebraic)f(o) o(v)o(erlapping)g(blo)q(c)o(k)g(Jacobi)h(preconditioner.)p eop %%Page: 78 90 89 bop 450 275 a FC(78)794 b Fr(CHAPTER)14 b(5.)32 b(REMAINING)14 b(TOPICS)450 391 y Fl(5.4.2)55 b(Non-o)n(v)n(erlapping)18 b(Sub)r(domain)g (Metho)r(ds)450 471 y FC(The)h(easiest)i(w)o(a)o(y)d(to)h(describ)q(e)i(this) e(approac)o(h)g(is)f(through)i(matrix)d(notation.)32 b(The)20 b(set)g(of)450 520 y(v)o(ertices)c(of)f Fy(T)674 526 y Fx(h)711 520 y FC(can)g(b)q(e)g(divided)g(in)o(to)f(t)o(w)o(o)h(groups.)22 b(The)15 b(set)h(of)f(in)o(terior)f(v)o(ertices)j Fy(I)h FC(of)d(all)e(\012) 1985 526 y Fx(i)450 570 y FC(and)e(the)i(set)f(of)f(v)o(ertices)i Fy(B)h FC(whic)o(h)d(lies)h(on)f(the)h(b)q(oundaries)1417 539 y Ft(S)1452 583 y Fx(i)1473 570 y Fy(@)r FC(\012)1527 555 y Fw(0)1527 581 y Fx(i)1552 570 y FC(of)f(the)i(coarse)f(triangles)f(\012)1985 555 y Fw(0)1985 581 y Fx(i)450 620 y FC(in)k Fy(T)524 626 y Fx(H)555 620 y FC(.)22 b(W)m(e)14 b(shall)h(re-order)h Fy(u)f FC(and)g Fy(f)k FC(as)c Fy(u)f Fu(\021)f FC(\()p Fy(u)1253 626 y Fx(I)1272 620 y Fy(;)7 b(u)1315 626 y Fx(B)1343 620 y FC(\))1359 605 y Fx(T)1400 620 y FC(and)15 b Fy(f)j Fu(\021)c FC(\()p Fy(f)1602 626 y Fx(I)1622 620 y Fy(;)7 b(f)1661 626 y Fx(B)1689 620 y FC(\))1705 605 y Fx(T)1746 620 y FC(corresp)q(onding)450 670 y(to)14 b(this)g(partition.)j(In)d(this)g(ordering,)f(equation)h(\(5.4\)) f(can)h(b)q(e)h(written)f(as)g(follo)o(ws:)554 722 y Ft(\022)618 755 y Fy(A)649 761 y Fx(I)723 755 y Fy(A)754 761 y Fx(I)r(B)605 805 y Fy(A)636 790 y Fx(T)636 817 y(I)r(B)732 805 y Fy(A)763 811 y Fx(B)821 722 y Ft(\023)6 b(\022)914 755 y Fy(u)938 761 y Fx(I)909 805 y Fy(u)933 811 y Fx(B)983 722 y Ft(\023)1025 781 y FC(=)1068 722 y Ft(\022)1125 755 y Fy(f)1145 761 y Fx(I)1120 805 y Fy(f)1140 811 y Fx(B)1189 722 y Ft(\023)1227 781 y Fy(:)675 b FC(\(5.5\))450 889 y(W)m(e)17 b(note)i(that)f(since)g(the)h(sub)q(domains)d (are)j(uncoupled)f(b)o(y)g(the)g(b)q(oundary)g(v)o(ertices,)i Fy(A)1930 895 y Fx(I)1967 889 y FC(=)450 939 y Fy(bl)q(ock)q(diag)q(onal)q FC(\()p Fy(A)748 945 y Fx(i)762 939 y FC(\))13 b(is)g(blo)q(c)o(k-diagonal)e (with)h(eac)o(h)i(blo)q(c)o(k)e Fy(A)1432 945 y Fx(i)1459 939 y FC(b)q(eing)h(the)h(sti\013ness)g(matrix)d(cor-)450 989 y(resp)q(onding)j (to)g(the)h(unkno)o(wns)f(b)q(elonging)e(to)i(the)h Fq(interior)i FC(v)o(ertices)e(of)e(sub)q(domain)g(\012)1883 995 y Fx(i)1896 989 y FC(.)512 1040 y(By)h(a)g(blo)q(c)o(k)g(LU-factorization)f(of)g Fy(A)p FC(,)g(the)i(system)f(in)f(\(5.5\))g(can)h(b)q(e)h(written)f(as:)554 1093 y Ft(\022)671 1125 y Fy(I)110 b FC(0)605 1177 y Fy(A)636 1162 y Fx(T)636 1188 y(I)r(B)682 1177 y Fy(A)713 1159 y Fw(\000)p Fv(1)713 1189 y Fx(I)799 1177 y Fy(I)841 1093 y Ft(\023)7 b(\022)930 1126 y Fy(A)961 1132 y Fx(I)1022 1126 y Fy(A)1053 1132 y Fx(I)r(B)945 1176 y FC(0)67 b Fy(S)1058 1182 y Fx(B)1119 1093 y Ft(\023)7 b(\022)1213 1126 y Fy(u)1237 1132 y Fx(I)1208 1176 y Fy(u)1232 1182 y Fx(B)1281 1093 y Ft(\023)1323 1151 y FC(=)1367 1093 y Ft(\022)1423 1126 y Fy(f)1443 1132 y Fx(I)1418 1176 y Fy(f)1438 1182 y Fx(B)1488 1093 y Ft(\023)1525 1151 y Fy(;)377 b FC(\(5.6\))450 1261 y(where)1000 1315 y Fy(S)1025 1321 y Fx(B)1066 1315 y Fu(\021)12 b Fy(A)1141 1321 y Fx(B)1178 1315 y Fu(\000)e Fy(A)1251 1298 y Fx(T)1251 1325 y(I)r(B)1297 1315 y Fy(A)1328 1297 y Fw(\000)p Fv(1)1328 1327 y Fx(I)1372 1315 y Fy(A)1403 1321 y Fx(I)r(B)450 1394 y FC(is)k(the)g(Sc)o(h)o(ur)h(complemen)o(t)c(of)j Fy(A)989 1400 y Fx(B)1031 1394 y FC(in)g Fy(A)p FC(.)512 1445 y(By)g(eliminating)d Fy(u)818 1451 y Fx(I)851 1445 y FC(in)i(\(5.6\),)g(w)o (e)h(arriv)o(e)g(at)g(the)g(follo)o(wing)d(equation)j(for)f Fy(u)1743 1451 y Fx(B)1772 1445 y FC(:)554 1531 y Fy(S)579 1537 y Fx(B)608 1531 y Fy(u)632 1537 y Fx(B)672 1531 y FC(=)e Fy(g)735 1537 y Fx(B)775 1531 y Fu(\021)h Fy(f)839 1537 y Fx(B)877 1531 y Fu(\000)e Fy(A)950 1537 y Fx(I)r(B)996 1531 y Fy(A)1027 1513 y Fw(\000)p Fv(1)1027 1543 y Fx(I)1071 1531 y Fy(f)1091 1537 y Fx(I)1111 1531 y Fy(:)791 b FC(\(5.7\))512 1618 y(W)m(e)14 b(note)g(the)g(follo)o(wing)e(prop)q(erties)j(of)e(this)h(Sc)o(h)o(ur)h (Complemen)o(t)c(system:)501 1705 y(1.)20 b Fy(S)579 1711 y Fx(B)622 1705 y FC(inherits)14 b(the)g(symmetric)f(p)q(ositiv)o(e)g (de\014niteness)j(of)d Fy(A)p FC(.)501 1794 y(2.)20 b Fy(S)579 1800 y Fx(B)625 1794 y FC(is)d(dense)h(in)f(general)g(and)g(computing)f(it)g (explicitly)g(requires)j(as)e(man)o(y)e(solv)o(es)i(on)554 1844 y(eac)o(h)d(sub)q(domain)e(as)i(there)i(are)e(p)q(oin)o(ts)g(on)f(eac)o (h)i(of)e(its)h(edges.)501 1932 y(3.)20 b(The)12 b(condition)e(n)o(um)o(b)q (er)h(of)g Fy(S)1035 1938 y Fx(B)1075 1932 y FC(is)g Fy(O)q FC(\()p Fy(h)1187 1917 y Fw(\000)p Fv(1)1232 1932 y FC(\),)g(an)g(impro)o(v)o (emen)o(t)e(o)o(v)o(er)j(the)g Fy(O)q FC(\()p Fy(h)1802 1917 y Fw(\000)p Fv(2)1846 1932 y FC(\))f(gro)o(wth)554 1982 y(for)i Fy(A)p FC(.)501 2071 y(4.)20 b(Giv)o(en)13 b(a)h(v)o(ector)g Fy(v)853 2077 y Fx(B)896 2071 y FC(de\014ned)h(on)f(the)g(b)q(oundary)g(v)o (ertices)i Fy(B)g FC(of)e Fy(T)1625 2077 y Fx(H)1656 2071 y FC(,)g(the)g(matrix-v)o(ector)554 2121 y(pro)q(duct)g Fy(S)734 2127 y Fx(B)763 2121 y Fy(v)783 2127 y Fx(B)824 2121 y FC(can)g(b)q(e)f (computed)g(according)g(to)g Fy(A)1413 2127 y Fx(B)1442 2121 y Fy(v)1462 2127 y Fx(B)1498 2121 y Fu(\000)8 b Fy(A)1569 2106 y Fx(T)1569 2132 y(I)r(B)1614 2121 y FC(\()p Fy(A)1661 2103 y Fw(\000)p Fv(1)1661 2133 y Fx(I)1706 2121 y FC(\()p Fy(A)1753 2127 y Fx(I)r(B)1799 2121 y Fy(v)1819 2127 y Fx(B)1848 2121 y FC(\)\))13 b(where)554 2171 y Fy(A)585 2153 y Fw(\000)p Fv(1)585 2183 y Fx(I)643 2171 y FC(in)o(v)o(olv)o(es)g Fy(p)h FC(indep)q(enden)o(t)h (sub)q(domain)d(solv)o(es)i(using)g Fy(A)1538 2153 y Fw(\000)p Fv(1)1538 2182 y Fx(i)1583 2171 y FC(.)501 2259 y(5.)20 b(The)c(righ)o(t)g (hand)g(side)h Fy(g)956 2265 y Fx(B)1000 2259 y FC(can)f(also)g(b)q(e)h (computed)e(using)h Fy(p)g FC(indep)q(enden)o(t)i(sub)q(domain)554 2309 y(solv)o(es.)450 2396 y(These)c(prop)q(erties)g(mak)o(e)d(it)h(p)q (ossible)h(to)f(apply)g(a)h(preconditioned)g(iterativ)o(e)f(metho)q(d)g(to)g (\(5.7\),)450 2446 y(whic)o(h)17 b(is)g(the)h(basic)g(metho)q(d)e(in)h(the)h (nono)o(v)o(erlapping)e(substructuring)j(approac)o(h.)28 b(W)m(e)17 b(will)450 2496 y(also)c(need)j(some)d(go)q(o)q(d)g(preconditioners)j(to)e (further)h(impro)o(v)o(e)d(the)j(con)o(v)o(ergence)g(of)f(the)h(Sc)o(h)o(ur) 450 2546 y(system.)512 2597 y(W)m(e)h(shall)g(\014rst)h(describ)q(e)h(the)f (Bram)o(ble-P)o(asciak-Sc)o(hatz)f(preconditioner)h([35)o(].)26 b(F)m(or)16 b(this,)450 2647 y(w)o(e)e(need)h(to)f(further)g(decomp)q(ose)g Fy(B)j FC(in)o(to)c(t)o(w)o(o)g(non-o)o(v)o(erlapping)g(index)h(sets:)554 2733 y Fy(B)g FC(=)e Fy(E)f Fu([)e Fy(V)746 2739 y Fx(H)1914 2733 y FC(\(5.8\))p eop %%Page: 79 91 90 bop 150 275 a Fr(5.4.)31 b(DOMAIN)14 b(DECOMPOSITION)h(METHODS)623 b FC(79)150 391 y(where)16 b Fy(V)295 397 y Fx(H)341 391 y FC(=)387 360 y Ft(S)422 404 y Fx(k)449 391 y Fy(V)473 397 y Fx(k)509 391 y FC(denote)g(the)g(set)h(of)d(no)q(des)i(corresp)q(onding)h(to) e(the)h(v)o(ertices)g Fy(V)1519 397 y Fx(k)1540 391 y FC('s)f(of)g Fy(T)1656 397 y Fx(H)1688 391 y FC(,)150 441 y(and)d Fy(E)i FC(=)318 410 y Ft(S)352 454 y Fx(i)373 441 y Fy(E)404 447 y Fx(i)430 441 y FC(denote)f(the)h(set)f(of)f(no)q(des)h(on)f(the)i(edges)f Fy(E)1124 447 y Fx(i)1138 441 y FC('s)f(of)g(the)h(coarse)h(triangles)e(in)g Fy(T)1656 447 y Fx(H)1688 441 y FC(,)150 491 y(excluding)i(the)g(v)o(ertices) h(b)q(elonging)e(to)h Fy(V)820 497 y Fx(H)852 491 y FC(.)212 541 y(In)i(addition)e(to)h(the)h(coarse)g(grid)f(in)o(terp)q(olation)f(and)h (restriction)h(op)q(erators)g Fy(R)1516 547 y Fx(H)1547 541 y Fy(;)7 b(R)1598 526 y Fx(T)1598 552 y(H)1644 541 y FC(de-)150 591 y(\014ned)18 b(b)q(efore,)g(w)o(e)g(shall)e(also)h(need)h(a)f(new)h(set)g (of)f(in)o(terp)q(olation)f(and)h(restriction)h(op)q(erators)150 640 y(for)c(the)g(edges)i Fy(E)428 646 y Fx(i)441 640 y FC('s.)j(Let)14 b Fy(R)606 646 y Fx(E)630 650 y Fj(i)659 640 y FC(b)q(e)h(the)g(p)q(oin)o(t)o (wise)e(restriction)i(op)q(erator)g(\(an)f Fy(n)1437 646 y Fx(E)1461 650 y Fj(i)1486 640 y Fu(\002)c Fy(n)k FC(matrix,)150 690 y(where)j Fy(n)297 696 y Fx(E)321 700 y Fj(i)352 690 y FC(is)e(the)h(n)o(um)o(b)q(er)f(of)g(v)o(ertices)i(on)e(the)h(edge)h Fy(E)1082 696 y Fx(i)1095 690 y FC(\))f(on)o(to)f(the)h(edge)g Fy(E)1422 696 y Fx(i)1451 690 y FC(de\014ned)h(b)o(y)e(its)150 740 y(action)f(\()p Fy(R)323 746 y Fx(E)347 750 y Fj(i)363 740 y Fy(u)387 746 y Fx(B)415 740 y FC(\))431 746 y Fx(j)462 740 y FC(=)f(\()p Fy(u)547 746 y Fx(B)576 740 y FC(\))592 746 y Fx(j)624 740 y FC(if)h Fy(j)i Fu(2)d Fy(E)768 746 y Fx(i)796 740 y FC(but)i(is)g(zero)g(otherwise.)22 b(The)15 b(action)g(of)f(its)h (transp)q(ose)150 790 y(extends)g(b)o(y)f(zero)h(a)e(function)h(de\014ned)h (on)f Fy(E)876 796 y Fx(i)903 790 y FC(to)g(one)g(de\014ned)h(on)e Fy(B)r FC(.)212 840 y(Corresp)q(onding)i(to)e(this)h(partition)f(of)h Fy(B)r FC(,)g Fy(S)i FC(can)e(b)q(e)h(written)f(in)g(the)g(blo)q(c)o(k)g (form:)254 945 y Fy(S)279 951 y Fx(B)319 945 y FC(=)363 887 y Ft(\022)428 920 y Fy(S)453 926 y Fx(E)536 920 y Fy(S)561 926 y Fx(E)r(V)414 970 y Fy(S)441 955 y Fx(T)439 981 y(E)r(V)549 970 y Fy(S)574 976 y Fx(V)637 887 y Ft(\023)675 945 y Fy(:)927 b FC(\(5.9\))150 1051 y(The)15 b(blo)q(c)o(k)f Fy(S)371 1057 y Fx(E)414 1051 y FC(can)g(again)g(b)q(e)h(blo)q(c)o(k)f(partitioned,)f(with) h(most)g(of)g(the)h(subblo)q(c)o(ks)f(b)q(eing)h(zero.)150 1101 y(The)d(diagonal)f(blo)q(c)o(ks)h Fy(S)547 1107 y Fx(E)571 1111 y Fj(i)599 1101 y FC(of)f Fy(S)669 1107 y Fx(E)709 1101 y FC(are)i(the)f(principal)f(submatrices)h(of)g Fy(S)j FC(corresp)q(onding)d (to)g Fy(E)1674 1107 y Fx(i)1688 1101 y FC(.)150 1151 y(Eac)o(h)g Fy(S)276 1157 y Fx(E)300 1161 y Fj(i)327 1151 y FC(represen)o(ts)i(the)e (coupling)f(of)g(no)q(des)h(on)f(in)o(terface)h Fy(E)1164 1157 y Fx(i)1189 1151 y FC(separating)g(t)o(w)o(o)f(neigh)o(b)q(ouring)150 1201 y(sub)q(domains.)212 1250 y(In)19 b(de\014ning)f(the)h(preconditioner,)g (the)g(action)f(of)g Fy(S)1081 1233 y Fw(\000)p Fv(1)1079 1263 y Fx(E)1103 1267 y Fj(i)1144 1250 y FC(is)g(needed.)33 b(Ho)o(w)o(ev)o(er,)19 b(as)g(noted)150 1300 y(b)q(efore,)14 b(in)f(general)h Fy(S)502 1306 y Fx(E)526 1310 y Fj(i)555 1300 y FC(is)g(a)f(dense)i(matrix)d(whic)o(h) h(is)g(also)g(exp)q(ensiv)o(e)i(to)f(compute,)e(and)i(ev)o(en)150 1350 y(if)g(w)o(e)g(had)g(it,)g(it)g(w)o(ould)g(b)q(e)h(exp)q(ensiv)o(e)g(to) f(compute)g(its)h(action)f(\(w)o(e)h(w)o(ould)e(need)i(to)g(compute)150 1400 y(its)i(in)o(v)o(erse)h(or)g(a)f(Cholesky)g(factorization\).)28 b(F)m(ortunately)m(,)17 b(man)o(y)e(e\016cien)o(tly)j(in)o(v)o(ertible)f(ap-) 150 1450 y(pro)o(ximations)10 b(to)i Fy(S)472 1456 y Fx(E)496 1460 y Fj(i)525 1450 y FC(ha)o(v)o(e)g(b)q(een)i(prop)q(osed)f(in)f(the)i (literature)f(\(see)g(Key)o(es)h(and)f(Gropp)f([136)o(]\))150 1500 y(and)g(w)o(e)h(shall)f(use)h(these)h(so-called)e(in)o(terface)h (preconditioners)h(for)e Fy(S)1271 1506 y Fx(E)1295 1510 y Fj(i)1324 1500 y FC(instead.)18 b(W)m(e)12 b(men)o(tion)150 1549 y(one)i(sp)q(eci\014c)h(preconditioner:)779 1599 y Fy(M)819 1605 y Fx(E)843 1609 y Fj(i)870 1599 y FC(=)d Fy(\013)941 1605 y Fx(E)965 1609 y Fj(i)980 1599 y Fy(K)1018 1582 y Fv(1)p Fx(=)p Fv(2)150 1674 y FC(where)20 b Fy(K)j FC(is)c(an)g Fy(n)468 1680 y Fx(E)492 1684 y Fj(i)521 1674 y Fu(\002)13 b Fy(n)591 1680 y Fx(E)615 1684 y Fj(i)650 1674 y FC(one)19 b(dimensional)e(Laplacian)h (matrix,)h(namely)e(a)i(tridiagonal)150 1724 y(matrix)10 b(with)h(2's)h(do)o (wn)f(the)i(main)d(diagonal)g(and)h Fu(\000)p FC(1's)h(do)o(wn)g(the)g(t)o(w) o(o)f(o\013-diagonals,)f(and)i Fy(\013)1660 1730 y Fx(E)1684 1734 y Fj(i)150 1774 y FC(is)h(tak)o(en)h(to)g(b)q(e)g(some)f(a)o(v)o(erage)g (of)g(the)h(co)q(e\016cien)o(t)h Fy(a)p FC(\()p Fy(x;)7 b(y)q FC(\))14 b(of)f(\(5.4\))f(on)i(the)g(edge)g Fy(E)1506 1780 y Fx(i)1520 1774 y FC(.)k(W)m(e)13 b(note)150 1824 y(that)k(since)g(the)g (eigen-decomp)q(osition)e(of)h Fy(K)k FC(is)c(kno)o(wn)g(and)g(computable)g (b)o(y)g(the)h(F)m(ast)f(Sine)150 1873 y(T)m(ransform,)c(the)i(action)g(of)f Fy(M)646 1879 y Fx(E)670 1883 y Fj(i)700 1873 y FC(can)h(b)q(e)g(e\016cien)o (tly)g(computed.)212 1923 y(With)f(these)i(notations,)e(the)h(Bram)o(ble-P)o (asciak-Sc)o(hatz)f(preconditioner)h(is)f(de\014ned)i(b)o(y)e(its)150 