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(found)262 1303 y(in)16 b([3)o(].)26 b(Here,)18 b(w)o(e)f(describ)q(e)h(the)f (soft)o(w)o(are)g(a)o(v)n(ailable)d(to)j(reduce)h(the)f(matrix)e(to)h(tridi-) 262 1353 y(agonal)e(form)h(and)h(to)h(compute)f(the)h(eigen)o(v)n(alues)f (and)g(eigen)o(v)o(ectors)i(of)d(the)i(resulting)262 1403 y(tridiagonal)11 b(matrix.)262 1540 y Fp(2)69 b(Initial)20 b(Appro)n(ximation)j(to)g(Eigen)n (v)l(alues)262 1640 y Fj(2.1)55 b(Reduction)18 b(to)g(T)-5 b(ridiagonal)19 b(F)-5 b(orm)262 1716 y Fo(The)16 b(algorithm)d(used)k(in)f (the)g(direct)h(reduction)g(to)e(tridiagonal)f(form)h(is)g(discussed)j(in)262 1766 y(detail)g(in)h([4].)34 b(The)19 b(algorithm)e(alternately)i(eliminates) f(columns)g(and)i(ro)o(ws)f(of)g(the)262 1816 y(matrix,)10 b(preserving)k(the)g(form)d(sho)o(wn)i(in)f(Figure)h(1.)18 b(Retaining)11 b(this)i(matrix)e(structure)262 1866 y(allo)o(ws)j(us)i(to)f (impro)o(v)o(e)f(the)j(o)o(v)o(erall)d(stabilit)o(y)h(of)g(the)h(algorithm)d (b)o(y)i(piv)o(oting)g(at)g(eac)o(h)262 1915 y(step.)733 2022 y Fi(0)733 2095 y(B)733 2119 y(B)733 2146 y(@)790 2054 y Fl(T)814 2060 y Fh(k)q Fg(\000)p Fk(1)p 971 2069 2 50 v 971 2119 V 993 2104 a Ff(\002)p 770 2120 417 2 v 919 2155 a(\002)p 971 2170 2 50 v 45 w Fl(\013)64 b(w)1118 2140 y Fh(T)p 971 2220 V 998 2205 a Fl(v)49 b(B)1098 2211 y Fh(n)p Fg(\000)p Fh(k)1186 2022 y Fi(1)1186 2095 y(C)1186 2119 y(C)1186 2146 y(A)657 2336 y Fo(Figure)14 b(1:)k(P)o(artially)12 b(reduced)k(matrix.)322 2436 y(A)o(t)21 b(the)h Fl(k)q Fo(th)g(stage)g(\(see)h(Figure)f(1\),)h(the)f (algorithm)d(applies)j(the)g(p)q(erm)o(utation)967 2574 y(2)p eop %%Page: 3 3 2 bop 262 307 a Fo(that)15 b(minim)o(i)o(zes)f(the)h(maxim)o(um)10 b(m)o(ultiplier)j(in)h(the)i(transformation)d(matrix)g Fl(N)1587 292 y Fg(\000)p Fk(1)1582 317 y Fh(r)1632 307 y Fl(N)1665 313 y Fh(c)1682 307 y Fo(.)262 357 y(Here,)g Fl(N)403 363 y Fh(r)435 357 y Fo(and)f Fl(N)547 363 y Fh(c)577 357 y Fo(are)h(elemen)o(tary)g (matrices)f(suc)o(h)h(that)g Fl(N)1236 363 y Fh(c)1253 357 y Fl(AN)1322 342 y Fg(\000)p Fk(1)1317 367 y Fh(c)1380 357 y Fo(reduces)i(column)c Fl(k)262 407 y Fo(and)19 b Fl(N)386 392 y Fg(\000)p Fk(1)381 417 y Fh(r)430 407 y Fo(\()p Fl(N)479 413 y Fh(c)497 407 y Fl(AN)566 392 y Fg(\000)p Fk(1)561 417 y Fh(c)610 407 y Fo(\))p Fl(N)659 413 y Fh(r)698 407 y Fo(reduces)j(ro)o(w)d Fl(k)q Fo(.)35 b(This)20 b(minim)o(izati)o(on)d(can)j(b)q(e)g(p)q(erformed) 262 457 y(in)d Fl(O)q Fo(\()p Fl(n)9 b Ff(\000)h Fl(k)q Fo(\))18 b(time)f(b)q(ecause)j(of)e(the)h(sp)q(ecial)f(structure)j(of)c Fl(N)1292 442 y Fg(\000)p Fk(1)1287 467 y Fh(r)1337 457 y Fl(N)1370 463 y Fh(c)1387 457 y Fo(.)31 b(Details)18 b(of)g(this)262 506 y(minim)o(i)o(zation)11 b(algorithm)g(can)j(b)q(e)h(found)e(in)h([)p Fe(?)p Fo(].)322 556 y(The)d(reduction)g(algorithm)e(ma)o(y)g(encoun)o(ter)j (a)f(piv)o(ot)f(that)h(is)f(zero)i(\(or)f(unacceptably)262 606 y(small\))h(regardless)k(of)e(the)h(p)q(erm)o(utation.)k(When)c(this)f(o) q(ccurs,)i(the)f(original)e(reduction)262 656 y(is)18 b(said)h(to)f(ha)o(v)o (e)h(brok)o(en)g(do)o(wn,)g(and)f(the)i(subprogram)d Fd(NEWSTR)h Fo(is)h(called.)32 