1973 y(action)h(on)f(a)h(v)o(ector)g Fy(v)511 1979 y Fx(B)554 1973 y FC(de\014ned)h(on)f Fy(B)i FC(as)e(follo)o(ws:)254 2060 y Fy(K)292 2043 y Fw(\000)p Fv(1)289 2073 y Fx(B)q(P)t(S)365 2060 y Fy(v)385 2066 y Fx(B)425 2060 y FC(=)e Fy(R)501 2043 y Fx(T)501 2071 y(H)532 2060 y Fy(A)563 2043 y Fw(\000)p Fv(1)563 2073 y Fx(H)608 2060 y Fy(R)640 2066 y Fx(H)671 2060 y Fy(v)691 2066 y Fx(B)729 2060 y FC(+)770 2021 y Ft(X)782 2110 y Fx(E)806 2114 y Fj(i)837 2060 y Fy(R)869 2043 y Fx(T)869 2071 y(E)893 2075 y Fj(i)908 2060 y Fy(M)953 2043 y Fw(\000)p Fv(1)948 2073 y Fx(E)972 2077 y Fj(i)998 2060 y Fy(R)1030 2066 y Fx(E)1054 2070 y Fj(i)1069 2060 y Fy(v)1089 2066 y Fx(B)1118 2060 y Fy(:)463 b FC(\(5.10\))212 2185 y(Analogous)15 b(to)g(the)i(additiv)o(e)d(Sc)o(h)o(w)o (arz)i(preconditioner,)h(the)f(computation)e(of)h(eac)o(h)h(term)150 2234 y(consists)g(of)e(the)h(three)h(steps)g(of)f(restriction-in)o(v)o (ersion-in)o(terp)q(olation)f(and)g(is)h(indep)q(enden)o(t)h(of)150 2284 y(the)e(others)h(and)f(th)o(us)g(can)h(b)q(e)f(carried)h(out)e(in)h (parallel.)212 2334 y(Bram)o(ble,)f(P)o(asciak)h(and)g(Sc)o(hatz)h([35)o(])f (pro)o(v)o(e)h(that)f(the)h(condition)e(n)o(um)o(b)q(er)h(of)g Fy(K)1530 2316 y Fw(\000)p Fv(1)1527 2346 y Fx(B)q(P)t(S)1603 2334 y Fy(S)1628 2340 y Fx(B)1671 2334 y FC(is)150 2384 y(b)q(ounded)h(b)o(y) g Fy(O)q FC(\(1)9 b(+)h(log)554 2365 y Fv(2)572 2384 y FC(\()p Fy(H)q(=h)p FC(\)\).)20 b(In)15 b(practice,)g(there)h(is)e(a)h(sligh)o(t)e (gro)o(wth)i(in)f(the)h(n)o(um)o(b)q(er)f(of)150 2434 y(iterations)h(as)g Fy(h)g FC(b)q(ecomes)g(small)d(\()p Fq(i.e.)p FC(,)i(as)h(the)h(\014ne)f (grid)g(is)f(re\014ned\))j(or)e(as)g Fy(H)i FC(b)q(ecomes)f(large)150 2483 y(\()p Fq(i.e.)p FC(,)d(as)h(the)g(coarse)h(grid)f(b)q(ecomes)g (coarser\).)212 2533 y(The)e(log)348 2515 y Fv(2)367 2533 y FC(\()p Fy(H)q(=h)p FC(\))e(gro)o(wth)h(is)g(due)g(to)g(the)g(coupling)f(of)h (the)g(unkno)o(wns)g(on)g(the)g(edges)h(inciden)o(t)150 2583 y(on)j(a)h(common)d(v)o(ertex)j Fy(V)565 2589 y Fx(k)586 2583 y FC(,)f(whic)o(h)h(is)f(not)h(accoun)o(ted)g(for)g(in)f Fy(K)1200 2589 y Fx(B)q(P)t(S)1276 2583 y FC(.)23 b(Smith)14 b([187)o(])h(prop)q(osed) 150 2633 y(a)g Fq(vertex)i(sp)n(ac)n(e)i FC(mo)q(di\014cation)13 b(to)j Fy(K)751 2639 y Fx(B)q(P)t(S)843 2633 y FC(whic)o(h)f(explicitly)g (accoun)o(ts)i(for)e(this)h(coupling)e(and)150 2683 y(therefore)20 b(eliminates)c(the)j(dep)q(endence)i(on)d Fy(H)j FC(and)d Fy(h)p FC(.)31 b(The)18 b(idea)g(is)g(to)g(in)o(tro)q(duce)h(further)150 2733 y(subsets)h(of)e Fy(B)j FC(called)d Fq(vertex)g(sp)n(ac)n(es)k Fy(X)h FC(=)890 2701 y Ft(S)924 2745 y Fx(k)952 2733 y Fy(X)986 2739 y Fx(k)1025 2733 y FC(with)18 b Fy(X)1158 2739 y Fx(k)1197 2733 y FC(consisting)g(of)g(a)g(small)e(set)j(of)p eop %%Page: 80 92 91 bop 450 275 a FC(80)794 b Fr(CHAPTER)14 b(5.)32 b(REMAINING)14 b(TOPICS)450 391 y FC(v)o(ertices)21 b(on)f(the)h(edges)g(inciden)o(t)f(on)g (the)g(v)o(ertex)h Fy(V)1326 397 y Fx(k)1367 391 y FC(and)f(adjacen)o(t)g(to) g(it.)36 b(Note)21 b(that)f Fy(X)450 441 y FC(o)o(v)o(erlaps)d(with)g Fy(E)j FC(and)d Fy(V)871 447 y Fx(H)903 441 y FC(.)28 b(Let)18 b Fy(S)1046 447 y Fx(X)1073 451 y Fj(k)1111 441 y FC(b)q(e)g(the)g(principal) f(submatrix)f(of)h Fy(S)1700 447 y Fx(B)1746 441 y FC(corresp)q(onding)450 491 y(to)e Fy(X)536 497 y Fx(k)557 491 y FC(,)g(and)g Fy(R)698 497 y Fx(X)725 501 y Fj(k)745 491 y Fy(;)7 b(R)796 476 y Fx(T)796 502 y(X)823 506 y Fj(k)857 491 y FC(b)q(e)16 b(the)f(corresp)q(onding)h (restriction)g(\(p)q(oin)o(t)o(wise\))g(and)f(extension)g(\(b)o(y)450 541 y(zero\))d(matrices.)17 b(As)11 b(b)q(efore,)h Fy(S)950 547 y Fx(X)977 551 y Fj(k)1009 541 y FC(is)f(dense)i(and)e(exp)q(ensiv)o(e)h (to)f(compute)f(and)h(factor/solv)o(e)g(but)450 591 y(e\016cien)o(tly)g(in)o (v)o(ertable)g(appro)o(ximations)d(\(some)i(using)h(v)n(arian)o(ts)f(of)h (the)g Fy(K)1642 576 y Fv(1)p Fx(=)p Fv(2)1706 591 y FC(op)q(erator)g (de\014ned)450 640 y(b)q(efore\))21 b(ha)o(v)o(e)g(b)q(een)g(dev)o(elop)q(ed) g(in)f(the)h(literature)g(\(see)h(Chan,)f(Mathew)g(and)f(Shao)g([50)o(]\).) 450 690 y(W)m(e)g(shall)g(let)g Fy(M)738 696 y Fx(X)765 700 y Fj(k)806 690 y FC(b)q(e)h(suc)o(h)h(a)e(preconditioner)h(for)f Fy(S)1387 696 y Fx(X)1414 700 y Fj(k)1435 690 y FC(.)38 b(Then)21 b(Smith's)d(V)m(ertex)j(Space)450 740 y(preconditioner)15 b(is)e(de\014ned)i (b)o(y:)554 829 y Fy(K)592 812 y Fw(\000)p Fv(1)589 842 y Fx(V)7 b(S)640 829 y Fy(v)660 835 y Fx(B)730 829 y FC(=)42 b Fy(R)836 812 y Fx(T)836 840 y(H)867 829 y Fy(A)898 812 y Fw(\000)p Fv(1)898 842 y Fx(H)943 829 y Fy(R)975 835 y Fx(H)1006 829 y Fy(v)1026 835 y Fx(B)1063 829 y FC(+)1105 790 y Ft(X)1116 879 y Fx(E)1140 883 y Fj(i)1172 829 y Fy(R)1204 812 y Fx(T)1204 840 y(E)1228 844 y Fj(i)1243 829 y Fy(M)1288 812 y Fw(\000)p Fv(1)1283 842 y Fx(E)1307 846 y Fj(i)1332 829 y Fy(R)1364 835 y Fx(E)1388 839 y Fj(i)1403 829 y Fy(v)1423 835 y Fx(B)813 948 y FC(+)855 908 y Ft(X)862 997 y Fx(X)889 1001 y Fj(k)921 948 y Fy(R)953 930 y Fx(T)953 958 y(X)980 962 y Fj(k)1000 948 y Fy(M)1045 930 y Fw(\000)p Fv(1)1040 960 y Fx(X)1067 964 y Fj(k)1090 948 y Fy(R)1122 954 y Fx(X)1149 958 y Fj(k)1169 948 y Fy(v)1189 954 y Fx(B)1217 948 y Fy(:)664 b FC(\(5.11\))450 1080 y(Smith)14 b([187)n(])h(pro)o(v)o(ed)h(that)f(the)h(condition)f(n)o(um)o(b)q(er)f(of)h Fy(K)1398 1062 y Fw(\000)p Fv(1)1395 1092 y Fx(V)7 b(S)1446 1080 y Fy(S)1471 1086 y Fx(B)1515 1080 y FC(is)15 b(b)q(ounded)h(indep)q (enden)o(t)h(of)450 1130 y Fy(H)g FC(and)c Fy(h)p FC(,)h(pro)o(vided)f(there) j(is)d(su\016cien)o(t)i(o)o(v)o(erlap)e(of)g Fy(X)1350 1136 y Fx(k)1385 1130 y FC(with)h Fy(B)r FC(.)450 1247 y Fl(5.4.3)55 b(Multiplicativ)n(e)16 b(Sc)n(h)n(w)n(arz)21 b(Metho)r(ds)450 1324 y FC(As)10 b(men)o(tioned)f(b)q(efore,)i(the)f(Additiv)o(e)g(Sc)o(h)o(w) o(arz)g(preconditioner)h(can)f(b)q(e)g(view)o(ed)g(as)g(an)g(o)o(v)o(erlap-) 450 1374 y(ping)g(blo)q(c)o(k)h(Jacobi)f(preconditioner.)18 b(Analogously)m(,)9 b(one)i(can)g(de\014ne)h(a)f Fq(multiplic)n(ative)i FC(Sc)o(h)o(w)o(arz)450 1424 y(preconditioner)g(whic)o(h)g(corresp)q(onds)h (to)f(a)f(symmetric)f(blo)q(c)o(k)h(Gauss-Seidel)h(v)o(ersion.)k(That)c(is,) 450 1474 y(the)k(solv)o(es)f(on)g(eac)o(h)g(sub)q(domain)f(are)h(p)q (erformed)g(sequen)o(tially)m(,)f(using)h(the)g(most)f(curren)o(t)j(it-)450 1523 y(erates)c(as)f(b)q(oundary)g(conditions)g(from)e(neigh)o(b)q(oring)h (sub)q(domains.)k(Ev)o(en)e(without)e(conjugate)450 1573 y(gradien)o(t)i (acceleration,)h(the)g(m)o(ultiplicativ)o(e)d(metho)q(d)h(can)i(tak)o(e)g (man)o(y)e(few)o(er)i(iterations)f(than)450 1623 y(the)j(additiv)o(e)e(v)o (ersion.)26 b(Ho)o(w)o(ev)o(er,)17 b(the)g(m)o(ultiplicati)o(v)o(e)d(v)o (ersion)i(is)g(not)h(as)f(parallelizable,)f(al-)450 1673 y(though)f(the)h (degree)g(of)f(parallelism)d(can)j(b)q(e)h(increased)h(b)o(y)d(trading)h (o\013)g(the)h(con)o(v)o(ergence)h(rate)450 1723 y(through)k(m)o (ulti-coloring)d(the)k(sub)q(domains.)37 b(The)21 b(theory)f(can)h(b)q(e)g (found)f(in)g(Bram)o(ble,)g Fq(et)450 1772 y(al.)13 b FC([36)o(].)450 1882 y Fs(Inexact)j(Solv)o(es)450 1959 y FC(The)i(exact)h(solv)o(es)f(in)o(v) o(olving)d Fy(A)987 1943 y Fw(0)999 1938 y(\000)p Fv(1)999 1969 y Fx(i)1043 1959 y Fy(;)7 b(A)1093 1941 y Fw(\000)p Fv(1)1093 1970 y Fx(i)1155 1959 y FC(and)18 b Fy(A)1271 1941 y Fw(\000)p Fv(1)1271 1971 y Fx(H)1333 1959 y FC(in)g Fy(K)1421 1965 y Fx(as)1457 1959 y Fy(;)7 b(K)1511 1965 y Fx(B)q(P)t(S)1587 1959 y Fy(;)g(K)1641 1965 y Fx(V)f(S)1709 1959 y FC(can)18 b(b)q(e)g(replaced)450 2022 y(b)o(y)e(inexact)h(solv)o(es)788 2011 y(~)777 2022 y Fy(A)808 2010 y Fw(0)820 1994 y(\000)p Fv(1)820 2032 y Fx(i)865 2022 y Fy(;)894 2011 y FC(~)884 2022 y Fy(A)915 2004 y Fw(\000)p Fv(1)915 2033 y Fx(i)975 2022 y FC(and)1069 2011 y(~)1058 2022 y Fy(A)1089 2004 y Fw(\000)1089 2034 y Fx(H)1121 2022 y FC(,)f(whic)o(h)g(can)h(b)q(e)g(standard)g(elliptic)e (preconditioners)450 2071 y(themselv)o(es)f(\(e.g.)k(m)o(ultigrid,)11 b(ILU,)i(SSOR,)g(etc.\).)512 2121 y(F)m(or)e(the)g(Sc)o(h)o(w)o(arz)g(metho)q (ds,)f(the)i(mo)q(di\014cation)c(is)i(straigh)o(tforw)o(ard)h(and)f(the)h Fq(Inexact)i(Solve)450 2171 y(A)n(dditive)h(Schwarz)h(Pr)n(e)n(c)n (onditioner)j FC(is)c(simply:)873 2285 y(~)862 2296 y Fy(K)900 2279 y Fw(\000)p Fv(1)897 2306 y Fx(as)945 2296 y Fy(v)f FC(=)f Fy(R)1054 2279 y Fx(T)1054 2306 y(H)1096 2285 y FC(~)1085 2296 y Fy(A)1116 2278 y Fw(\000)p Fv(1)1116 2308 y Fx(H)1161 2296 y Fy(R)1193 2302 y Fx(H)1224 2296 y Fy(v)f FC(+)1318 2242 y Fx(p)1296 2256 y Ft(X)1299 2345 y Fx(i)p Fv(=1)1363 2296 y Fy(R)1395 2279 y Fx(T)1395 2306 y(i)1432 2285 y FC(~)1421 2296 y Fy(A)1452 2284 y Fw(0)1464 2268 y(\000)p Fv(1)1464 2306 y Fx(i)1508 2296 y Fy(R)1540 2302 y Fx(i)1554 2296 y Fy(v)q(:)512 2424 y FC(The)19 b(Sc)o(h)o(ur)f(Complemen)o(t)d(metho)q(ds)j(require)g(more) f(c)o(hanges)i(to)e(accommo)q(date)f(inexact)450 2474 y(solv)o(es.)k(By)15 b(replacing)g Fy(A)862 2456 y Fw(\000)p Fv(1)862 2486 y Fx(H)921 2474 y FC(b)o(y)990 2463 y(~)979 2474 y Fy(A)1010 2456 y Fw(\000)p Fv(1)1010 2486 y Fx(H)1069 2474 y FC(in)g(the)g(de\014nitions)f(of)g Fy(K)1475 2480 y Fx(B)q(P)t(S)1566 2474 y FC(and)h Fy(K)1683 2480 y Fx(V)7 b(S)1734 2474 y FC(,)14 b(w)o(e)h(can)g(easily)450 2529 y(obtain)9 b(inexact)i(preconditioners)1014 2518 y(~)1003 2529 y Fy(K)1038 2535 y Fx(B)q(P)t(S)1125 2529 y FC(and)1213 2518 y(~)1202 2529 y Fy(K)1237 2535 y Fx(V)c(S)1298 2529 y FC(for)j Fy(S)1383 2535 y Fx(B)1412 2529 y FC(.)17 b(The)11 b(main)d(di\016cult)o(y)h(is,)h(ho)o(w)o(ev)o(er,)450 2578 y(that)i(the)g(ev)n(aluation)f(of)g(the)h(pro)q(duct)h Fy(S)1098 2584 y Fx(B)1127 2578 y Fy(v)1147 2584 y Fx(B)1187 2578 y FC(requires)g (exact)g(sub)q(domain)d(solv)o(es)i(in)f Fy(A)1851 2561 y Fw(\000)p Fv(1)1851 2591 y Fx(I)1896 2578 y FC(.)17 b(One)450 2633 y(w)o(a)o(y)12 b(to)h(get)g(around)g(this)g(is)f(to)h(use)h(an)f Fq(inner)k FC(iteration)12 b(using)1488 2622 y(~)1477 2633 y Fy(A)1508 2639 y Fx(i)1535 2633 y FC(as)h(a)f(preconditioner)i(for)e Fy(A)1985 2639 y Fx(i)450 2683 y FC(in)i(order)i(to)e(compute)h(the)g(action) g(of)f Fy(A)1107 2665 y Fw(\000)p Fv(1)1107 2695 y Fx(I)1151 2683 y FC(.)21 b(An)15 b(alternativ)o(e)g(is)f(to)h(p)q(erform)f(the)h (iteration)f(on)450 2733 y(the)h(larger)f(system)h(\(5.5\))e(and)h(construct) i(a)f(preconditioner)g(from)d(the)j(factorization)f(in)g(\(5.6\))p eop %%Page: 81 93 92 bop 150 275 a Fr(5.5.)31 b(MUL)m(TIGRID)12 b(METHODS)937 b FC(81)150 391 y(b)o(y)15 b(replacing)f(the)i(terms)e Fy(A)607 397 y Fx(I)626 391 y Fy(;)7 b(S)670 397 y Fx(B)714 391 y FC(b)o(y)783 381 y(~)772 391 y Fy(A)803 397 y Fx(I)822 391 y Fy(;)848 381 y FC(~)841 391 y Fy(S)866 397 y Fx(B)910 391 y FC(resp)q(ectiv)o(ely)m(,)15 b(where)1273 381 y(~)1266 391 y Fy(S)1291 397 y Fx(B)1335 391 y FC(can)g(b)q(e)g(either)1599 381 y(~)1588 391 y Fy(K)1623 397 y Fx(B)q(P)t(S)150 441 y FC(or)212 431 y(~)201 441 y Fy(K)236 447 y Fx(V)7 b(S)287 441 y FC(.)18 b(Care)c(m)o(ust)e(b)q(e)j(tak)o(en)f(to)f (scale)848 431 y(~)837 441 y Fy(A)868 447 y Fx(H)914 441 y FC(and)1005 431 y(~)994 441 y Fy(A)1025 447 y Fx(i)1053 441 y FC(so)g(that)h(they)g(are)h(as)e(close)i(to)e Fy(A)1587 447 y Fx(H)1632 441 y FC(and)150 491 y Fy(A)181 497 y Fx(i)208 491 y FC(as)g(p)q(ossible)h(resp)q(ectiv)o(ely)g(|)f(it)g(is)g(not)g (su\016cien)o(t)g(that)h(the)f(condition)g(n)o(um)o(b)q(er)f(of)1572 480 y(~)1561 491 y Fy(A)1592 473 y Fw(\000)p Fv(1)1592 503 y Fx(H)1637 491 y Fy(A)1668 497 y Fx(H)150 546 y FC(and)242 535 y(~)231 546 y Fy(A)262 528 y Fw(\000)p Fv(1)262 557 y Fx(i)307 546 y Fy(A)338 552 y Fx(i)366 546 y FC(b)q(e)j(close)g(to)f(unit)o(y)m(,)f(b) q(ecause)j(the)f(scaling)f(of)f(the)i(coupling)f(matrix)e Fy(A)1506 552 y Fx(I)r(B)1566 546 y FC(ma)o(y)h(b)q(e)150 595 y(wrong.)150 712 y Fs(Nonsymmetric)h(Problems)150 792 y FC(The)31 b(preconditioners)h(giv) o(en)d(ab)q(o)o(v)o(e)i(extend)g(naturally)f(to)g(nonsymmetric)e Fy(A)p FC('s)j(\()p Fq(e.g.)p FC(,)150 842 y(con)o(v)o(ection-di\013usion)13 b(problems\),)g(at)g(least)h(when)g(the)h(nonsymmetric)d(part)h(is)h(not)g (to)q(o)f(large.)150 892 y(The)j(nice)g(theoretical)g(con)o(v)o(ergence)h (rates)f(can)g(b)q(e)g(retained)g(pro)o(vided)g(that)f(the)h(coarse)h(grid) 150 942 y(size)d Fy(H)j FC(is)c(c)o(hosen)i(small)d(enough)h(\(dep)q(ending)i (on)e(the)h(size)h(of)e(the)h(nonsymmetric)e(part)i(of)f Fy(A)p FC(\))150 991 y(\(see)i(Cai)e(and)h(Widlund)f([42)o(]\).)150 1108 y Fs(Choice)i(of)h(Coarse)f(Grid)f(Size)h Fy(H)150 1188 y FC(Giv)o(en)k Fy(h)p FC(,)g(it)g(has)h(b)q(een)g(observ)o(ed)g(empirically) d(\(see)k(Gropp)d(and)i(Key)o(es)g([110)o(]\))f(that)g(there)150 1238 y(often)c(exists)g(an)g(optimal)d(v)n(alue)i(of)g Fy(H)k FC(whic)o(h)d(minim)o(i)o(zes)e(the)j(total)e(computational)e(time)i(for)150 1288 y(solving)i(the)h(problem.)26 b(A)17 b(small)d Fy(H)20 b FC(pro)o(vides)d(a)g(b)q(etter,)i(but)e(more)e(exp)q(ensiv)o(e,)k(coarse)f (grid)150 1337 y(appro)o(ximation,)11 b(and)j(requires)i(solving)d(more,)h (but)g(smaller,)f(sub)q(domain)g(solv)o(es.)20 b(A)14 b(large)g Fy(H)150 1387 y FC(has)j(the)h(opp)q(osite)g(e\013ect.)29 b(F)m(or)17 b(mo)q(del)f(problems,)h(the)g(optimal)e Fy(H)20 b FC(can)d(b)q(e)h (determined)g(for)150 1437 y(b)q(oth)e(sequen)o(tial)f(and)h(parallel)e (implemen)o(tations)f(\(see)k(Chan)e(and)h(Shao)f([51)o(]\).)23 b(In)16 b(practice,)150 1487 y(it)c(ma)o(y)f(pa)o(y)i(to)f(determine)h(a)g (near)g(optimal)d(v)n(alue)i(of)g Fy(H)k FC(empirically)10 b(if)i(the)i(preconditioner)f(is)150 1537 y(to)j(b)q(e)g(re-used)h(man)o(y)d (times.)22 b(Ho)o(w)o(ev)o(er,)16 b(there)i(ma)o(y)13 b(also)i(b)q(e)i (geometric)e(constrain)o(ts)h(on)g(the)150 1587 y(range)e(of)f(v)n(alues)h (that)g Fy(H)j FC(can)d(tak)o(e.)150 1733 y Fp(5.5)70 b(Multigrid)21 b(Metho)r(ds)150 1827 y FC(Simple)8 b(iterativ)o(e)j(metho)q(ds)e(\(suc)o(h)i (as)g(the)f(Jacobi)g(metho)q(d\))g(tend)h(to)f(damp)e(out)i(high)g(frequency) 150 1877 y(comp)q(onen)o(ts)15 b(of)f(the)i(error)f(fastest)h(\(see)h Fu(x)p FC(2.2.1\).)j(This)14 b(has)i(led)f(p)q(eople)g(to)g(dev)o(elop)g (metho)q(ds)150 1927 y(based)g(on)e(the)i(follo)o(wing)c(heuristic:)201 2015 y(1.)20 b(P)o(erform)13 b(some)g(steps)i(of)e(a)h(basic)g(metho)q(d)f (in)g(order)i(to)f(smo)q(oth)e(out)i(the)h(error.)201 2105 y(2.)20 b(Restrict)g(the)g(curren)o(t)h(state)f(of)f(the)h(problem)e(to)i(a)f (subset)i(of)d(the)i(grid)f(p)q(oin)o(ts,)i(the)254 2154 y(so-called)13 b(\\coarse)i(grid",)e(and)g(solv)o(e)h(the)h(resulting)f(pro)r(jected)h (problem.)201 2244 y(3.)20 b(In)o(terp)q(olate)g(the)g(coarse)g(grid)f (solution)g(bac)o(k)h(to)f(the)h(original)e(grid,)i(and)f(p)q(erform)f(a)254 2294 y(n)o(um)o(b)q(er)13 b(of)g(steps)i(of)f(the)g(basic)g(metho)q(d)f (again.)150 2382 y(Steps)j(1)g(and)f(3)g(are)h(called)f(\\pre-smo)q(othing")f (and)h(\\p)q(ost-smo)q(othing")e(resp)q(ectiv)o(ely;)k(b)o(y)f(ap-)150 2432 y(plying)9 b(this)i(metho)q(d)e(recursiv)o(ely)i(to)g(step)g(2)f(it)g(b) q(ecomes)h(a)f(true)h(\\m)o(ultigrid")c(metho)q(d.)16 b(Usually)150 2482 y(the)e(generation)h(of)e(subsequen)o(tly)i(coarser)g(grids)f(is)g (halted)f(at)h(a)g(p)q(oin)o(t)f(where)i(the)g(n)o(um)o(b)q(er)e(of)150 2532 y(v)n(ariables)g(b)q(ecomes)h(small)e(enough)i(that)g(direct)h(solution) e(of)g(the)h(linear)g(system)f(is)h(feasible.)212 2583 y(The)k(metho)q(d)f (outlined)g(ab)q(o)o(v)o(e)h(is)f(said)h(to)f(b)q(e)i(a)e(\\V-cycle")h(metho) q(d,)f(since)i(it)e(descends)150 2633 y(through)f(a)g(sequence)i(of)d (subsequen)o(tly)i(coarser)g(grids,)f(and)g(then)g(ascends)i(this)e(sequence) i(in)150 2683 y(rev)o(erse)g(order.)26 b(A)17 b(\\W-cycle")f(metho)q(d)f (results)i(from)e(visiting)g(the)i(coarse)g(grid)f Fq(twic)n(e)p FC(,)g(with)150 2733 y(p)q(ossibly)e(some)f(smo)q(othing)f(steps)j(in)e(b)q (et)o(w)o(een.)p eop %%Page: 82 94 93 bop 450 275 a FC(82)794 b Fr(CHAPTER)14 b(5.)32 b(REMAINING)14 b(TOPICS)512 391 y FC(An)f(analysis)e(of)h(m)o(ultigrid)e(metho)q(ds)h(is)i (relativ)o(ely)e(straigh)o(tforw)o(ard)h(in)g(the)h(case)g(of)f(simple)450 441 y(di\013eren)o(tial)17 b(op)q(erators)g(suc)o(h)h(as)f(the)g(P)o(oisson)g (op)q(erator)g(on)f(tensor)i(pro)q(duct)g(grids.)26 b(In)17 b(that)450 491 y(case,)h(eac)o(h)f(next)g(coarse)h(grid)e(is)h(tak)o(en)g(to) f(ha)o(v)o(e)h(the)g(double)g(grid)f(spacing)g(of)h(the)g(previous)450 541 y(grid.)22 b(In)15 b(t)o(w)o(o)g(dimensions,)e(a)i(coarse)i(grid)d(will)g (ha)o(v)o(e)h(one)h(quarter)g(of)e(the)i(n)o(um)o(b)q(er)f(of)f(p)q(oin)o(ts) 450 591 y(of)i(the)h(corresp)q(onding)h(\014ne)f(grid.)25 b(Since)17 b(the)g(coarse)h(grid)e(is)h(again)e(a)h(tensor)i(pro)q(duct)f(grid,)450 640 y(a)12 b(F)m(ourier)g(analysis)f(\(see)i(for)f(instance)g(Briggs)g([41)o (]\))g(can)g(b)q(e)h(used.)18 b(F)m(or)12 b(the)g(more)f(general)h(case)450 690 y(of)17 b(self-adjoin)o(t)g(elliptic)g(op)q(erators)i(on)e(arbitrary)h (domains)e(a)i(more)e(sophisticated)j(analysis)450 740 y(is)d(needed)i(\(see) f(Hac)o(kbusc)o(h)g([116)o(],)f(McCormic)o(k)e([145)o(]\).)25 b(Man)o(y)15 b(m)o(ultigrid)f(metho)q(ds)h(can)i(b)q(e)450 790 y(sho)o(wn)c(to)f(ha)o(v)o(e)g(an)h(\(almost\))e(optimal)e(n)o(um)o(b)q (er)j(of)g(op)q(erations,)h(that)f(is,)h(the)g(w)o(ork)f(in)o(v)o(olv)o(ed)f (is)450 840 y(prop)q(ortional)i(to)h(the)g(n)o(um)o(b)q(er)f(of)h(v)n (ariables.)512 889 y(F)m(rom)9 b(the)i(ab)q(o)o(v)o(e)f(description)h(it)f (is)g(clear)h(that)g(iterativ)o(e)f(metho)q(ds)g(pla)o(y)f(a)i(role)f(in)g(m) o(ultigrid)450 939 y(theory)g(as)g(smo)q(others)f(\(see)i(Kettler)g([133)n (]\).)17 b(Con)o(v)o(ersely)m(,)9 b(m)o(ultigrid-lik)n(e)e(metho)q(ds)i(can)h (b)q(e)g(used)450 989 y(as)j(preconditioners)h(in)e(iterativ)o(e)h(metho)q (ds.)k(The)c(basic)g(idea)g(here)h(is)e(to)h(partition)f(the)h(matrix)450 1039 y(on)h(a)f(giv)o(en)h(grid)f(to)h(a)f(2)c Fu(\002)h FC(2)j(structure)554 1157 y Fy(A)585 1140 y Fv(\()p Fx(i)p Fv(\))636 1157 y FC(=)680 1086 y Ft( )720 1129 y Fy(A)751 1107 y Fv(\()p Fx(i)p Fv(\))751 1140 y(1)p Fx(;)p Fv(1)837 1129 y Fy(A)868 1107 y Fv(\()p Fx(i)p Fv(\))868 1140 y(1)p Fx(;)p Fv(2)720 1193 y Fy(A)751 1172 y Fv(\()p Fx(i)p Fv(\))751 1204 y(2)p Fx(;)p Fv(1)837 1193 y Fy(A)868 1172 y Fv(\()p Fx(i)p Fv(\))868 1204 y(2)p Fx(;)p Fv(2)921 1086 y Ft(!)450 1275 y FC(with)e(the)h(v)n(ariables)e(in)h(the)h (second)g(blo)q(c)o(k)f(ro)o(w)g(corresp)q(onding)h(to)f(the)h(coarse)g(grid) f(no)q(des.)18 b(The)450 1325 y(matrix)12 b(on)i(the)g(next)h(grid)e(is)h (then)g(an)g(incomplete)f(v)o(ersion)h(of)f(the)i(Sc)o(h)o(ur)f(complemen)o (t)554 1418 y Fy(A)585 1401 y Fv(\()p Fx(i)p Fv(+1\))678 1418 y Fu(\031)e Fy(S)749 1401 y Fv(\()p Fx(i)p Fv(\))801 1418 y FC(=)g Fy(A)876 1397 y Fv(\()p Fx(i)p Fv(\))876 1429 y(2)p Fx(;)p Fv(2)930 1418 y Fu(\000)e Fy(A)1003 1397 y Fv(\()p Fx(i)p Fv(\))1003 1429 y(2)p Fx(;)p Fv(1)1048 1418 y Fy(A)1079 1397 y Fv(\()p Fx(i)p Fv(\))1117 1384 y Fb(\000)p Fk(1)1079 1429 y Fv(1)p Fx(;)p Fv(1)1158 1418 y Fy(A)1189 1397 y Fv(\()p Fx(i)p Fv(\))1189 1429 y(1)p Fx(;)p Fv(2)1234 1418 y Fy(:)450 1501 y FC(The)15 b(coarse)g(grid)e(is)i(t)o(ypically)d(formed)h(based)i(on)f(a)g (red-blac)o(k)g(or)g(cyclic)h(reduction)g(ordering;)450 1551 y(see)g(for)f(instance)g(Ro)q(drigue)g(and)f(W)m(olitzer)h([175)n(],)f(and)h (Elman)e([89)o(].)512 1601 y(Some)f(m)o(ultigrid)e(preconditioners)k(try)f (to)g(obtain)f(optimalit)o(y)e(results)k(similar)c(to)j(those)h(for)450 1651 y(the)g(full)f(m)o(ultigrid)e(metho)q(d.)17 b(Here)c(w)o(e)g(will)f (merely)g(supply)g(some)g(p)q(oin)o(ters)h(to)g(the)g(literature:)450 1700 y(Axelsson)h(and)f(Eijkhout)f([16)o(],)g(Axelsson)i(and)f(V)m (assilevski)g([22)o(,)f(21],)g(Braess)j([34)o(],)d(Maitre)h(and)450 1750 y(Musy)h([142)o(],)f(McCormic)o(k)g(and)g(Thomas)g([146)n(],)g(Yseren)o (tan)o(t)j([213)o(])d(and)h(W)m(esseling)f([210)o(].)450 1888 y Fp(5.6)70 b(Ro)n(w)22 b(Pro)t(jection)g(Metho)r(ds)450 1979 y FC(Most)d(iterativ)o(e)f(metho)q(ds)g(dep)q(end)i(on)e(sp)q(ectral)h(prop)q 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2410 y Fx(T)928 2437 y(i)955 2427 y Fy(x:)450 2510 y FC(These)k(metho)q(ds)f(ha)o (v)o(e)g(go)q(o)q(d)g(parallel)e(prop)q(erties)k(and)e(seem)g(to)g(b)q(e)g (robust)h(in)f(handling)f(non-)450 2560 y(symmetric)f(and)i(inde\014nite)g (problems.)512 2609 y(Ro)o(w)f(pro)r(jection)i(metho)q(ds)f(can)g(b)q(e)h (used)g(as)f(preconditioners)h(in)f(the)g(conjugate)g(gradien)o(t)450 2659 y(metho)q(d.)26 b(In)17 b(that)g(case,)h(there)g(is)f(a)g(theoretical)g (connection)g(with)g(the)g(conjugate)g(gradien)o(t)450 2709 y(metho)q(d)c(on)h(the)g(normal)e(equations)i(\(see)h Fu(x)q FC(2.3.3\).)p eop %%Page: 83 95 94 bop 150 712 a Fz(App)s(endix)34 b(A)150 929 y FB(Obtaining)42 b(the)e(Soft)m(w)m(are)150 1154 y FC(A)13 b(large)g(b)q(o)q(dy)h(of)e(n)o (umerical)g(soft)o(w)o(are)i(is)f(freely)g(a)o(v)n(ailable)e(24)i(hours)h(a)f (da)o(y)g(via)f(an)h(electronic)150 1204 y(service)k(called)e Fq(Netlib)p FC(.)21 b(In)16 b(addition)e(to)h(the)h(template)f(material,)e (there)k(are)e(dozens)i(of)e(other)150 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b(measure)g(of)f(the)i(error) g(decreases)h(b)o(y)e(a)f(gro)o(wing)g(factor.)904 2838 y(87)p eop %%Page: 88 100 99 bop 450 275 a FC(88)973 b Fr(APPENDIX)15 b(C.)27 b(GLOSSAR)m(Y)604 391 y Fu(\017)20 b FC(Smo)q(oth:)15 b(the)d(measure)f(of)f(the)i(error)g (decreases)i(in)c(all)g(or)h(most)f(iterations,)h(though)645 441 y(not)j(necessarily)h(b)o(y)f(the)g(same)f(factor.)604 514 y Fu(\017)20 b FC(Irregular:)c(the)10 b(measure)g(of)f(the)h(error)g (decreases)i(in)d(some)g(iterations)g(and)h(increases)645 564 y(in)16 b(others.)26 b(This)17 b(observ)n(ation)f(unfortunately)g(do)q(es)h (not)f(imply)e(an)o(ything)h(ab)q(out)645 613 y(the)g(ultimate)d(con)o(v)o (ergence)j(of)e(the)i(metho)q(d.)604 686 y Fu(\017)20 b FC(Stalled:)d(the)c (measure)g(of)e(the)j(error)f(sta)o(ys)g(more)f(or)g(less)h(constan)o(t)g (during)g(a)f(n)o(um-)645 736 y(b)q(er)f(of)f(iterations.)17 b(As)11 b(ab)q(o)o(v)o(e,)g(this)f(do)q(es)h(not)g(imply)d(an)o(ything)i(ab)q (out)g(the)h(ultimate)645 786 y(con)o(v)o(ergence)16 b(of)d(the)h(metho)q(d.) 450 881 y Fs(Dense)h(matrix)k FC(Matrix)12 b(for)g(whic)o(h)h(the)g(n)o(um)o (b)q(er)f(of)g(zero)h(elemen)o(ts)g(is)f(to)q(o)h(small)d(to)i(w)o(arran)o(t) 554 931 y(sp)q(ecialized)i(algorithms)e(to)i(exploit)f(these)i(zeros.)450 1026 y Fs(Diagonally)e(dominan)o(t)g(matrix)19 b FC(See:)g(Matrix)14 b(prop)q(erties)450 1122 y Fs(Direct)g(metho)q(d)19 b FC(An)13 b(algorithm)e(that)i(pro)q(duces)h(the)f(solution)g(to)f(a)h(system)g(of)f (linear)g(equa-)554 1172 y(tions)17 b(in)g(a)g(n)o(um)o(b)q(er)f(of)h(op)q (erations)h(that)f(is)g(determined)g(a)g(priori)g(b)o(y)g(the)h(size)g(of)f (the)554 1222 y(system.)33 b(In)19 b(exact)h(arithmetic,)e(a)h(direct)h (metho)q(d)e(yields)h(the)h(true)g(solution)e(to)h(the)554 1271 y(system.)f(See:)h(Iterativ)o(e)14 b(metho)q(d.)450 1367 y Fs(Distribu)o(ted)e(memory)20 b FC(See:)f(P)o(arallel)13 b(computer.)450 1462 y Fs(Div)o(ergence)18 b FC(An)12 b(iterativ)o(e)f(metho) q(d)g(is)g(said)g(to)g(div)o(erge)h(if)e(it)h(do)q(es)h(not)g(con)o(v)o(erge) g(in)f(a)g(reason-)554 1512 y(able)k(n)o(um)o(b)q(er)h(of)f(iterations,)h(or) f(if)g(some)g(measure)h(of)f(the)i(error)g(gro)o(ws)e(unacceptably)m(.)554 1562 y(Ho)o(w)o(ev)o(er,)i(gro)o(wth)g(of)f(the)h(error)h(as)f(suc)o(h)g(is)g (no)g(sign)f(of)g(div)o(ergence:)25 b(a)17 b(metho)q(d)f(with)554 1612 y(irregular)f(con)o(v)o(ergence)i(b)q(eha)o(vior)e(ma)o(y)e(ultimately)g (con)o(v)o(erge,)j(ev)o(en)f(though)g(the)h(error)554 1662 y(gro)o(ws)e(during)f(some)g(iterations.)450 1757 y Fs(Domain)h(decomp)q (osition)e(metho)q(d)20 b FC(Solution)d(metho)q(d)h(for)g(linear)h(systems)g (based)g(on)f(a)554 1807 y(partitioning)12 b(of)g(the)i(ph)o(ysical)f(domain) e(of)h(the)i(di\013eren)o(tial)f(equation.)18 b(Domain)10 b(decom-)554 1857 y(p)q(osition)17 b(metho)q(ds)h(t)o(ypically)f(in)o(v)o(olv)o(e)g(\(rep) q(eated\))j(indep)q(enden)o(t)g(system)e(solution)f(on)554 1906 y(the)e(sub)q(domains,)f(and)g(some)g(w)o(a)o(y)g(of)h(com)o(bining)d (data)j(from)e(the)i(sub)q(domains)f(on)h(the)554 1956 y(separator)f(part)g (of)g(the)g(domain.)450 2052 y Fs(Field)g(of)h(v)m(alues)20 b FC(Giv)o(en)15 b(a)g(matrix)f Fy(A)p FC(,)h(the)h(\014eld)g(of)f(v)n(alues) g(is)g(the)h(set)h Fu(f)p Fy(x)1692 2037 y Fx(T)1717 2052 y Fy(Ax)d FC(:)g Fy(x)1836 2037 y Fx(T)1862 2052 y Fy(x)g FC(=)g(1)p Fu(g)p FC(.)554 2102 y(F)m(or)f(symmetric)f(matrices)i(this)g(is)f(the)i (range)f([)p Fy(\025)1340 2108 y Fv(min)1397 2102 y FC(\()p Fy(A)p FC(\))p Fy(;)7 b(\025)1503 2108 y Fv(max)1566 2102 y FC(\()p Fy(A)p FC(\)].)450 2197 y