b Fd(NEWSTR)262 706 y Fo(applies)13 b(a)g(random)f (Householder)i(similarit)o(y)d(transformation)h(to)h(the)h(original)e (matrix.)262 756 y(The)17 b(original)e(matrix)h(m)o(ust)g(therefore)j(b)q(e)e (sa)o(v)o(ed,)h(but)g(this)f(is)g(already)g(necessary)i(in)262 805 y(order)12 b(to)f(apply)g(the)i(iterativ)o(e)e(re\014nemen)o(t)h(metho)q (d)f(describ)q(ed)i(in)f(Section)g(3.2.)k Fd(NEWSTR)262 855 y Fo(is)f(only)f(applied)h(once)h(in)f(our)h(sc)o(heme.)22 b(If)15 b(the)h(reduction)g(of)f(the)h(second)h(matrix)c(also)262 905 y(breaks)f(do)o(wn,)f(then)h(the)h(algorithm)c(returns)k(an)e(error)i(co) q(de.)18 b(This)12 b(o)q(ccurrence)i(has)e(not)262 955 y(y)o(et)i(b)q(een)h (observ)o(ed)g(in)e(practice.)322 1005 y(The)h(transformations)e(used)j(in)e (annihilating)e(eac)o(h)k(column)d(and)h(ro)o(w)h(are)g(sa)o(v)o(ed)g(in)262 1054 y(the)h(lo)q(cations)g(made)f(a)o(v)n(ailable)f(b)o(y)i(the)h (eliminations)d(at)i(eac)o(h)h(step.)23 b(These)17 b(transfor-)262 1104 y(mations)c(are)j(needed)h(for)f(the)g(calculation)e(of)h(the)h(eigen)o (v)o(ectors)h(during)e(the)i(iterativ)o(e)262 1154 y(re\014nemen)o(t)d(step.) 262 1270 y Fj(2.2)55 b(Eigen)n(v)m(alues)17 b(of)i(a)g(T)-5 b(ridiagonal)19 b(Matrix)262 1347 y Fo(One)12 b(of)e(the)i(most)e(e\016cien)o (t)i(metho)q(ds)f(of)g(calculating)f(all)g(the)i(eigen)o(v)n(alues)f(of)g(a)g (nonsym-)262 1397 y(metric)i(tridiagonal)f(matrix)g(is)h(LR)h(iteration.)j (Most)e(of)e(the)h(impro)o(v)o(emen)o(ts)e(that)i(ha)o(v)o(e)262 1446 y(b)q(een)g(incorp)q(orated)g(in)o(to)f(QR)g(iteration)g(o)o(v)o(er)h (the)g(y)o(ears)g([9)o(],)e(suc)o(h)j(as)e(implicit)e(double-)262 1496 y(shift)h(iterations,)h(de\015ation,)f(splitting,)g(and)g(arbitrary)h (shifts,)g(can)g(also)f(b)q(e)i(used)g(in)e(the)262 1546 y(con)o(text)i(of)f (LR)h(iteration.)322 1596 y(An)g(implemen)o(tation)e(of)i(LR)g(iteration)h (that)g(is)f(sp)q(eci\014cally)h(tailored)g(for)f(tridiago-)262 1646 y(nal)e(matrices,)h(called)h Fd(TLR)p Fo(,)e(has)i(b)q(een)h(dev)o(elop) q(ed.)j(The)c(user)h(supplies)f(the)h(tridiagonal)262 1696 y(matrix)g(as)h(three)j(v)o(ectors.)28 b(In)17 b(the)g(\014rst)h(step,)g(the) f(matrix)e(is)i(scaled)h(so)f(that)g(its)f(su-)262 1745 y(p)q(erdiagonal)d (elemen)o(ts)h(are)g(equal)g(to)g(one.)k(This)c(reduces)i(the)f(op)q(eration) f(coun)o(t,)g(since)262 1795 y(LR)e(iteration)h(preserv)o(es)i(the)f(form)d (of)h(suc)o(h)i(a)f(tridiagonal)e(matrix.)16 b(Moreo)o(v)o(er,)d(the)g(su-) 262 1845 y(p)q(erdiagonal)g(v)o(ector)i(is)f(no)o(w)g(free)h(for)f(use)h(as)g (a)f(w)o(orking)f(arra)o(y)m(.)19 b(Implicit)12 b(double-shift)262 1895 y(iterations)j(and)g(splitting)g(are)h(implemen)o(ted)d(just)j(as)f (they)i(are)e(in)h Fs(EISP)m(A)o(CK)f Fo(for)g Fd(HQR)p Fo(.)262 1945 y(Splitting)h(due)i(to)f(either)h(negligable)f(sub)q(diagonal)f(elemen)o (ts)h(or)h(to)f(t)o(w)o(o)g(consecutiv)o(e)262 1994 y(small)10 b(sub)q(diagonal)h(elemen)o(ts)i(are)g(implemen)o(ted.)i(The)e(criteria)g(w)o (e)g(use)g(for)g(negligable)262 2044 y(and)g(small)f(en)o(tries)j(are)f(the)h (same)e(as)h(in)f Fd(HQR)p Fo(.)322 2094 y(Implicit)c(arbitrary)j(shifts)g (are)h(in)o(v)o(ok)o(ed)e(in)h(t)o(w)o(o)f(di\