Fs(Fill)19 b FC(A)13 b(p)q(osition)g(that)g (is)h(zero)g(in)f(the)h(original)e(matrix)f Fy(A)j FC(but)f(not)h(in)f(an)g (exact)h(factorization)554 2247 y(of)f Fy(A)p FC(.)18 b(In)c(an)g(incomplete) f(factorization,)f(some)h(\014ll)g(elemen)o(ts)h(are)g(discarded.)450 2342 y Fs(F)l(orw)o(ard)h(error)20 b FC(The)15 b(di\013erence)j(b)q(et)o(w)o 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Fs(P)o(ositiv)o(e)f(de\014nite)k FC(if)c(it)h(satis\014es)h Fy(x)867 2094 y Fx(T)893 2109 y Fy(Ax)c(>)h FC(0)i(for)f(all)g(nonzero)h(v)o(ectors)h Fy(x)p FC(.)254 2176 y Fs(Diagonally)e(dominan)o(t)18 b FC(if)j Fy(a)778 2182 y Fx(i;i)838 2176 y Fy(>)895 2145 y Ft(P)938 2188 y Fx(j)r Fw(6)p Fv(=)p Fx(i)1000 2176 y Fu(j)p Fy(a)1034 2182 y Fx(i;j)1073 2176 y Fu(j)p FC(;)j(the)f(excess)g(amoun)o(t)d(min)1561 2182 y Fx(i)1574 2176 y Fu(f)p Fy(a)1617 2182 y Fx(i;i)1667 2176 y Fu(\000)345 2194 y Ft(P)389 2238 y Fx(j)r Fw(6)p Fv(=)p Fx(i)451 2226 y Fu(j)p Fy(a)485 2232 y Fx(i;j)523 2226 y Fu(jg)13 b FC(is)h(called)g(the)g(diagonal)e(dominance)h(of)g(the)h(matrix.)254 2298 y Fs(An)h Fy(M)5 b Fs(-matrix)19 b FC(if)14 b Fy(a)614 2304 y Fx(i;j)665 2298 y Fu(\024)f FC(0)h(for)g Fy(i)f Fu(6)p FC(=)f Fy(j)r FC(,)j(and)f(it)g(is)g(nonsingular)f(with)i(\()p Fy(A)1458 2283 y Fw(\000)p Fv(1)1502 2298 y FC(\))1518 2304 y Fx(i;j)1570 2298 y Fu(\025)d FC(0)j(for)345 2348 y(all)e Fy(i)p FC(,)g Fy(j)r FC(.)150 2432 y Fs(Message)j(passing)j FC(See:)f(P)o(arallel)13 b(computer.)150 2516 y Fs(Multigrid)f(metho)q(d)19 b FC(Solution)13 b(metho)q(d)g(for)g(linear)h(systems)g(based)g(on)g (restricting)g(and)g(ex-)254 2565 y(trap)q(olating)f(solutions)g(b)q(et)o(w)o (een)i(a)f(series)h(of)e(nested)j(grids.)150 2649 y Fs(Mo)q(di\014ed)e (incomplete)f(factorization)18 b FC(See:)g(Incomplete)c(factorization.)150 2733 y Fs(Natural)h(ordering)j FC(See:)h(Ordering)14 b(of)f(unkno)o(wns.)p eop %%Page: 90 102 101 bop 450 275 a FC(90)973 b Fr(APPENDIX)15 b(C.)27 b(GLOSSAR)m(Y)450 391 y Fs(Nonstationary)14 b(iterativ)o(e)f(metho)q(d)19 b FC(Iterativ)o(e)i (metho)q(d)f(that)h(has)g(iteration-dep)q(enden)o(t)554 441 y(co)q(e\016cien)o(ts.)450 531 y Fs(Normal)15 b(equations)k FC(F)m(or)g(a)f(nonsymmetric)g(or)h(inde\014nite)h(\(but)f(nonsingular\))g (system)g(of)554 581 y(equations)c Fy(Ax)e FC(=)h Fy(b)p FC(,)h(either)h(of)e (the)i(related)g(symmetric)d(systems)i(\()p Fy(A)1685 566 y Fx(T)1712 581 y Fy(Ax)e FC(=)h Fy(A)1857 566 y Fx(T)1883 581 y Fy(b)p FC(\))h(and)554 631 y(\()p Fy(AA)632 616 y Fx(T)658 631 y Fy(y)j FC(=)d Fy(b)p FC(;)i Fy(x)f FC(=)f Fy(A)908 616 y Fx(T)935 631 y Fy(y)q FC(\).)26 b(F)m(or)16 b(complex)f Fy(A)p FC(,)h Fy(A)1342 616 y Fx(T)1385 631 y FC(is)g(replaced)h(with)f Fy(A)1723 616 y Fx(H)1771 631 y FC(in)g(the)g(ab)q(o)o(v)o(e)554 681 y(expressions.)450 771 y Fs(Norms)k FC(A)14 b(function)g Fy(f)i FC(:)11 b Fy(R)909 756 y Fx(n)943 771 y Fu(!)g Fy(R)j FC(is)g(called)f(a)h(v)o(ector)g(norm)f(if)604 861 y Fu(\017)20 b Fy(f)t FC(\()p Fy(x)p FC(\))12 b Fu(\025)g FC(0)i(for)f(all)g Fy(x)p FC(,)g(and)h Fy(f)t FC(\()p Fy(x)p FC(\))e(=)g(0)i(only)f(if)g Fy(x)e FC(=)h(0.)604 931 y Fu(\017)20 b Fy(f)t FC(\()p Fy(\013x)p FC(\))12 b(=)g Fu(j)p Fy(\013)p Fu(j)p Fy(f)t FC(\()p Fy(x)p FC(\))h(for)h(all)f Fy(\013)p FC(,)g Fy(x)p FC(.)604 1001 y Fu(\017)20 b Fy(f)t FC(\()p Fy(x)10 b FC(+)f Fy(y)q FC(\))k Fu(\024)f Fy(f)t FC(\()p Fy(x)p FC(\))e(+)f Fy(f)t FC(\()p Fy(y)q FC(\))16 b(for)e(all)e Fy(x)p FC(,)h Fy(y)q FC(.)554 1092 y(The)h(same)g(prop)q(erties)i(hold)d(for)h(matrix)f(norms.)18 b(A)c(matrix)f(norm)f(and)j(a)f(v)o(ector)h(norm)554 1141 y(\(b)q(oth)i (denoted)g Fu(k)11 b(\001)g(k)p FC(\))17 b(are)g(called)f(a)h(m)o(utually)d (consisten)o(t)k(pair)e(if)g(for)g(all)g(matrices)g Fy(A)554 1191 y FC(and)d(v)o(ectors)j Fy(x)658 1298 y Fu(k)p Fy(Ax)p Fu(k)10 b(\024)i(k)p Fy(A)p Fu(k)7 b(k)p Fy(x)p Fu(k)p Fy(:)450 1405 y Fs(Ordering)13 b(of)j(unkno)o(wns)i FC(F)m(or)c(linear)g(systems)g (deriv)o(ed)h(from)d(a)i(partial)f(di\013eren)o(tial)h(equa-)554 1455 y(tion,)f(eac)o(h)i(unkno)o(wn)f(corresp)q(onds)j(to)d(a)g(no)q(de)h(in) f(the)h(discretization)g(mesh.)k(Di\013eren)o(t)554 1505 y(orderings)13 b(of)g(the)g(unkno)o(wns)g(corresp)q(ond)i(to)e(p)q(erm)o(utations)f(of)g (the)i(co)q(e\016cien)o(t)g(matrix.)554 1554 y(The)i(con)o(v)o(ergence)h(sp)q (eed)g(of)e(iterativ)o(e)g(metho)q(ds)g(ma)o(y)f(dep)q(end)j(on)e(the)h (ordering)g(used,)554 1604 y(and)10 b(often)g(the)g(parallel)f(e\016ciency)i (of)f(a)g(metho)q(d)f(on)h(a)f(parallel)g(computer)h(is)g(strongly)g(de-)554 1654 y(p)q(enden)o(t)k(on)f(the)h(ordering)f(used.)18 b(Some)12 b(common)e(orderings)k(for)f(rectangular)g(domains)554 1704 y(are:)604 1794 y Fu(\017)20 b FC(The)25 b(natural)e(ordering;)29 b(this)24 b(is)g(the)h(consecutiv)o(e)g(n)o(um)o(b)q(ering)e(b)o(y)h(ro)o(ws) g(and)645 1844 y(columns.)604 1914 y Fu(\017)c FC(The)14 b(red/blac)o(k)f (ordering;)g(this)g(is)g(the)g(n)o(um)o(b)q(ering)f(where)i(all)e(no)q(des)i (with)e(co)q(ordi-)645 1964 y(nates)i(\()p Fy(i;)7 b(j)r FC(\))13 b(for)g(whic)o(h)g Fy(i)7 b FC(+)h Fy(j)16 b FC(is)c(o)q(dd)h(are)h(n)o(um)o (b)q(ered)e(b)q(efore)i(those)g(for)f(whic)o(h)f Fy(i)c FC(+)g Fy(j)645 2014 y FC(is)14 b(ev)o(en.)604 2084 y Fu(\017)20 b FC(The)14 b(ordering)f(b)o(y)h(diagonals;)d(this)j(is)f(the)h(ordering)f (where)i(no)q(des)f(are)g(group)q(ed)g(in)645 2133 y(lev)o(els)e(for)f(whic)o (h)g Fy(i)t FC(+)t Fy(j)k FC(is)d(constan)o(t.)18 b(All)10 b(no)q(des)i(in)f(one)h(lev)o(el)f(are)h(n)o(um)o(b)q(ered)f(b)q(efore)645 2183 y(the)k(no)q(des)f(in)g(the)g(next)h(lev)o(el.)554 2273 y(F)m(or)i(matrices)h(from)e(problems)h(on)h(less)g(regular)g(domains,)f (some)g(common)e(orderings)554 2323 y(are:)604 2413 y Fu(\017)20 b FC(The)c(Cuthill-McKee)g(ordering;)f(this)h(starts)g(from)e(one)h(p)q(oin)o (t,)g(then)h(n)o(um)o(b)q(ers)f(its)645 2463 y(neigh)o(b)q(ors,)j(and)g(con)o (tin)o(ues)g(n)o(um)o(b)q(ering)e(p)q(oin)o(ts)i(that)g(are)g(neigh)o(b)q (ors)g(of)f(already)645 2513 y(n)o(um)o(b)q(ered)g(p)q(oin)o(ts.)27 b(The)17 b(Rev)o(erse)h(Cuthill-McKee)g(ordering)f(then)g(rev)o(erses)j(the) 645 2563 y(n)o(um)o(b)q(ering;)h(this)f(ma)o(y)e(reduce)k(the)e(amoun)o(t)e (of)h(\014ll)g(in)h(a)f(factorization)g(of)h(the)645 2613 y(matrix.)604 2683 y Fu(\017)g FC(The)c(Minim)o(um)c(Degree)17 b(ordering;)f(this)g(orders) h(the)g(matrix)d(ro)o(ws)h(b)o(y)h(increasing)645 2733 y(n)o(um)o(b)q(ers)e (of)f(nonzeros.)p eop %%Page: 91 103 102 bop 1658 275 a FC(91)150 391 y Fs(P)o(arallel)13 b(computer)19 b FC(Computer)f(with)h(m)o(ultiple)e(indep)q(enden)o(t)j(pro)q(cessing)h (units.)34 b(If)19 b(the)254 441 y(pro)q(cessors)g(ha)o(v)o(e)d(immediate)e (access)19 b(to)d(the)i(same)e(memory)m(,)e(the)j(memory)d(is)j(said)f(to)254 491 y(b)q(e)f(shared;)g(if)f(pro)q(cessors)i(ha)o(v)o(e)e(priv)n(ate)h (memory)d(that)i(is)g(not)h(imm)o(ediately)d(visible)h(to)254 541 y(other)i(pro)q(cessors,)h(the)f(memory)d(is)i(said)g(to)g(b)q(e)h (distributed.)20 b(In)15 b(that)f(case,)h(pro)q(cessors)254 591 y(comm)o(unicate)c(b)o(y)j(message-passing.)150 678 y Fs(Pip)q(elini)o(n) o(g)k FC(See:)h(V)m(ector)c(computer.)150 765 y Fs(P)o(ositiv)o(e)e (de\014nite)g(matrix)19 b FC(See:)g(Matrix)14 b(prop)q(erties.)150 852 y Fs(Preconditi)o(oner)j FC(An)i(auxiliary)d(matrix)h(in)h(an)g(iterativ) o(e)h(metho)q(d)e(that)i(appro)o(ximates)e(in)254 902 y(some)d(sense)j(the)e (co)q(e\016cien)o(t)h(matrix)e(or)h(its)g(in)o(v)o(erse.)22 b(The)16 b(preconditioner,)f(or)g(precon-)254 951 y(ditioning)d(matrix,)g(is) h(applied)h(in)f(ev)o(ery)i(step)g(of)e(the)h(iterativ)o(e)g(metho)q(d.)150 1039 y Fs(Red/blac)o(k)g(ordering)k FC(See:)h(Ordering)14 b(of)g(unkno)o (wns.)150 1126 y Fs(Reduced)g(system)20 b FC(Linear)15 b(system)g(obtained)g (b)o(y)g(eliminating)d(certain)k(v)n(ariables)e(from)g(an-)254 1175 y(other)j(linear)f(system.)27 b(Although)16 b(the)h(n)o(um)o(b)q(er)f (of)g(v)n(ariables)g(is)g(smaller)g(than)g(for)h(the)254 1225 y(original)12 b(system,)h(the)i(matrix)d(of)h(a)h(reduced)i(system)d (generally)h(has)g(more)f(nonzero)i(en-)254 1275 y(tries.)i(If)10 b(the)g(original)f(matrix)f(w)o(as)i(symmetric)e(and)i(p)q(ositiv)o(e)f (de\014nite,)i(then)g(the)f(reduced)254 1325 y(system)j(has)h(a)g(smaller)e (condition)i(n)o(um)o(b)q(er.)150 1412 y Fs(Relaxed)h(incomplete)d (factorization)18 b FC(See:)h(Incomplete)13 b(factorization.)150 1499 y Fs(Residual)18 b FC(If)12 b(an)g(iterativ)o(e)h(metho)q(d)f(is)g (emplo)o(y)o(ed)f(to)h(solv)o(e)h(for)f Fy(x)g FC(in)g(a)g(linear)g(system)h Fy(Ax)e FC(=)h Fy(b)p FC(,)254 1549 y(then)i(the)h(residual)f(corresp)q (onding)h(to)e(a)h(v)o(ector)g Fy(y)i FC(is)e Fy(Ay)d Fu(\000)f Fy(b)p FC(.)150 1636 y Fs(Searc)o(h)15 b(direction)i FC(V)m(ector)e(that)f (is)f(used)i(to)f(up)q(date)g(an)g(iterate.)150 1723 y Fs(Shared)g(memory)20 b FC(See:)f(P)o(arallel)13 b(computer.)150 1810 y Fs(Sim)o(ultan)o(eous)f (displacemen)o(t)o(s,)h(metho)q(d)i(of)20 b FC(Jacobi)14 b(metho)q(d.)150 1897 y Fs(Sparse)h(matrix)k FC(Matrix)d(for)g(whic)o(h)g(the)h(n)o(um)o(b)q (er)e(of)h(zero)h(elemen)o(ts)f(is)g(large)g(enough)g(that)254 1947 y(algorithms)c(a)o(v)o(oiding)g(op)q(erations)j(on)f(zero)h(elemen)o(ts) f(pa)o(y)g(o\013.)19 b(Matrices)c(deriv)o(ed)g(from)254 1997 y(partial)g(di\013eren)o(tial)g(equations)h(t)o(ypically)e(ha)o(v)o(e)i(a)g (n)o(um)o(b)q(er)f(of)g(nonzero)h(elemen)o(ts)g(that)254 2047 y(is)e(prop)q(ortional)g(to)g(the)h(matrix)d(size,)j(while)f(the)h(total)f(n) o(um)o(b)q(er)g(of)f(matrix)g(elemen)o(ts)h(is)254 2097 y(the)g(square)h(of)e (the)i(matrix)d(size.)150 2184 y Fs(Sp)q(ectral)i(condition)f(n)o(um)o(b)q (er)18 b FC(The)c(pro)q(duct)358 2321 y Fu(k)p Fy(A)p Fu(k)431 2327 y Fv(2)458 2321 y Fu(\001)9 b(k)p Fy(A)531 2304 y Fw(\000)p Fv(1)575 2321 y Fu(k)596 2327 y Fv(2)626 2321 y FC(=)675 2293 y Fy(\025)699 2272 y Fv(1)p Fx(=)p Fv(2)699 2298 y(max)763 2293 y FC(\()p Fy(A)810 2278 y Fx(T)836 2293 y Fy(A)p FC(\))p 675 2312 209 2 v 678 2360 a Fy(\025)702 2338 y Fv(1)p Fx(=)p Fv(2)702 2372 y(min)759 2360 y FC(\()p Fy(A)806 2348 y Fx(T)833 2360 y Fy(A)p FC(\))888 2321 y Fy(;)254 2459 y FC(where)17 b Fy(\025)400 2465 y Fv(max)479 2459 y FC(and)f Fy(\025)586 2465 y Fv(min)660 2459 y FC(denote)g(the)h(largest)f(and)g(smallest)f(eigen)o (v)n(alues,)h(resp)q(ectiv)o(ely)m(.)254 2509 y(F)m(or)10 b(linear)g(systems) h(deriv)o(ed)g(from)e(partial)h(di\013eren)o(tial)g(equations)h(in)f(2D,)g (the)h(condition)254 2558 y(n)o(um)o(b)q(er)i(is)h(prop)q(ortional)f(to)g (the)i(n)o(um)o(b)q(er)e(of)g(unkno)o(wns.)150 2645 y Fs(Sp)q(ectral)h (radius)19 b FC(The)14 b(sp)q(ectral)h(radius)f(of)f(a)h(matrix)e Fy(A)i FC(is)g(max)n Fu(fj)p Fy(\025)p FC(\()p Fy(A)p FC(\))p Fu(jg)p FC(.)150 2733 y Fs(Sp)q(ectrum)19 b FC(The)14 b(set)h(of)e(all)g (eigen)o(v)n(alues)h(of)f(a)g(matrix.)p eop %%Page: 92 104 103 bop 450 275 a FC(92)973 b Fr(APPENDIX)15 b(C.)27 b(GLOSSAR)m(Y)450 391 y Fs(Stationary)13 b(iterativ)o(e)h(metho)q(d)19 b FC(Iterativ)o(e)13 b(metho)q(d)g(that)g(p)q(erforms)g(in)g(eac)o(h)h(iteration)f(the)554 441 y(same)g(op)q(erations)h(on)g(the)g(curren)o(t)h(iteration)f(v)o(ectors.) 450 523 y Fs(Stopping)f(criterion)k FC(Since)g(an)e(iterativ)o(e)h(metho)q(d) f(computes)h(successiv)o(e)i(appro)o(ximations)554 573 y(to)11 b(the)h(solution)f(of)f(a)i(linear)f(system,)g(a)g(practical)g(test)i(is)e (needed)i(to)e(determine)g(when)h(to)554 623 y(stop)g(the)g(iteration.)k (Ideally)11 b(this)h(test)g(w)o(ould)f(measure)g(the)h(distance)g(of)f(the)h (last)g(iterate)554 673 y(to)i(the)i(true)f(solution,)f(but)h(this)f(is)h (not)f(p)q(ossible.)21 b(Instead,)15 b(v)n(arious)f(other)h(metrics)g(are)554 723 y(used,)f(t)o(ypically)f(in)o(v)o(olving)e(the)k(residual.)450 805 y Fs(Storage)f(sc)o(heme)20 b FC(The)15 b(w)o(a)o(y)e(elemen)o(ts)i(of)e (a)h(matrix)f(are)i(stored)g(in)f(the)h(memory)d(of)i(a)g(com-)554 855 y(puter.)31 b(F)m(or)18 b(dense)h(matrices,)f(this)h(can)f(b)q(e)g(the)h (decision)f(to)g(store)h(ro)o(ws)f(or)g(columns)554 904 y(consecutiv)o(ely)m (.)k(F)m(or)14 b(sparse)i(matrices,)f(common)d(storage)j(sc)o(hemes)h(a)o(v)o (oid)e(storing)g(zero)554 954 y(elemen)o(ts;)j(as)f(a)g(result)h(they)f(in)o (v)o(olv)o(e)f(indices,)i(stored)g(as)f(in)o(teger)h(data,)f(that)g(indicate) 554 1004 y(where)f(the)f(stored)h(elemen)o(ts)f(\014t)g(in)o(to)f(the)i (global)d(matrix.)450 1086 y Fs(Successiv)o(e)i(displacemen)n(ts,)f(metho)q (d)h(of)21 b FC(Gauss-Seidel)14 b(metho)q(d.)450 1168 y Fs(Symmetric)g (matrix)19 b FC(See:)g(Matrix)14 b(prop)q(erties.)450 1251 y Fs(T)l(emplate)19 b FC(Description)c(of)f(an)g(algorithm,)d(abstracting)k (a)o(w)o(a)o(y)e(from)g(implemen)o(tati)o(onal)e(de-)554 1300 y(tails.)450 1383 y Fs(T)l(une)20 b FC(Adapt)15 b(soft)o(w)o(are)g(for)f(a)g (sp)q(eci\014c)j(application)c(and)h(computing)g(en)o(vironmen)o(t)f(in)h (order)554 1432 y(to)g(obtain)f(b)q(etter)i(p)q(erformance)f(in)f(that)h (case)h(only)m(.)i(itemize)450 1515 y Fs(Upp)q(er)e(triangul)o(ar)e(matrix)20 b FC(Matrix)13 b Fy(A)h FC(for)g(whic)o(h)f Fy(a)1370 1521 y Fx(i;j)1421 1515 y FC(=)f(0)h(if)g Fy(j)h(<)e(i)p FC(.)450 1597 y Fs(V)l(ector)j(computer)k FC(Computer)12 b(that)h(is)g(able)f(to)h (pro)q(cess)i(consecutiv)o(e)f(iden)o(tical)e(op)q(erations)554 1647 y(\(t)o(ypically)h(additions)h(or)g(m)o(ultiplications\))e(sev)o(eral)j (times)f(faster)h(than)f(in)o(termixed)g(op-)554 1696 y(erations)19 b(of)g(di\013eren)o(t)i(t)o(yp)q(es.)35 b(Pro)q(cessing)21 b(iden)o(tical)d(op)q(erations)i(this)g(w)o(a)o(y)e(is)h(called)554 1746 y(`pip)q(elining')11 b(the)k(op)q(erations.)450 1828 y Fs(V)l(ector)g(norms)20 b FC(See:)f(Norms.)450 1965 y Fp(C.1)69 b(Notation)450 2056 y FC(In)12 b(this)h(section,)g(w)o(e)f(presen)o(t)i(some) e(of)g(the)g(notation)g(w)o(e)h(use)g(throughout)f(the)h(b)q(o)q(ok.)k(W)m(e) 12 b(ha)o(v)o(e)450 2106 y(tried)i(to)g(use)h(standard)f(notation)f(that)h(w) o(ould)g(b)q(e)g(found)g(in)f(an)o(y)h(curren)o(t)h(publication)e(on)h(the) 450 2156 y(sub)r(jects)i(co)o(v)o(ered.)512 2206 y(Throughout,)e(w)o(e)g (follo)o(w)e(sev)o(eral)i(con)o(v)o(en)o(tions:)512 2287 y Fu(\017)21 b FC(Matrices)15 b(are)f(denoted)h(b)o(y)e(capital)g(letters.)512 2369 y Fu(\017)21 b FC(V)m(ectors)15 b(are)f(denoted)h(b)o(y)e(lo)o(w)o (ercase)i(letters.)512 2452 y Fu(\017)21 b FC(Lo)o(w)o(ercase)15 b(greek)f(letters)i(usually)d(denote)h(scalars.)512 2533 y(W)m(e)g(de\014ne)h (matrix)d Fy(A)i FC(of)f(dimension)f Fy(m)e Fu(\002)f Fy(n)14 b FC(as)g(follo)o(ws:)554 2671 y Fy(A)d FC(=)640 2575 y Ft(2)640 2648 y(6)640 2675 y(4)695 2607 y Fy(a)717 2613 y Fv(1)p Fx(;)p Fv(1)810 2607 y Fu(\001)c(\001)g(\001)46 b Fy(a)928 2613 y Fv(1)p Fx(;n)723 2651 y FC(.)723 2668 y(.)723 2684 y(.)936 2651 y(.)936 2668 y(.)936 2684 y(.)689 2734 y Fy(a)711 2740 y Fx(m;)p Fv(1)810 2734 y Fu(\001)7 b(\001)g(\001)40 b Fy(a)922 2740 y Fx(m;n)1005 2575 y Ft(3)1005 2648 y(7)1005 2675 y(5)1122 2671 y Fy(a)1144 2677 y Fx(i;j)1195 2671 y Fu(2)11 b(R)p Fy(:)p eop %%Page: 93 105 104 bop 150 275 a Fr(C.1.)31 b(NOT)m(A)m(TION)1176 b FC(93)212 391 y(W)m(e)14 b(de\014ne)h(v)o(ector)f Fy(x)g FC(of)f(dimension)f Fy(n)i FC(as)g(follo)o(ws:)254 528 y Fy(x)d FC(=)333 432 y Ft(2)333 505 y(6)333 531 y(4)383 464 y Fy(x)407 470 y Fv(1)399 508 y FC(.)399 524 y(.)399 541 y(.)381 591 y Fy(x)405 597 y Fx(n)448 432 y Ft(3)448 505 y(7)448 531 y(5)566 528 y Fy(x)590 534 y Fx(i)615 528 y 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y(enc)n(e)f(matric)n(es)p FC(,)e(SIAM)h(J.)g(Sci.)f(Statist.)h(Comput.,)d(12)j(\(1991\),)e(pp.)i (1373{1400.)471 847 y([17])19 b Fa(O.)i(Axelsson)h(and)f(I.)g(Gust)m(afsson)p FC(,)g Fq(Iter)n(ative)d(solution)h(for)g(the)g(solution)g(of)g(the)556 897 y(Navier)13 b(e)n(quations)g(of)h(elasticity)p FC(,)d(Comput.)f(Metho)q (ds)j(Appl.)e(Mec)o(h.)h(Engrg.,)g(15)f(\(1978\),)556 946 y(pp.)j(241{258.) 471 1032 y([18])19 b Fa(O.)f(Axelsson)g(and)h(G.)e(Lindsk)o(og)p FC(,)f Fq(On)h(the)f(eigenvalue)h(distribution)f(of)g(a)g(class)g(of)556 1082 y(pr)n(e)n(c)n(onditioning)f(matric)n(es)p FC(,)e(Numer.)g(Math.,)g(48)g (\(1986\),)g(pp.)g(479{498.)471 1167 y([19])p 556 1160 V 115 w(,)26 b Fq(On)f(the)g(r)n(ate)f(of)g(c)n(onver)n(genc)n(e)i(of)e(the)h(pr)n (e)n(c)n(onditione)n(d)g(c)n(onjugate)g(gr)n(adient)556 1217 y(metho)n(d)p FC(,)14 b(Numer.)f(Math.,)g(48)g(\(1986\),)g(pp.)g(499{523.)471 1302 y([20])19 b Fa(O.)24 b(Axelsson)i(and)f(B.)f(Polman)p FC(,)g 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y(line)n(ar)f(systems)p FC(,)g(SIAM)g(J.)f(Sci.)h(Statist.)f(Comput.,)f (13)h(\(1992\),)g(pp.)g(168{193.)171 1796 y([38])19 b Fa(C.)d(Brezinski)f (and)i(H.)f(Sadok)p FC(,)f Fq(A)o(voiding)g(br)n(e)n(akdown)g(in)g(the)f(CGS) i(algorithm)p FC(,)c(Nu-)256 1846 y(mer.)h(Alg.,)f(1)i(\(1991\),)e(pp.)i (199{206.)171 1936 y([39])19 b Fa(C.)k(Brezinski,)i(M.)e(Za)o(glia,)i(and)f (H.)f(Sadok)p FC(,)h Fq(A)o(voiding)d(br)n(e)n(akdown)g(and)h(ne)n(ar)256 1985 y(br)n(e)n(akdown)15 b(in)g(Lanczos)h(typ)n(e)f(algorithms)p FC(,)d(Numer.)h(Alg.,)g(1)g(\(1991\),)g(pp.)g(261{284.)171 2075 y([40])p 256 2068 V 115 w(,)i Fq(A)g(br)n(e)n(akdown)i(fr)n(e)n(e)e(L)n (anczos)i(typ)n(e)f(algorithm)f(for)h(solving)g(line)n(ar)f(systems)p FC(,)g(Nu-)256 2125 y(mer.)e(Math.,)g(63)g(\(1992\),)g(pp.)g(29{38.)171 2214 y([41])19 b Fa(W.)d(Briggs)p FC(,)c Fq(A)j(Multigrid)f(T)m(utorial)p FC(,)e(SIAM,)i(Philadelphia,)e(1977.)171 2304 y([42])19 b Fa(X.-C.)14 b(Cai)g(and)i(O.)e(Widlund)p FC(,)f 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Fa(T.)i(Chan,)i(E.)e(Gallopoulo)q(s,)k(V.)c(Simoncini,)i(T.)d(Szeto,)j(and) f(C.)f(Tong)p FC(,)e Fq(A)556 441 y(quasi-minimal)c(r)n(esidual)g(variant)h (of)f(the)h(Bi-CGST)m(AB)f(algorithm)g(for)g(nonsymmetric)556 491 y(systems)p FC(,)k(T)m(ec)o(h.)e(Rep)q(ort)i(CAM)f(92-26,)f(UCLA,)h (Dept.)g(of)g(Math.,)g(Los)g(Angeles,)h(CA)556 541 y(90024-1555,)11 b(1992.)17 b(SIAM)d(J.)g(Sci.)f(Comput.,)f(to)h(app)q(ear.)471 624 y([46])19 b Fa(T.)h(Chan,)j(R.)e(Glo)o(winski,)i(,)f(J.)e(P)1204 621 y(\023)1204 624 y(eria)o(ux,)j(and)f(O.)e(Widlund)p FC(,)g(eds.,)f Fq(Domain)556 674 y(De)n(c)n(omp)n(osition)e(Metho)n(ds)p FC(,)f (Philadelphia,)f(1989,)f(SIAM.)24 b(Pro)q(ceedings)17 b(of)e(the)h(Second)556 723 y(In)o(ternational)10 b(Symp)q(osium)e(on)j(Domain)d(Decomp)q(osition)h (Metho)q(ds,)j(Los)f(Angeles,)h(CA,)556 773 y(Jan)o(uary)i(14)f(-)h(16,)f (1988.)471 856 y([47])p 556 849 96 2 v 115 w(,)18 b(eds.,)g Fq(Domain)h(De)n(c)n(omp)n(osition)g(Metho)n(ds)p FC(,)g(Philadelphia,)e (1990,)g(SIAM.)29 b(Pro-)556 906 y(ceedings)11 b(of)e(the)i(Third)f(In)o (ternational)f(Symp)q(osium)e(on)i(Domain)f(Decomp)q(osition)g(Meth-)556 956 y(o)q(ds,)14 b(Houston,)f(TX,)h(1989.)471 1039 y([48])p 556 1032 V 115 w(,)k(eds.,)g Fq(Domain)i(De)n(c)n(omp)n(osition)f(Metho)n(ds) p FC(,)f(SIAM,)g(Philadelphia,)f(1991.)28 b(Pro-)556 1089 y(ceedings)23 b(of)f(the)g(Fourth)g(In)o(ternational)g(Symp)q(osium)d(on)j(Domain)d(Decomp) q(osition)556 1139 y(Metho)q(ds,)c(Mosco)o(w,)e(USSR,)g(1990.)471 1222 y([49])19 b Fa(T.)13 b(Chan)h(and)f(C.-C.)g(J.)g(Kuo)p FC(,)f Fq(Two-c)n(olor)f(Fourier)h(analysis)g(of)h(iter)n(ative)e(algorithms) 556 1271 y(for)k(el)r(liptic)f(pr)n(oblems)h(with)g(r)n(e)n(d/black)g(or)n (dering)p FC(,)f(SIAM)g(J.)h(Sci.)e(Statist.)i(Comput.,)d(11)556 1321 y(\(1990\),)h(pp.)g(767{793.)471 1404 y([50])19 b Fa(T.)14 b(F.)g(Chan,)h(T.)f(P.)g(Ma)m(thew,)h(and)g(J.)f(P.)g(Sha)o(o)p FC(,)f Fq(E\016cient)h(variants)f(of)g(the)g(vertex)556 1454 y(sp)n(ac)n(e)i(domain)f(de)n(c)n(omp)n(osition)h(algorithm)p FC(,)d(T)m(ec)o(h.)h(Rep)q(ort)g(CAM)g(92-07,)f(UCLA,)g(Dept.)556 1504 y(of)g(Math.,)h(Los)f(Angeles,)i(CA)f(90024-1555,)d(1992.)15 b(SIAM)f(J.)e(Sci.)h(Comput.,)d(to)j(app)q(ear.)471 1587 y([51])19 b Fa(T.)h(F.)h(Chan)g(and)g(J.)g(Sha)o(o)p FC(,)f Fq(On)g(the)f(choic)n(e)g (of)g(c)n(o)n(arse)g(grid)g(size)f(in)h(domain)h(de-)556 1637 y(c)n(omp)n(osition)e(metho)n(ds)p FC(,)g(T)m(ec)o(h.)g(Rep)q(ort,)g(UCLA,)f (Dept.)g(of)g(Math.,)g(Los)h(Angeles,)g(CA)556 1686 y(90024-1555,)11 b(1993.)17 b(to)d(app)q(ear.)471 1769 y([52])19 b Fa(D.)g(Chazan)g(and)g(W.)f (Miranker)p FC(,)f Fq(Chaotic)f(r)n(elaxation)p FC(,)g(Linear)g(Algebra)g (Appl.,)f(2)556 1819 y(\(1969\),)e(pp.)g(199{222.)471 1902 y([53])19 b Fa(A.)12 b(Chr)o(onopoulos)i(and)e(C.)g(Gear)p FC(,)f Fy(s)p Fq(-step)g(iter)n(ative)f(metho)n(ds)i(for)f(symmetric)f(line)n (ar)556 1952 y(systems)p FC(,)j(J.)h(Comput.)e(Appl.)h(Math.,)g(25)g (\(1989\),)g(pp.)h(153{168.)471 2035 y([54])19 b Fa(P.)d(Concus)h(and)g(G.)g (Golub)p FC(,)f Fq(A)f(gener)n(alize)n(d)g(c)n(onjugate)h(gr)n(adient)f (metho)n(d)h(for)f(non-)556 2085 y(symmetric)c(systems)g(of)h(line)n(ar)f(e)n (quations)p FC(,)g(in)e(Computer)h(metho)q(ds)g(in)f(Applied)h(Sciences)556 2135 y(and)18 b(Engineering,)h(Second)g(In)o(ternational)f(Symp)q(osium,)d (Dec)k(15{19,)f(1975;)g(Lecture)556 2185 y(Notes)i(in)e(Economics)h(and)f (Mathematical)g(Systems,)h(V)m(ol.)f(134,)h(Berlin,)h(New)f(Y)m(ork,)556 2234 y(1976,)12 b(Springer-V)m(erlag.)471 2317 y([55])19 b Fa(P.)c(Concus,)g(G.)g(Golub,)i(and)f(G.)f(Meurant)p FC(,)e Fq(Blo)n(ck)h(pr)n(e)n(c)n(onditioning)h(for)e(the)h(c)n(on-)556 2367 y(jugate)h(gr)n(adient)g(metho)n(d)p FC(,)f(SIAM)g(J.)f(Sci.)h(Statist.) f(Comput.,)f(6)h(\(1985\),)g(pp.)h(220{252.)471 2450 y([56])19 b Fa(P.)g(Concus,)i(G.)e(Golub,)j(and)e(D.)f(O'Lear)m(y)p FC(,)h Fq(A)d(gener)n(alize)n(d)h(c)n(onjugate)g(gr)n(adient)556 2500 y(metho)n(d)j(for)f(the)h(numeric)n(al)g(solution)g(of)f(el)r(liptic)g(p)n (artial)g(di\013er)n(ential)h(e)n(quations)p FC(,)h(in)556 2550 y(Sparse)13 b(Matrix)e(Computations,)f(J.)i(Bunc)o(h)h(and)f(D.)f(Rose,) h(eds.,)g(Academic)f(Press,)j(New)556 2600 y(Y)m(ork,)f(1976,)f(pp.)i (309{332.)471 2683 y([57])19 b Fa(E.)d(Cuthill)g(and)g(J.)g(McKee)p FC(,)d Fq(R)n(e)n(ducing)j(the)f(b)n(andwidth)g(of)f(sp)n(arse)h(symmetric)f (ma-)556 2733 y(tric)n(es)p FC(,)f(in)g(A)o(CM)h(Pro)q(ceedings)h(of)e(the)i (24th)f(National)e(Conference,)j(1969.)p eop %%Page: 99 111 110 bop 150 275 a Fr(BIBLIOGRAPHY)1177 b FC(99)171 391 y([58])19 b Fa(E.)f(D'Azevedo,)j(V.)d(Eijkhout,)h(and)h(C.)e(R)o(omine)p FC(,)f Fq(LAP)m(A)o(CK)f(working)h(note)g(56:)256 441 y(R)n(e)n(ducing)c(c)n (ommunic)n(ation)g(c)n(osts)f(in)h(the)f(c)n(onjugate)h(gr)n(adient)f (algorithm)g(on)h(distribute)n(d)256 491 y(memory)k(multipr)n(o)n(c)n(essor)p FC(,)e(T)m(ec)o(h.)h(Rep)q(ort,)h(Computer)e(Science)j(Departmen)o(t,)d(Univ) o(er-)256 541 y(sit)o(y)f(of)f(T)m(ennessee,)j(Kno)o(xville,)c(TN,)h(1993.) 171 626 y([59])19 b Fa(E.)i(D'Azevedo)h(and)f(C.)g(R)o(omine)p FC(,)f Fq(R)n(e)n(ducing)g(c)n(ommunic)n(ation)g(c)n(osts)f(in)h(the)f(c)n (on-)256 676 y(jugate)e(gr)n(adient)f(algorithm)g(on)g(distribute)n(d)g (memory)g(multipr)n(o)n(c)n(essors)p FC(,)f(T)m(ec)o(h.)g(Rep)q(ort)256 726 y(ORNL/TM-12192,)d(Oak)i(Ridge)f(National)f(Lab,)h(Oak)h(Ridge,)f(TN,)g (1992.)171 811 y([60])19 b Fa(E.)g(de)g(Sturler)p FC(,)e Fq(A)h(p)n(ar)n(al)r (lel)e(r)n(estructur)n(e)n(d)g(version)i(of)f(GMRES\(m\))p FC(,)h(T)m(ec)o(h.)f(Rep)q(ort)256 861 y(91-85,)12 b(Delft)i(Univ)o(ersit)o (y)f(of)h(T)m(ec)o(hnology)m(,)e(Delft,)h(The)h(Netherlands,)h(1991.)171 946 y([61])k Fa(E.)e(de)f(Sturler)i(and)f(D.)g(R.)g(F)o(okkema)p FC(,)f Fq(Neste)n(d)f(Krylov)g(metho)n(ds)h(and)h(pr)n(eserving)256 996 y(the)g(ortho)n(gonality)p FC(,)e(T)m(ec)o(h.)g(Rep)q(ort)h(Preprin)o(t)h 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FC(,)f(Appl.)256 1587 y(Numer.)f(Math.,)g(7)g(\(1991\),)g(pp.)h(417{436.)171 1672 y([66])19 b Fa(J.)e(Dongarra,)j(J.)d(DuCr)o(oz,)h(I.)f(Duff,)j(and)e(S.) f(Hammarling)p FC(,)f Fq(A)g(set)g(of)g(level)g(3)256 1722 y(Basic)11 b(Line)n(ar)g(Algebr)n(a)g(Subpr)n(o)n(gr)n(ams)p FC(,)f(A)o(CM)g(T)m(rans.)f(Math.)g(Soft.,)h(16)f(\(1990\),)g(pp.)h(1{17.)171 1807 y([67])19 b Fa(J.)d(Dongarra,)i(J.)e(DuCr)o(oz,)h(S.)g(Hammarling,)g (and)g(R.)g(Hanson)p FC(,)e Fq(A)o(n)g(extende)n(d)256 1857 y(set)g(of)g(F)o(OR)m(TRAN)f(Basic)h(Line)n(ar)g(Algebr)n(a)f(Subpr)n(o)n(gr) n(ams)p FC(,)f(A)o(CM)h(T)m(rans.)f(Math.)h(Soft.,)256 1907 y(14)f(\(1988\),)g(pp.)h(1{32.)171 1992 y([68])19 b Fa(J.)i(Dongarra,)k(I.)d (Duff,)i(D.)e(Sorensen,)j(and)d(H.)g(v)l(an)h(der)e(V)o(orst)p FC(,)f Fq(Solving)256 2042 y(Line)n(ar)14 b(Systems)g(on)g(V)m(e)n(ctor)f (and)i(Shar)n(e)n(d)f(Memory)g(Computers)p FC(,)e(SIAM,)g(Philadelphia,)256 2092 y(P)m(A,)h(1991.)171 2177 y([69])19 b Fa(J.)14 b(Dongarra)j(and)e(E.)g (Gr)o(osse)p FC(,)e Fq(Distribution)h(of)f(mathematic)n(al)h(softwar)n(e)f (via)h(ele)n(c-)256 2227 y(tr)n(onic)g(mail)p FC(,)f(Comm.)e(A)o(CM,)i(30)g (\(1987\),)g(pp.)h(403{407.)171 2313 y([70])19 b Fa(J.)f(Dongarra,)j(C.)e (Moler,)h(J.)e(Bunch,)i(and)g(G.)f(Stew)l(ar)m(t)p FC(,)e Fq(LINP)m(A)o(CK)f (Users')256 2363 y(Guide)p FC(,)e(SIAM,)f(Philadelphia,)f(1979.)171 2448 y([71])19 b Fa(J.)h(Dongarra)i(and)f(H.)f(V)-5 b(an)21 b(der)g(V)o(orst)p FC(,)d Fq(Performanc)n(e)h(of)f(various)h(c)n(omputers)256 2498 y(using)f(standar)n(d)f(sp)n(arse)g(line)n(ar)g(e)n(quations)g(solving)h (te)n(chniques)p FC(,)f(in)f(Computer)f(Benc)o(h-)256 2548 y(marks,)j(J.)f(Dongarra)h(and)g(W.)f(Gen)o(tzsc)o(h,)j(eds.,)f(Elsevier)g (Science)g(Publishers)g(B.V.,)256 2597 y(New)14 b(Y)m(ork,)f(1993,)g(pp.)g (177{188.)171 2683 y([72])19 b Fa(F.)g(Dorr)p FC(,)f Fq(The)g(dir)n(e)n(ct)f (solution)h(of)f(the)h(discr)n(ete)f(Poisson)i(e)n(quation)f(on)g(a)g(r)n(e)n (ctangle)p FC(,)256 2733 y(SIAM)c(Rev.,)f(12)g(\(1970\),)g(pp.)g(248{263.)p eop %%Page: 100 112 111 bop 450 275 a FC(100)1154 b Fr(BIBLIOGRAPHY)471 391 y FC([73])19 b Fa(M.)f(Dr)m(yja)h(and)g(O.)f(B.)f(Widlund)p FC(,)g Fq(T)m(owar)n(ds)e(a)i (uni\014e)n(d)h(the)n(ory)e(of)h(domain)g(de)n(c)n(om-)556 441 y(p)n(osition)f(algorithms)f(for)g(el)r(liptic)g(pr)n(oblems)p FC(,)f(T)m(ec)o(h.)h(Rep)q(ort)g(486,)f(also)h(Ultracomputer)556 491 y(Note)f(167,)f(Departmen)o(t)g(of)h(Computer)f(Science,)i(Couran)o(t)e (Institute,)i(1989.)471 580 y([74])k Fa(D.)g(Dubois,)h(A.)e(Greenba)o(um,)j (and)e(G.)g(R)o(odrigue)p FC(,)d Fq(Appr)n(oximating)h(the)g(inverse)556 630 y(of)f(a)g(matrix)f(for)g(use)h(in)g(iter)n(ative)f(algorithms)g(on)h(ve) n(ctor)f(pr)n(o)n(c)n(essors)p FC(,)f(Computing,)f(22)556 680 y(\(1979\),)g(pp.)g(257{268.)471 768 y([75])19 b Fa(I.)c(Duff,)h(R.)f (Grimes,)f(and)i(J.)e(Lewis)p FC(,)e Fq(Sp)n(arse)i(matrix)f(test)g(pr)n (oblems)p FC(,)f(A)o(CM)h(T)m(rans.)556 818 y(Math.)g(Soft.,)g(15)g (\(1989\),)g(pp.)h(1{14.)471 907 y([76])19 b Fa(I.)c(Duff)i(and)f(G.)g (Meurant)p FC(,)e Fq(The)g(e\013e)n(ct)h(of)g(or)n(dering)f(on)h(pr)n(e)n(c)n (onditione)n(d)g(c)n(onjugate)556 957 y(gr)n(adients)p FC(,)e(BIT,)h(29)f (\(1989\),)g(pp.)h(635{657.)471 1046 y([77])19 b Fa(I.)c(S.)g(Duff,)i(A.)e (M.)g(Erisman,)h(and)f(J.K.Reid)p FC(,)e Fq(Dir)n(e)n(ct)g(metho)n(ds)i(for)e (sp)n(arse)h(matri-)556 1096 y(c)n(es)p FC(,)f(Oxford)h(Univ)o(ersit)o(y)g (Press,)h(London,)e(1986.)471 1185 y([78])19 b Fa(T.)d(Dupont,)j(R.)e(Kend)o (all,)i(and)e(H.)f(Ra)o(chf)o(ord)p FC(,)g Fq(A)o(n)g(appr)n(oximate)g (factorization)556 1235 y(pr)n(o)n(c)n(e)n(dur)n(e)h(for)f(solving)h (self-adjoint)g(el)r(liptic)f(di\013er)n(enc)n(e)i(e)n(quations)p FC(,)f(SIAM)f(J.)g(Numer.)556 1284 y(Anal.,)c(5)i(\(1968\),)f(pp.)g(559{573.) 471 1373 y([79])19 b Fa(E.)f(D'Y)l(ak)o(ono)o(v)q FC(,)h Fq(The)e(metho)n(d)g (of)g(variable)f(dir)n(e)n(ctions)g(in)h(solving)g(systems)g(of)g(\014nite) 556 1423 y(di\013er)n(enc)n(e)d(e)n(quations)p FC(,)f(So)o(viet)f(Math.)g (Dokl.,)f(2)h(\(1961\),)g(pp.)g(577{580.)i(TOM)f(138,)e(271{)556 1473 y(274.)471 1562 y([80])19 b Fa(L.)d(Ehrlich)p FC(,)d Fq(A)o(n)i(A)n (d-Ho)n(c)f(SOR)i(metho)n(d)p FC(,)d(J.)g(Comput.)f(Ph)o(ys.,)h(43)g (\(1981\),)g(pp.)g(31{45.)471 1651 y([81])19 b Fa(M.)j(Eiermann)g(and)g(R.)g (V)-5 b(ar)o(ga)p FC(,)21 b Fq(Is)e(the)h(optimal)g Fy(!)h Fq(b)n(est)f(for)f(the)h(SOR)g(iter)n(ation)556 1701 y(metho)n(d?)p FC(,)14 b(Linear)g(Algebra)g(Appl.,)e(182)h(\(1993\),)g(pp.)h(257{277.)471 1790 y([82])19 b Fa(V.)d(Eijkhout)p FC(,)d Fq(A)o(nalysis)h(of)h(p)n(ar)n(al) r(lel)f(inc)n(omplete)g(p)n(oint)h(factorizations)p FC(,)e(Linear)h(Alge-)556 1839 y(bra)g(Appl.,)f(154{156)f(\(1991\),)g(pp.)i(723{740.)471 1928 y([83])p 556 1921 96 2 v 115 w(,)j Fq(Bewar)n(e)g(of)h(unp)n(erturb)n(e) n(d)g(mo)n(di\014e)n(d)h(inc)n(omplete)e(p)n(oint)h(factorizations)p FC(,)f(in)g(Pro-)556 1978 y(ceedings)c(of)e(the)h(IMA)o(CS)g(In)o (ternational)e(Symp)q(osium)f(on)i(Iterativ)o(e)i(Metho)q(ds)f(in)f(Linear) 556 2028 y(Algebra,)i(Brussels,)j(Belgium,)11 b(R.)i(Beau)o(w)o(ens)i(and)f (P)m(.)f(de)h(Gro)q(en,)g(eds.,)g(1992.)471 2117 y([84])p 556 2110 V 115 w(,)20 b Fq(LAP)m(A)o(CK)g(working)g(note)g(50:)30 b(Distribute)n(d)20 b(sp)n(arse)g(data)h(structur)n(es)e(for)h(lin-)556 2167 y(e)n(ar)14 b(algebr)n(a)f(op)n(er)n(ations)p FC(,)g(T)m(ec)o(h.)g(Rep)q (ort)g(CS)g(92-169,)e(Computer)h(Science)i(Departmen)o(t,)556 2217 y(Univ)o(ersit)o(y)g(of)f(T)m(ennessee,)j(Kno)o(xville,)c(TN,)h(1992.) 471 2306 y([85])p 556 2299 V 115 w(,)k Fq(LAP)m(A)o(CK)g(working)h(note)g (51:)26 b(Qualitative)17 b(pr)n(op)n(erties)g(of)h(the)g(c)n(onjugate)h(gr)n (a-)556 2355 y(dient)h(and)h(L)n(anczos)g(metho)n(ds)f(in)g(a)g(matrix)g(fr)n (amework)p FC(,)f(T)m(ec)o(h.)g(Rep)q(ort)h(CS)f(92-170,)556 2405 y(Computer)13 b(Science)j(Departmen)o(t,)c(Univ)o(ersit)o(y)i(of)f(T)m (ennessee,)j(Kno)o(xville,)c(TN,)i(1992.)471 2494 y([86])19 b Fa(V.)e(Eijkhout)f(and)i(B.)e(Polman)p FC(,)g Fq(De)n(c)n(ay)g(r)n(ates)f (of)h(inverses)f(of)h(b)n(ande)n(d)g Fy(M)5 b Fq(-matric)n(es)556 2544 y(that)17 b(ar)n(e)f(ne)n(ar)h(to)g(To)n(eplitz)f(matric)n(es)p FC(,)f(Linear)h(Algebra)g(Appl.,)f(109)g(\(1988\),)h(pp.)f(247{)556 2594 y(277.)471 2683 y([87])k Fa(S.)g(Eisenst)m(a)m(t)p FC(,)e Fq(E\016cient)h(implementation)f(of)h(a)g(class)f(of)h(pr)n(e)n(c)n (onditione)n(d)g(c)n(onjugate)556 2733 y(gr)n(adient)d(metho)n(ds)p FC(,)f(SIAM)g(J.)f(Sci.)h(Statist.)f(Comput.,)f(2)h(\(1981\),)g(pp.)g(1{4.)p eop %%Page: 101 113 112 bop 150 275 a Fr(BIBLIOGRAPHY)1156 b FC(101)171 391 y([88])19 b Fa(R.)14 b(Elkin)p FC(,)f Fq(Conver)n(genc)n(e)g(the)n(or)n(ems)g(for)g (Gauss-Seidel)g(and)h(other)f(minimization)g(algo-)256 441 y(rithms)p FC(,)g(T)m(ec)o(h.)i(Rep)q(ort)g(68-59,)e(Computer)h(Science)i (Cen)o(ter,)f(Univ)o(ersit)o(y)f(of)g(Maryland,)256 491 y(College)f(P)o(ark,) h(MD,)f(Jan.)g(1968.)171 576 y([89])19 b Fa(H.)f(Elman)p FC(,)e Fq(Appr)n(oximate)h(Schur)g(c)n(omplement)f(pr)n(e)n(c)n(onditioners)h(on)g (serial)e(and)j(p)n(ar-)256 626 y(al)r(lel)c(c)n(omputers)p FC(,)g(SIAM)g(J.)f(Sci.)h(Statist.)f(Comput.,)f(10)h(\(1989\),)g(pp.)g (581{605.)171 712 y([90])19 b Fa(H.)h(Elman)g(and)g(M.)g(Schul)m(tz)p FC(,)e Fq(Pr)n(e)n(c)n(onditioning)h(by)f(fast)g(dir)n(e)n(ct)f(metho)n(ds)i (for)e(non)256 761 y(self-adjoint)h(nonsep)n(ar)n(able)g(el)r(liptic)f(e)n (quations)p FC(,)h(SIAM)g(J.)f(Numer.)f(Anal.,)h(23)f(\(1986\),)256 811 y(pp.)e(44{57.)171 897 y([91])19 b Fa(L.)24 b(Elsner)p FC(,)f Fq(A)e(note)h(on)f(optimal)g(blo)n(ck-sc)n(aling)h(of)f(matric)n(es)p FC(,)g(Numer.)f(Math.,)i(44)256 946 y(\(1984\),)13 b(pp.)g(127{128.)171 1032 y([92])19 b Fa(V.)h(F)-5 b(aber)20 b(and)h(T.)e(Manteuffel)p FC(,)i Fq(Ne)n(c)n(essary)d(and)h(su\016cient)f(c)n(onditions)h(for)f(the)256 1082 y(existenc)n(e)i(of)g(a)h(c)n(onjugate)f(gr)n(adient)g(metho)n(d)p FC(,)h(SIAM)e(J.)h(Numer.)e(Anal.,)h(21)g(\(1984\),)256 1131 y(pp.)14 b(315{339.)171 1217 y([93])19 b Fa(G.)e(F)-5 b(air)l(wea)m(ther,)17 b(A.)g(Gourla)m(y,)i(and)e(A.)f(Mitchell)p FC(,)e Fq(Some)i(high)g(ac)n(cur)n (acy)g(dif-)256 1267 y(fer)n(enc)n(e)g(schemes)i(with)e(a)g(splitting)g(op)n (er)n(ator)h(for)f(e)n(quations)h(of)g(p)n(ar)n(ab)n(olic)f(and)h(el)r (liptic)256 1316 y(typ)n(e)p FC(,)d(Numer.)e(Math.,)h(10)h(\(1967\),)e(pp.)i (56{66.)171 1402 y([94])19 b Fa(R.)c(Fletcher)p FC(,)e Fq(Conjugate)h(gr)n (adient)g(metho)n(ds)h(for)e(inde\014nite)i(systems)p FC(,)e(in)f(Numerical) 256 1452 y(Analysis)f(Dundee)h(1975,)f(G.)f(W)m(atson,)h(ed.,)g(Berlin,)h (New)g(Y)m(ork,)f(1976,)f(Springer)i(V)m(erlag,)256 1501 y(pp.)i(73{89.)171 1587 y([95])p 256 1580 96 2 v 115 w(,)19 b Fq(Conjugate)i(gr)n(adient)e (metho)n(ds)h(for)f(inde\014nite)h(systems)p FC(,)g(v)o(ol.)e(506)g(of)g (Lecture)256 1637 y(Notes)d(Math.,)e(Springer-V)m(erlag,)g(Berlin,)g(New)i(Y) m(ork,)e(1976,)f(pp.)i(73{89.)171 1722 y([96])19 b Fa(G.)j(F)o(orsythe)f(and) h(E.)f(Stra)o(uss)p FC(,)g Fq(On)f(b)n(est)f(c)n(onditione)n(d)i(matric)n(es) p FC(,)e(Pro)q(c.)g(Amer.)256 1772 y(Math.)13 b(So)q(c.,)h(6)f(\(1955\),)g (pp.)h(340{345.)171 1857 y([97])19 b Fa(R.)26 b(Freund)p FC(,)g Fq(Conjugate)e(gr)n(adient-typ)n(e)f(metho)n(ds)h(for)f(line)n(ar)g(systems)g (with)g(c)n(om-)256 1907 y(plex)17 b(symmetric)g(c)n(o)n(e\016cient)g(matric) n(es)p FC(,)e(SIAM)i(J.)f(Sci.)g(Statist.)g(Comput.,)f(13)g(\(1992\),)256 1957 y(pp.)f(425{448.)171 2042 y([98])19 b Fa(R.)f(Freund,)g(G.)f(Golub,)j (and)e(N.)f(Na)o(chtigal)p FC(,)d Fq(Iter)n(ative)i(solution)g(of)g(line)n (ar)f(sys-)256 2092 y(tems)p FC(,)e(T)m(ec)o(h.)h(Rep)q(ort)g(NA-91-05,)e (Stanford)i(Univ)o(ersit)o(y)m(,)f(Stanford,)g(CA,)h(1991.)171 2177 y([99])19 b Fa(R.)d(Freund,)g(M.)g(Gutknecht,)h(and)f(N.)g(Na)o(chtigal) p FC(,)d Fq(A)o(n)i(implementation)f(of)h(the)256 2227 y(lo)n(ok-ahe)n(ad)j (L)n(anczos)h(algorithm)d(for)h(non-Hermitian)h(matric)n(es)p FC(,)e(SIAM)h(J.)g(Sci.)f(Com-)256 2277 y(put.,)d(14)h(\(1993\),)e(pp.)i (137{158.)150 2363 y([100])19 b Fa(R.)f(Freund)h(and)f(N.)g(Na)o(chtigal)p FC(,)d Fq(QMR:)i(A)f(quasi-minimal)h(r)n(esidual)f(metho)n(d)h(for)256 2412 y(non-Hermitian)e(line)n(ar)f(systems)p FC(,)f(Numer.)g(Math.,)g(60)h (\(1991\),)e(pp.)i(315{339.)150 2498 y([101])p 256 2491 V 115 w(,)e Fq(A)o(n)i(implementation)g(of)g(the)f(QMR)i(metho)n(d)f(b)n(ase)n(d)g (on)g(c)n(ouple)n(d)h(two-term)d(r)n(e)n(cur-)256 2548 y(r)n(enc)n(es)p FC(,)h(T)m(ec)o(h.)h(Rep)q(ort)g(92.15,)e(RIA)o(CS,)h(NASA)h(Ames,)f(Ames,)g (CA,)h(1992.)150 2633 y([102])19 b Fa(R.)j(Freund)g(and)g(T.)f(Szeto)p FC(,)g Fq(A)f(quasi-minimal)f(r)n(esidual)h(squar)n(e)n(d)g(algorithm)f(for) 256 2683 y(non-Hermitian)d(line)n(ar)f(systems)p FC(,)f(T)m(ec)o(h.)h(Rep)q (ort,)g(RIA)o(CS,)e(NASA)j(Ames,)e(Ames,)g(CA,)256 2733 y(1991.)p eop %%Page: 102 114 113 bop 450 275 a FC(102)1154 b Fr(BIBLIOGRAPHY)450 391 y FC([103])19 b Fa(R.)h(W.)g(Freund)p FC(,)f Fq(A)g(tr)n(ansp)n(ose-fr)n(e)n(e)e 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y(phia,)15 b(1988.)23 b(Pro)q(ceedings)18 b(of)d(the)i(First)f(In)o (ternational)f(Symp)q(osium)e(on)j(Domain)d(De-)556 1007 y(comp)q(osition)g (Metho)q(ds)i(for)g(P)o(artial)e(Di\013eren)o(tial)h(Equations,)g(P)o(aris,)g (F)m(rance,)h(Jan)o(uary)556 1057 y(1987.)450 1146 y([107])k Fa(G.)14 b(Golub)i(and)e(D.)h(O'Lear)m(y)p FC(,)f Fq(Some)g(history)e(of)i (the)f(c)n(onjugate)h(gr)n(adient)f(and)h(L)n(anc-)556 1195 y(zos)h(metho)n(ds)p FC(,)f(SIAM)g(Rev.,)f(31)g(\(1989\),)g(pp.)g(50{102.)450 1284 y([108])19 b Fa(G.)13 b(Golub)g(and)g(C.)e(V)-5 b(an)13 b(Lo)o(an)p FC(,)f Fq(Matrix)f(Computations,)i FC(second)e(edition,)f(The)h (Johns)556 1334 y(Hopkins)j(Univ)o(ersit)o(y)g(Press,)h(Baltimore,)c(1989.) 450 1423 y([109])19 b Fa(A.)h(Greenba)o(um)h(and)f(Z.)g(Strak)o(os)p FC(,)f Fq(Pr)n(e)n(dicting)f(the)g(b)n(ehavior)g(of)g(\014nite)h(pr)n(e)n (cision)556 1473 y(Lanczos)h(and)g(c)n(onjugate)g(gr)n(adient)e(c)n (omputations)p FC(,)i(SIAM)f(J.)f(Mat.)g(Anal.)f(Appl.,)i(13)556 1523 y(\(1992\),)13 b(pp.)g(121{137.)450 1612 y([110])19 b Fa(W.)f(D.)h(Gr)o(opp)g(and)g(D.)f(E.)g(Keyes)p FC(,)f Fq(Domain)h(de)n(c)n (omp)n(osition)f(with)f(lo)n(c)n(al)h(mesh)g(r)n(e-)556 1662 y(\014nement)p FC(,)d(SIAM)g(J.)g(Sci.)g(Statist.)f(Comput.,)f(13)h (\(1992\),)g(pp.)g(967{993.)450 1751 y([111])19 b Fa(I.)i(Gust)m(afsson)p FC(,)g Fq(A)e(class)g(of)g(\014rst-or)n(der)g(factorization)f(metho)n(ds)p FC(,)i(BIT,)e(18)g(\(1978\),)556 1800 y(pp.)c(142{156.)450 1889 y([112])19 b Fa(M.)h(H.)g(Gutknecht)p FC(,)f Fq(A)f(c)n(omplete)n(d)h (the)n(ory)f(of)g(the)h(unsymmetric)f(Lanczos)i(pr)n(o)n(c)n(ess)556 1939 y(and)c(r)n(elate)n(d)e(algorithms,)g(p)n(art)g(II)p FC(,)g(T)m(ec)o(h.) f(Rep)q(ort)i(90-16,)d(IPS)i(Researc)o(h)h(Rep)q(ort,)f(ETH)556 1989 y(Z)q(\177)-22 b(uric)o(h,)14 b(Switzerland,)g(1990.)450 2078 y([113])p 556 2071 96 2 v 115 w(,)d Fq(The)i(unsymmetric)f(L)n(anczos)i (algorithms)e(and)h(their)f(r)n(elations)g(to)h(P\023)-21 b(ade)13 b(appr)n(oxi-)556 2128 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Fa(T.)h(C.)h(Oppe,)h(W.)f(D.)g(Jouber)m(t,)h(and)f(D.)g(R.) g(Kincaid)p FC(,)d Fq(NSPCG)i(user's)e(guide,)256 2633 y(version)12 b(1.0:)17 b(A)12 b(p)n(ackage)h(for)e(solving)h(lar)n(ge)g(sp)n(arse)f(line)n (ar)h(systems)f(by)h(various)g(iter)n(ative)256 2683 y(metho)n(ds)p FC(,)j(T)m(ec)o(h.)g(Rep)q(ort)h(CNA{216,)e(Cen)o(ter)i(for)f(Numerical)e (Analysis,)i(Univ)o(ersit)o(y)g(of)256 2733 y(T)m(exas)f(at)g(Austin,)f (Austin,)h(TX,)f(April)h(1988.)p eop %%Page: 106 118 117 bop 450 275 a FC(106)1154 b Fr(BIBLIOGRAPHY)450 391 y FC([162])19 b Fa(J.)14 b(M.)h(Or)m(tega)p FC(,)d Fq(Intr)n(o)n(duction)i(to)g(Par)n(al)r (lel)e(and)j(V)m(e)n(ctor)d(Solution)i(of)g(Line)n(ar)f(Systems)p FC(,)556 441 y(Plen)o(um)g(Press,)i(New)f(Y)m(ork)g(and)f(London,)g(1988.)450 530 y([163])19 b Fa(C.)d(P)l(aige,)f(B.)h(P)l(arlett,)g(and)h(H.)e(V)-5 b(an)17 b(der)e(V)o(orst)p FC(,)e Fq(Appr)n(oximate)i(solutions)f(and)556 580 y(eigenvalue)i(b)n(ounds)f(fr)n(om)g(Krylov)f(subsp)n(ac)n(es)p FC(,)g(Linear)g(Algebra)g(Appl.,)e(to)i(app)q(ear.)450 669 y([164])19 b Fa(C.)h(P)l(aige)g(and)g(M.)h(Sa)o(unders)p FC(,)f Fq(Solution)f(of)f(sp)n(arse)h(inde\014nite)g(systems)f(of)h(line)n(ar)556 719 y(e)n(quations)p FC(,)14 b(SIAM)g(J.)g(Numer.)f(Anal.,)f(12)h(\(1975\),)g (pp.)h(617{629.)450 808 y([165])19 b Fa(C.)i(C.)f(P)l(aige)g(and)i(M.)f(A.)f (Sa)o(unders)p FC(,)h Fq(LSQR:)f(An)f(algorithm)f(for)h(sp)n(arse)g(line)n (ar)556 857 y(e)n(quations)e(and)g(sp)n(arse)g(le)n(ast)f(squar)n(es)p FC(,)f(A)o(CM)h(T)m(rans.)f(Math.)g(Soft.,)g(8)g(\(1982\),)g(pp.)g(43{)556 907 y(71.)450 996 y([166])k Fa(G.)g(P)l(a)o(olini)g(and)g(G.)f(Radica)m(ti)g (di)f(Br)o(ozolo)p FC(,)h Fq(Data)f(structur)n(es)f(to)h(ve)n(ctorize)f(CG) 556 1046 y(algorithms)e(for)h(gener)n(al)f(sp)n(arsity)h(p)n(atterns)p FC(,)e(BIT,)g(29)h(\(1989\),)e(pp.)i(703{718.)450 1135 y([167])19 b Fa(B.)d(P)l(arlett)p FC(,)e Fq(The)g(symmetric)g(eigenvalue)i(pr)n(oblem)p FC(,)d(Pren)o(tice-Hall,)h(London,)f(1980.)450 1224 y([168])19 b Fa(B.)i(N.)f(P)l(arlett,)h(D.)g(R.)g(T)l(a)m(ylor,)i(and)e(Z.)g(A.)f(Liu)p FC(,)f Fq(A)g(lo)n(ok-ahe)n(ad)g(L)n(anczos)h(al-)556 1274 y(gorithm)g(for)h(unsymmetric)g(matric)n(es)p FC(,)f(Mathematics)g(of)g (Computation,)f(44)h(\(1985\),)556 1324 y(pp.)14 b(105{124.)450 1412 y([169])19 b Fa(D.)h(Pea)o(ceman)g(and)g(j.)g(H.H.)f(Ra)o(chf)o(ord)p FC(,)g Fq(The)f(numeric)n(al)g(solution)g(of)g(p)n(ar)n(ab)n(olic)556 1462 y(and)d(el)r(liptic)e(di\013er)n(ential)g(e)n(quations)p FC(,)g(J.)g(So)q(c.)f(Indust.)h(Appl.)f(Math.,)g(3)h(\(1955\),)f(pp.)g(28{) 556 1512 y(41.)450 1601 y([170])19 b Fa(C.)12 b(Pommerell)p FC(,)g Fq(Solution)g(of)f(lar)n(ge)g(unsymmetric)h(systems)f(of)h(line)n(ar)f (e)n(quations)p FC(,)g(PhD)556 1651 y(thesis,)j(Swiss)h(F)m(ederal)f (Institute)g(of)g(T)m(ec)o(hnology)m(,)e(Z)q(\177)-22 b(uric)o(h,)13 b(Switzerland,)h(1992.)450 1740 y([171])19 b Fa(E.)i(Poole)g(and)h(J.)e(Or)m (tega)p FC(,)g Fq(Multic)n(olor)e(ICCG)h(metho)n(ds)g(for)g(ve)n(ctor)g(c)n (omputers)p FC(,)556 1790 y(T)m(ec)o(h.)h(Rep)q(ort)h(RM)f(86-06,)g (Departmen)o(t)f(of)h(Applied)g(Mathematics,)h(Univ)o(ersit)o(y)f(of)556 1839 y(Virginia,)12 b(Charlottesville,)h(V)-5 b(A,)14 b(1986.)450 1928 y([172])19 b Fa(A.)c(Quar)m(ter)o(oni)p FC(,)f(ed.,)e Fq(Domain)j(De)n(c)n(omp)n(osition)f(Metho)n(ds,)g(Pr)n(o)n(c)n(e)n(e)n (dings)g(of)g(the)g(Sixth)556 1978 y(International)23 b(Symp)n(osium)h(on)g (Domain)g(De)n(c)n(omp)n(osition)g(Metho)n(ds,)h(Como,)g(Italy,)p FC(,)556 2028 y(Pro)o(vidence,)14 b(RI,)f(1993,)g(AMS.)18 b(to)c(app)q(ear.) 450 2117 y([173])19 b Fa(G.)e(Radica)m(ti)e(di)h(Br)o(ozolo)i(and)f(Y.)f(R)o (ober)m(t)p FC(,)e Fq(V)m(e)n(ctor)h(and)h(p)n(ar)n(al)r(lel)e(CG-like)g (algo-)556 2167 y(rithms)k(for)f(sp)n(arse)i(non-symmetric)f(systems)p FC(,)g(T)m(ec)o(h.)f(Rep)q(ort)h(681-M,)f(IMA)o(G/TIM3,)556 2217 y(Grenoble,)d(F)m(rance,)g(1987.)450 2306 y([174])19 b Fa(J.)f(Reid)p FC(,)d Fq(On)i(the)g(metho)n(d)g(of)g(c)n(onjugate)g(gr)n 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(metho)n(ds)g(on)h(sup)n(er)n(c)n(omputers)p FC(,)d(SIAM)h(J.)f(Sci.)h (Statist.)f(Com-)256 1022 y(put.,)f(10)h(\(1989\),)e(pp.)i(1200{1232.)150 1105 y([182])p 256 1098 V 115 w(,)e Fq(SP)m(ARSKIT:)i(A)f(b)n(asic)h(to)n(ol) f(kit)h(for)f(sp)n(arse)g(matrix)g(c)n(omputation)p FC(,)g(T)m(ec)o(h.)g(Rep) q(ort)256 1155 y(CSRD)g(TR)h(1029,)e(CSRD,)h(Univ)o(ersit)o(y)g(of)h (Illinois,)d(Urbana,)j(IL,)f(1990.)150 1238 y([183])p 256 1231 V 115 w(,)20 b Fq(A)f(\015exible)i(inner-outer)e(pr)n(e)n(c)n(onditione)n(d)h (GMRES)h(algorithm)p FC(,)f(SIAM)g(J.)f(Sci.)256 1288 y(Comput.,)12 b(14)h(\(1993\),)g(pp.)g(461{469.)150 1371 y([184])19 b Fa(Y.)14 b(Saad)i(and)e(M.)h(Schul)m(tz)p FC(,)e Fq(Conjugate)h(gr)n(adient-like)e (algorithms)h(for)g(solving)h(non-)256 1421 y(symmetric)d(line)n(ar)h (systems)p FC(,)f(Mathematics)f(of)g(Computation,)f(44)i(\(1985\),)f(pp.)g (417{424.)150 1504 y([185])p 256 1497 V 115 w(,)17 b Fq(GMRES:)h(A)f(gener)n 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Fa(C.)d(Tong)p FC(,)e Fq(A)g(c)n(omp)n(ar)n(ative)h(study)h (of)f(pr)n(e)n(c)n(onditione)n(d)g(L)n(anczos)h(metho)n(ds)f(for)f(nonsym-) 256 2683 y(metric)d(line)n(ar)f(systems)p FC(,)h(T)m(ec)o(h.)f(Rep)q(ort)g (SAND91-8240,)f(Sandia)g(Nat.)h(Lab.,)g(Liv)o(ermore,)256 2733 y(CA,)k(1992.)p eop %%Page: 108 120 119 bop 450 275 a FC(108)1154 b Fr(BIBLIOGRAPHY)450 391 y FC([193])19 b Fa(A.)h(v)l(an)h(der)e(Sluis)p FC(,)f Fq(Condition)h(numb)n(ers)f(and)h(e)n (quilibr)n(ation)e(of)h(matric)n(es)p FC(,)f(Numer.)556 441 y(Math.,)c(14)g(\(1969\),)g(pp.)h(14{23.)450 523 y([194])19 b Fa(A.)c(v)l(an)i(der)e(Sluis)h(and)f(H.)g(v)l(an)i(der)e(V)o(orst)p FC(,)e Fq(The)h(r)n(ate)g(of)g(c)n(onver)n(genc)n(e)h(of)f(c)n(onju-)556 573 y(gate)h(gr)n(adients)p FC(,)e(Numer.)g(Math.,)g(48)g(\(1986\),)g(pp.)h (543{560.)450 655 y([195])19 b Fa(H.)14 b(v)l(an)h(der)f(V)o(orst)p FC(,)d Fq(Iter)n(ative)i(solution)g(metho)n(ds)g(for)g(c)n(ertain)g(sp)n (arse)g(line)n(ar)f(systems)556 704 y(with)h(a)h(non-symmetric)g(matrix)g (arising)f(fr)n(om)g(PDE-pr)n(oblems)p FC(,)g(J.)f(Comput.)f(Ph)o(ys.,)i(44) 556 754 y(\(1981\),)g(pp.)g(1{19.)450 836 y([196])p 556 829 96 2 v 115 w(,)e Fq(A)h(ve)n(ctorizable)g(variant)h(of)f(some)h(ICCG)f(metho) n(ds)p FC(,)g(SIAM)f(J.)g(Sci.)g(Statist.)g(Com-)556 886 y(put.,)i(3)h (\(1982\),)f(pp.)g(350{356.)450 968 y([197])p 556 961 V 115 w(,)g Fq(L)n(ar)n(ge)i(tridiagonal)g(and)g(blo)n(ck)g(tridiagonal)g(line)n (ar)g(systems)f(on)i(ve)n(ctor)f(and)h(p)n(ar-)556 1018 y(al)r(lel)e(c)n (omputers)p FC(,)g(P)o(arallel)e(Comput.,)g(5)h(\(1987\),)g(pp.)h(45{54.)450 1099 y([198])p 556 1092 V 115 w(,)23 b Fq(\(M\)ICCG)g(for)f(2D)h(pr)n(oblems) f(on)h(ve)n(ctor)g(c)n(omputers)p FC(,)g(in)f(Sup)q(ercomputing,)556 1149 y(A.Lic)o(hnewsky)10 b(and)g(C.Saguez,)g(eds.,)g(North-Holland,)f(1988.) h(Also)g(as)g(Rep)q(ort)g(No.A-17,)556 1199 y(Data)j(Pro)q(cessing)j(Cen)o (ter,)e(Ky)o(oto)g(Univ)o(ersit)o(y)m(,)f(Ky)o(oto,)g(Japan,)g(Decem)o(b)q (er)h(17,)f(1986.)450 1281 y([199])p 556 1274 V 115 w(,)22 b Fq(High)f(p)n(erformanc)n(e)g(pr)n(e)n(c)n(onditioning)p FC(,)i(SIAM)e(J.)g(Sci.)g(Statist.)f(Comput.,)h(10)556 1331 y(\(1989\),)13 b(pp.)g(1174{1185.)450 1412 y([200])p 556 1405 V 115 w(,)j Fq(ICCG)h(and)i(r)n(elate)n(d)d(metho)n(ds)i(for)f(3D)h(pr)n (oblems)f(on)h(ve)n(ctor)f(c)n(omputers)p FC(,)g(Com-)556 1462 y(puter)g(Ph)o(ysics)f(Comm)o(unicatio)o(ns,)d(\(1989\),)i(pp.)h(223{235.)21 b(Also)16 b(as)g(Rep)q(ort)g(No.A-18,)556 1512 y(Data)d(Pro)q(cessing)j(Cen)o (ter,)e(Ky)o(oto)g(Univ)o(ersit)o(y)m(,)f(Ky)o(oto,)g(Japan,)g(Ma)o(y)g(30,)g (1987.)450 1594 y([201])p 556 1587 V 115 w(,)e Fq(The)j(c)n(onver)n(genc)n(e) f(b)n(ehavior)g(of)g(pr)n(e)n(c)n(onditione)n(d)h(CG)f(and)h(CG-S)f(in)g(the) g(pr)n(esenc)n(e)556 1644 y(of)g(r)n(ounding)g(err)n(ors)p FC(,)d(in)h(Preconditioned)h(Conjugate)f(Gradien)o(t)g(Metho)q(ds,)h(O.)g (Axelsson)556 1694 y(and)17 b(L.)f(Y.)g(Kolotilina,)e(eds.,)k(v)o(ol.)d(1457) g(of)h(Lecture)j(Notes)e(in)f(Mathematics,)g(Berlin,)556 1743 y(New)e(Y)m(ork,)f(1990,)g(Springer-V)m(erlag.)450 1825 y([202])p 556 1818 V 115 w(,)k Fq(Bi-CGST)m(AB:)h(A)g(fast)g(and)h(smo)n(othly)f(c)n (onver)n(ging)h(variant)f(of)g(Bi-CG)g(for)f(the)556 1875 y(solution)k(of)g (nonsymmetric)h(line)n(ar)e(systems)p FC(,)i(SIAM)f(J.)f(Sci.)g(Statist.)h (Comput.,)f(13)556 1925 y(\(1992\),)13 b(pp.)g(631{644.)450 2007 y([203])19 b Fa(H.)k(v)l(an)h(der)f(V)o(orst)f(and)i(J.)e(Melissen)p FC(,)g Fq(A)f(Petr)n(ov-Galerkin)f(typ)n(e)h(metho)n(d)h(for)556 2057 y(solving)f Fy(Ax)i FC(=)h Fy(b)d Fq(wher)n(e)f Fy(A)i Fq(is)f(symmetric)f(c)n(omplex)p FC(,)i(IEEE)g(T)m(rans.)e(Magnetics,)j(26) 556 2106 y(\(1990\),)13 b(pp.)g(706{708.)450 2188 y([204])19 b Fa(H.)j(v)l(an)i(der)e(V)o(orst)g(and)h(C.)f(Vuik)p FC(,)f Fq(GMRESR:)h(A)e(family)g(of)g(neste)n(d)h(GMRES)556 2238 y(metho)n(ds)p FC(,)c(T)m(ec)o(h.)g(Rep)q(ort)g(91-80,)e(Delft)h(Univ)o(ersit)o(y)h(of)f(T)m (ec)o(hnology)m(,)f(F)m(acult)o(y)h(of)g(T)m(ec)o(h.)556 2288 y(Math.,)d(Delft,)g(The)h(Netherlands,)h(1991.)450 2370 y([205])k Fa(J.)e(V)-5 b(an)18 b(R)o(osend)o(ale)p FC(,)e Fq(Minimizing)g(inner)g(pr)n (o)n(duct)g(data)g(dep)n(endencies)i(in)e(c)n(onjugate)556 2419 y(gr)n(adient)f(iter)n(ation)p FC(,)e(T)m(ec)o(h.)h(Rep)q(ort)h(172178,) d(ICASE,)i(NASA)h(Langley)f(Researc)o(h)h(Cen-)556 2469 y(ter,)f(1983.)450 2551 y([206])19 b Fa(R.)e(V)-5 b(ar)o(ga)p FC(,)15 b Fq(Matrix)g(Iter)n (ative)f(A)o(nalysis)p FC(,)g(Pren)o(tice-Hall)h(Inc.,)f(Englew)o(o)q(o)q(d)g (Cli\013s,)g(NJ,)556 2601 y(1962.)450 2683 y([207])19 b Fa(P.)k(V)-5 b(assilevski)p FC(,)23 b Fq(Pr)n(e)n(c)n(onditioning)e(nonsymmetric)h(and)g (inde\014nite)g(\014nite)f(element)556 2733 y(matric)n(es)p FC(,)13 b(J.)g(Numer.)g(Alg.)g(Appl.,)g(1)g(\(1992\),)g(pp.)g(59{76.)p eop %%Page: 109 121 120 bop 150 275 a Fr(BIBLIOGRAPHY)1156 b FC(109)150 391 y([208])19 b Fa(V.)f(V)o(oev)o(odin)p FC(,)f Fq(The)g(pr)n(oblem)f(of)h (non-self-adjoint)g(gener)n(alization)g(of)g(the)g(c)n(onjugate)256 441 y(gr)n(adient)12 b(metho)n(d)h(is)f(close)n(d)p FC(,)f(U.S.S.R.)e (Comput.)g(Maths.)i(and)g(Math.)g(Ph)o(ys.,)g(23)f(\(1983\),)256 491 y(pp.)k(143{144.)150 574 y([209])19 b Fa(H.)11 b(F.)g(W)-5 b(alker)p FC(,)11 b Fq(Implementation)g(of)g(the)g(GMRES)g(metho)n(d)h(using) f(Householder)g(tr)n(ans-)256 624 y(formations)p FC(,)i(SIAM)h(J.)g(Sci.)f (Statist.)h(Comput.,)d(9)j(\(1988\),)f(pp.)g(152{163.)150 707 y([210])19 b Fa(P.)c(Wesseling)p FC(,)e Fq(A)o(n)h(Intr)n(o)n(duction)g(to)g (Multigrid)g(Metho)n(ds)p FC(,)f(Wiley)m(,)e(Chic)o(hester,)k(1991.)150 790 y([211])k Fa(O.)14 b(Widlund)p FC(,)f Fq(A)g(Lanczos)i(metho)n(d)e(for)g (a)h(class)f(of)h(non-symmetric)f(systems)g(of)h(line)n(ar)256 840 y(e)n(quations)p FC(,)g(SIAM)g(J.)g(Numer.)f(Anal.,)f(15)h(\(1978\),)g (pp.)h(801{812.)150 923 y([212])19 b Fa(D.)13 b(Young)p FC(,)e Fq(Iter)n(ative)f(solution)i(of)f(lar)n(ge)g(line)n(ar)g(systems)p FC(,)f(Academic)f(Press,)j(New)f(Y)m(ork,)256 972 y(1971.)150 1056 y([213])19 b Fa(H.)h(Yserent)m(ant)p FC(,)g Fq(On)f(the)g(multilevel)e (splitting)i(of)f(\014nite)h(element)g(sp)n(ac)n(es)p FC(,)g(Numer.)256 1105 y(Math.,)13 b(49)g(\(1986\),)g(pp.)h(379{412.)p eop %%Page: 110 122 121 bop 450 629 a FB(Index)450 837 y FC(ad)15 b(ho)q(c)g(SOR)f(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)c(ad)616 886 y(ho)q(c)g(SOR)450 937 y(async)o(hronous)j(metho)q(d,)e Fq(se)n(e)20 b FC(metho)q(d,)616 987 y(async)o(hronous)450 1099 y(Bi-CGST)m(AB)c(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)616 1149 y(Bi-CGST)m(AB)450 1200 y(Bi-Conjugate)d(Gradien)o(t)g(Stabilized)616 1250 y(metho)q(d,)j Fq(se)n(e)j FC(metho)q(d,)616 1300 y(Bi-CGST)m(AB)450 1351 y(bi-orthogonalit)o(y)533 1402 y(in)13 b(BiCG,)g(21)533 1453 y(in)g(QMR,)h(23)450 1504 y(BiCG)f(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)12 b(BiCG)450 1555 y(BiConjugate)k(Gradien)o(t)g(metho)q(d,)g Fq(se)n(e)616 1604 y FC(metho)q(d,)d(BiCG)450 1655 y FA(BLAS)p FC(,)g(2,)g(67)450 1706 y(blo)q(c)o(k)h(metho)q(ds,)f(74{75)450 1757 y(breakdo)o(wn)533 1808 y(a)o(v)o(oiding)f(b)o(y)i(lo)q(ok-ahead,)e(22) 533 1859 y(in)h(Bi-CGST)m(AB,)g(28)533 1910 y(in)g(BiCG,)g(22,)g(23)533 1961 y(in)g(CG)h(for)f(inde\014nite)i(systems,)e(17)450 2073 y(CG)g(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)c(CG)450 2124 y(CGNE)h(metho)q(d,)e Fq(se)n(e)17 b FC(metho)q(d,)c(CGNE)450 2175 y(CGNR)g(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)12 b(CGNR)450 2226 y(CGS)h(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)c(CGS)450 2277 y(c)o(haotic)20 b(metho)q(d,)h Fq(se)n(e)j FC(metho)q(d,)616 2327 y(async)o(hronous)450 2378 y(Cheb)o(yshev)18 b(iteration,)g Fq(se)n(e)i FC(metho)q(d,)616 2428 y(Cheb)o(yshev)15 b(iteration)450 2479 y(co)q(des)533 2530 y FA(FORTRAN)p FC(,)d(2)533 2581 y FA(MATLAB)p FC(,)g(2)450 2632 y(complex)h(systems,)g(51)450 2683 y(Conjugate)18 b(Gradien)o(t)f(metho)q(d,)h Fq(se)n(e)616 2733 y FC(metho)q(d,)13 b(CG)1297 837 y(Conjugate)h(Gradien)o(t)h(Squared)g (metho)q(d,)1463 886 y Fq(se)n(e)i FC(metho)q(d,)c(CGS)1297 937 y(con)o(v)o(ergence)1380 988 y(irregular,)h(88)1422 1039 y(of)f(BiCG,)g(22{23,)f(25)1422 1090 y(of)h(CGS,)g(26,)g(27)1380 1141 y(linear,)g(87)1380 1192 y(of)h(Bi-CGST)m(AB,)e(27{28)1380 1242 y(of)i(BiCG,)e(22{23)1380 1293 y(of)i(CG,)e(15{16)1380 1344 y(of)i(CGNR)f(and)g(CGNE,)h(18)1380 1395 y(of)g(CGS,)f(26)1380 1446 y(of)h(Cheb)o(yshev)h(iteration,)e(29)1380 1497 y(of)h(Gauss-Seidel,)f (10)1380 1548 y(of)h(Jacobi,)f(8{9)1380 1598 y(of)h(MINRES,)f(17)1380 1649 y(of)h(QMR,)f(23{25)1380 1700 y(of)h(SSOR,)f(12)1380 1751 y(smo)q(oth,)f(88)1422 1802 y(of)h(Bi-CGST)m(AB,)g(27)1380 1853 y(stalled,)h(88)1422 1904 y(of)f(BiCG,)g(25)1422 1955 y(of)g(GMRES,)g(19)1380 2005 y(sup)q(erlinear,)i(87)1422 2056 y(of)e(BiCG,)g(29)1422 2107 y(of)g(CG,)g(16)1422 2158 y(of)g(GMRES,)g(29)1297 2268 y(data)h(structures,)i(57{67)1297 2319 y(domain)c(decomp)q(osition)1380 2369 y(m)o(ultiplicativ)o(e)f(Sc)o(h)o(w)o(arz,)j(80{81)1380 2420 y(non-o)o(v)o(erlapping)i(sub)q(domains,)1463 2470 y(78{80)1380 2521 y(o)o(v)o(erlapping)d(sub)q(domains,)f(76{77)1380 2572 y(Sc)o(h)o(ur)j(complemen)o(t,)c(76)1380 2623 y(Sc)o(h)o(w)o(arz,)j(76)1297 2733 y FA(FORTRAN)f FC(co)q(des,)h Fq(se)n(e)j FC(co)q(des,)e FA(FORTRAN)1193 2838 y FC(110)p eop %%Page: 111 123 122 bop 150 275 a Fr(INDEX)1350 b FC(111)150 391 y(Gauss-Seidel)17 b(metho)q(d,)f Fq(se)n(e)k FC(metho)q(d,)316 441 y(Gauss-Seidel)150 491 y(Generalized)12 b(Minimal)c(Residual)j(metho)q(d,)316 541 y Fq(se)n(e)17 b FC(metho)q(d,)c(GMRES)150 591 y(GMRES)d(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)9 b(GMRES)150 689 y(ill-conditioned)j(systems)233 739 y(using)i(GMRES)f(on,)g(19)150 789 y(implemen)o(tatio)o(n)233 839 y(of)g(Bi-CGST)m(AB,)g(28)233 889 y(of)g(BiCG,)g(23)233 939 y(of)g(CG,)g(16)233 990 y(of)g(CGS,)g(26{27)233 1040 y(of)g(Cheb)o(yshev) i(iteration,)e(29)233 1090 y(of)g(GMRES,)g(19{21)233 1140 y(of)g(QMR,)h(25) 150 1190 y FA(IMSL)p FC(,)f(1)150 1240 y(inner)h(pro)q(ducts)233 1290 y(as)g(b)q(ottlenec)o(ks,)h(16,)e(28{29)233 1341 y(a)o(v)o(oiding)f (with)h(Cheb)o(yshev,)i(28,)e(29)150 1391 y(irregular)f(con)o(v)o(ergence,)i Fq(se)n(e)h FC(con)o(v)o(ergence,)316 1441 y(irregular)150 1491 y FA(ITPACK)p FC(,)d(11)150 1588 y(Jacobi)i(metho)q(d,)e Fq(se)n(e)17 b FC(metho)q(d,)c(Jacobi)150 1686 y(Krylo)o(v)g(subspace,)i(15) 150 1783 y(Lanczos)233 1833 y(and)f(CG,)f(15,)g(73{74)150 1883 y FA(LAPACK)p FC(,)f(1)150 1933 y(linear)j(con)o(v)o(ergence,)j Fq(se)n(e)g FC(con)o(v)o(ergence,)316 1983 y(linear)150 2033 y FA(LINPACK)p FC(,)12 b(1)150 2131 y FA(MATLAB)h FC(co)q(des,)h Fq(se)n(e)j FC(co)q(des,)e FA(MATLAB)150 2181 y FC(metho)q(d)233 2231 y(ad)f(ho)q(c)g(SOR,)f(13)233 2281 y(adaptiv)o(e)g(Cheb)o(yshev,)i(28) 233 2331 y(async)o(hronous,)f(12)233 2381 y(Bi-CGST)m(AB,)f(3,)g(7,)g Fq(27{28)233 2432 y FC(Bi-CGST)m(AB2,)g(28)233 2482 y(BiCG,)g(3,)g(7,)g Fq(21{23)233 2532 y FC(CG,)g(3,)g(6,)g Fq(14{16)275 2582 y FC(blo)q(c)o(k)g(v)o(ersion,)h(75)233 2632 y(CGNE,)f(3,)g(6,)h Fq(18)233 2682 y FC(CGNR,)f(3,)g(6,)g Fq(18)233 2733 y FC(CGS,)g(3,)g(7,)g Fq(25{27)1080 391 y FC(c)o(haotic,)22 b(12,)f Fq(se)n(e)i FC(metho)q(d,)1163 441 y(async)o(hronous)1080 491 y(Cheb)o(yshev)c(iteration,)e(3,)h(5,)g(7,) 1163 541 y Fq(28{29)1122 591 y FC(comparison)9 b(with)h(other)h(metho)q(ds,) 1163 641 y(28{29)1122 691 y(sp)q(ectral)17 b(information)12 b(required)1163 741 y(b)o(y)m(,)h(28)1080 791 y(domain)f(decomp)q(osition,)g Fq(76{81)1080 841 y FC(Gauss-Seidel,)i(3,)f(5,)g(7,)g Fq(9{10)1080 891 y FC(GMRES,)g(3,)g(6,)g Fq(18{21)1080 941 y FC(Jacobi,)h(3,)f(5,)g(7,)g Fq(8{9)1080 991 y FC(MINRES,)h(3,)f(6,)g Fq(16{18)1080 1041 y FC(of)j(sim)o(ultaneous)f(displacemen)o(ts,)1163 1091 y Fq(se)n(e)i FC(metho)q(d,)c(Jacobi)1080 1141 y(of)i(successiv)o(e)j(displacemen)o(ts,)e Fq(se)n(e)1163 1191 y FC(metho)q(d,)d(Gauss-Seidel)1080 1241 y(QMR,)h(3,)f(7,)g Fq(23{25)1080 1291 y FC(relaxation,)g(12,)g(13)1080 1341 y(SOR,)h(3,)f(6,)g(7,)g Fq(10{12)1122 1392 y FC(c)o(ho)q(osing)g Fy(!)j FC(in,)d(11{12)1080 1442 y(SSOR,)h(3,)f(6,)g(7,)g Fq(12)1080 1492 y FC(SYMMLQ,)h(3,)f(6,)g Fq(16{18)997 1542 y FC(minimi)o(zation)e(prop)q (ert)o(y)1080 1592 y(in)j(Bi-CGST)m(AB,)f(27)1080 1642 y(in)h(CG,)f(14,)g(17) 1080 1692 y(in)h(MINRES,)f(17)997 1742 y(MINRES)19 b(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)1163 1792 y(MINRES)997 1842 y(m)o(ultigrid,)11 b(81{82)997 1939 y FA(NAG)p FC(,)i(1)997 1989 y(nonstationary)h(metho)q(ds,)f (14{29)997 2039 y(normal)f(equations,)h(6)997 2135 y(o)o(v)o(errelaxation,)g (11)997 2232 y(parallelism,)e(68{72)1080 2282 y(in)j(BiCG,)f(23)1080 2332 y(in)h(CG,)f(16)1080 2382 y(in)h(Cheb)o(yshev)h(iteration,)e(29)1080 2432 y(in)h(GMRES,)f(21)1080 2482 y(in)h(QMR,)f(25)1080 2532 y(inner)h(pro)q(ducts,)h(68{70)1080 2582 y(matrix-v)o(ector)e(pro)q(ducts,)i (70)1080 2632 y(v)o(ector)g(up)q(dates,)f(70)997 2682 y(preconditioners,)h (39{50)1080 2733 y(ADI,)e(50)p eop %%Page: 112 124 123 bop 450 275 a FC(112)1349 b Fr(INDEX)575 391 y FC(parallelism)11 b(in,)i(50)533 441 y(blo)q(c)o(k)h(factorizations,)f(45{47)533 492 y(blo)q(c)o(k)h(tridiagonal,)d(46{47)533 542 y(cen)o(tral)j (di\013erences,)i(43{44)533 592 y(cost,)e(39{40)533 642 y(fast)g(solv)o(ers,) g(49{50)533 692 y(incomplete)f(factorization,)g(42{48)533 742 y(left,)g(40)533 792 y(p)q(oin)o(t)i(incomplete)g(factorizations,)616 842 y(43{45)575 892 y(mo)q(di\014ed,)d(44{45)575 942 y(parallelism)f(in,)i (45)533 992 y(p)q(oin)o(t)g(Jacobi,)h(40{41)533 1042 y(p)q(olynomial,)c (48{49)533 1092 y(reduced)16 b(system,)d(75{76)533 1142 y(righ)o(t,)g(40)533 1192 y(SSOR,)g(41{42)575 1243 y(parallelism)e(in,)i(42)533 1293 y(symmetric)f(part,)i(49)450 1389 y(QMR)g(metho)q(d,)e Fq(se)n(e)17 b FC(metho)q(d,)c(QMR)450 1439 y(Quasi-Minimal)f(Residual)i (metho)q(d,)h Fq(se)n(e)616 1489 y FC(metho)q(d,)e(QMR)450 1585 y(relaxation)18 b(metho)q(d,)g Fq(se)n(e)k FC(metho)q(d,)616 1635 y(relaxation)450 1685 y(residuals)533 1735 y(in)13 b(BiCG,)g(21)533 1785 y(in)g(CG,)g(14)575 1835 y(orthogonalit)o(y)f(of,)g(14)533 1885 y(in)h(SYMMLQ)575 1935 y(orthogonalit)o(y)f(of,)g(17)450 1985 y(restarting)533 2036 y(in)h(BiCG,)g(22)533 2086 y(in)g(GMRES,)g(18{19,) f(21)450 2136 y(ro)o(w)i(pro)r(jection)g(metho)q(ds,)f(82)450 2232 y(searc)o(h)i(directions)533 2282 y(in)e(BiCG,)g(21)533 2332 y(in)g(CG,)g(14{15)575 2382 y(A-orthogonalit)o(y)f(of,)g(14)450 2432 y(smo)q(oth)h(con)o(v)o(ergence,)j Fq(se)n(e)i FC(con)o(v)o(ergence,)616 2482 y(smo)q(oth)450 2532 y(soft)o(w)o(are)533 2582 y(obtaining,)12 b Fq(83{84)450 2632 y FC(SOR)i(metho)q(d,)e Fq(se)n(e)17 b FC(metho)q(d,)c(SOR)450 2682 y(sparse)i(matrix)d(storage,)i(58{62)533 2733 y(BCRS,)f(59)1380 391 y(CCS,)h(59)1380 441 y(CDS,)f(59{61)1380 491 y(CRS,)g(58{59)1380 541 y(JDS,)h(61{62)1380 591 y(SKS,)g(62)1297 640 y(SSOR)g(metho)q(d,)f Fq(se)n(e)k FC(metho)q(d,)12 b(SSOR)1297 690 y(stalled)j(con)o(v)o(ergence,)i Fq(se)n(e)h FC(con)o(v)o(ergence,)1463 740 y(stalled)1297 790 y(Stationary)13 b(metho)q(ds,)g(7{13)1297 840 y(stopping)h(criteria,)g(51{57)1297 889 y(Successiv)o(e)i(Ov)o (errelaxation)e(metho)q(d,)f Fq(se)n(e)1463 939 y FC(metho)q(d,)g(SOR)1297 989 y(sup)q(erlinear)k(con)o(v)o(ergence,)g Fq(se)n(e)i FC(con)o(v)o(er-)1463 1039 y(gence,)c(sup)q(erlinear)1297 1089 y(Symmetric)f(LQ)h(metho)q(d,)f Fq(se)n(e)19 b FC(metho)q(d,)1463 1139 y(SYMMLQ)1297 1188 y(Symmetric)13 b(Successiv)o(e)k(Ov)o(errelaxation)1463 1238 y(metho)q(d,)c Fq(se)n(e)k FC(metho)q(d,)12 b(SSOR)1297 1288 y(SYMMLQ)19 b(metho)q(d,)f Fq(se)n(e)j FC(metho)q(d,)1463 1338 y(SYMMLQ)1297 1429 y(template,)13 b(1)1297 1479 y(three-term)i(recurrence)1380 1529 y(in)f(CG,)f(15)1297 1579 y(t)o(w)o(o-term)g(recurrence,)j(25)1297 1670 y(underrelaxation,)e(11)p eop %%Page: 113 125 124 bop 150 351 1503 2 v 149 401 2 50 v 462 401 V 462 401 V 532 386 a FC(Op)q(erations)14 b(p)q(er)h(Iteration)p 1044 401 V 183 w(Storage)p 1409 401 V 1651 401 V 149 451 V 175 436 a(Metho)q(d)p 462 451 V 180 w Fy(x)519 421 y Fx(T)545 436 y Fy(y)p 598 451 V 59 w(\013x)8 b FC(+)i Fy(y)p 771 451 V 55 w(Ax)p 883 451 V 54 w(M)954 421 y Fw(\000)p Fv(1)998 436 y Fy(y)p 1044 451 V 85 w FC(Requiremen)o(ts)p 1409 451 V 83 w(Commen)o(ts)p 1651 451 V 150 453 1503 2 v 149 503 2 50 v 175 488 a(JA)o(COBI)p 462 503 V 598 503 V 771 503 V 475 w(1)828 473 y Fx(a)p 883 503 V 1044 503 V 1118 488 a FC(matrix)e(+)h(3)p Fy(n)p 1409 503 V 154 w FC(E,)14 b(P)p 1651 503 V 149 552 V 175 537 a(GS)p 462 552 V 598 552 V 444 w(1)p 771 552 V 111 w(1)828 522 y Fx(a)p 883 552 V 1044 552 V 1118 537 a FC(matrix)8 b(+)h(2)p Fy(n)p 1409 552 V 181 w FC(E)p 1651 552 V 149 602 V 175 587 a(SOR)p 462 602 V 598 602 V 414 w(1)p 771 602 V 111 w(1)828 572 y Fx(a)p 883 602 V 1044 602 V 1118 587 a FC(matrix)f(+)h(2)p Fy(n)p 1409 602 V 179 w FC(K)p 1651 602 V 149 652 V 175 637 a(CG)p 462 652 V 283 w(2)p 598 652 V 133 w(3)p 771 652 V 121 w(1)p 883 652 V 116 w(1)p 1044 652 V 143 w(matrix)f(+)h(6)p Fy(n)p 1409 652 V 183 w FC(S)p 1651 652 V 149 702 V 175 687 a(GMRES)p 462 702 V 160 w Fy(i)h FC(+)f(1)p 598 702 V 68 w Fy(i)h FC(+)f(1)p 771 702 V 89 w(1)p 883 702 V 116 w(1)p 1044 702 V 95 w(matrix)e(+)j(\()p Fy(i)f FC(+)h(5\))p Fy(n)p 1409 702 V 99 w FC(M,)j(N)p 1651 702 V 149 752 V 175 737 a(BiCG)p 462 752 V 242 w(2)p 598 752 V 133 w(5)p 771 752 V 101 w(1/1)p 883 752 V 73 w(1/1)p 1044 752 V 112 w(matrix)7 b(+)j(10)p Fy(n)p 1409 752 V 120 w FC(I,)k(N,)f(T)p 1651 752 V 149 802 2 51 v 175 787 a(QMR)p 462 802 V 245 w(2)p 598 802 V 91 w(8+4)707 772 y Fx(bc)p 771 802 V 797 787 a FC(1/1)p 883 802 V 73 w(1/1)p 1044 802 V 104 w(matrix)7 b(+)i(16)p Fy(n)1338 772 y Fx(c)p 1409 802 V 1488 787 a FC(N,)k(T)p 1651 802 V 149 852 2 50 v 175 837 a(CGS)p 462 852 V 260 w(2)p 598 852 V 133 w(6)p 771 852 V 121 w(2)p 883 852 V 116 w(2)p 1044 852 V 133 w(matrix)7 b(+)j(11)p Fy(n)p 1409 852 V 148 w FC(I,)j(N)p 1651 852 V 149 902 V 175 887 a(Bi-CGST)m(AB)p 462 902 V 118 w(4)p 598 902 V 133 w(6)p 771 902 V 121 w(2)p 883 902 V 116 w(2)p 1044 902 V 133 w(matrix)7 b(+)j(10)p Fy(n)p 1409 902 V 168 w FC(N)p 1651 902 V 149 951 V 175 936 a(CHEBYSHEV)p 462 951 V 598 951 V 238 w(2)p 771 951 V 121 w(1)p 883 951 V 116 w(1)p 1044 951 V 143 w(matrix)e(+)h(5)p Fy(n)p 1409 951 V 150 w FC(K,)14 b(N)p 1651 951 V 150 953 1503 2 v 150 1179 a(T)m(able)j(C.1:)26 b(Summary)15 b(of)j(Op)q(erations)h(for)e(Iteration)h Fy(i)p FC(.)31 b(\\1/1")17 b(means)g(iteration)h(requires)150 1229 y(b)q(oth)j(a)f(matrix)e(times)i(v)o(ector)h(and)g(matrix)d(transp)q(ose)k (times)e(v)o(ector)h(op)q(eration.)602 1426 y(Key)14 b(to)g(Commen)o(ts)p 150 1443 1234 2 v 149 1493 2 50 v 175 1478 a(E)60 b(Easy)14 b(to)g(use)p 1383 1493 V 149 1542 V 175 1527 a(I)73 b(Irregular)14 b(con)o(v)o(ergence)i(b)q(eha)o(vior)p 1383 1542 V 149 1592 V 175 1577 a(K)56 b(Kno)o(wledge)14 b(of)f(the)i(sp)q(ectrum)f(required)p 1383 1592 V 149 1642 V 175 1627 a(M)50 b(Requires)14 b(signi\014can)o(tly)f (more)g(memory)e(than)j(the)h(other)f(metho)q(ds)p 1383 1642 V 149 1692 V 175 1677 a(N)57 b(W)m(orks)13 b(with)h(nonsymmetric)e(matrices)p 1383 1692 V 149 1742 V 175 1727 a(P)60 b(P)o(o)q(or)14 b(con)o(v)o(ergence)p 1383 1742 V 149 1791 V 175 1776 a(S)65 b(W)m(orks)13 b(with)h(symmetric)e(p)q (ositiv)o(e)h(de\014nite)i(matrices)p 1383 1791 V 149 1841 V 175 1826 a(T)58 b(Requires)14 b(b)q(oth)g Fy(Ax)g FC(and)g Fy(A)712 1811 y Fx(T)738 1826 y Fy(x)p 1383 1841 V 150 1843 1234 2 v 150 1870 620 2 v 195 1896 a Fi(a)214 1908 y Fn(This)25 b(metho)q(d)f(p)q(erforms)f(no)i(real)g(matrix)f(v)o(ector)g(pro)q(duct)f(or) i(precondition)o(er)e(solv)o(e,)k(but)e(the)150 1947 y(n)o(um)o(b)q(er)g(of)i (op)q(erations)e(is)i(equiv)n(alen)o(t)e(to)i(a)g(matrix-v)o(ecto)o(r)e(m)o (ultiply)m(.)198 1976 y Fi(b)214 1988 y Fn(T)m(rue)37 b Fm(SAXPY)e Fn(op)q(erations)g(+)j(v)o(ector)e(scalings.)198 2016 y Fi(c)214 2028 y Fn(Less)22 b(for)g(implemen)o(ta)o(tion)o(s)e(that)i(do)g(not)g (recursiv)o(ely)e(up)q(date)g(the)i(residual.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF .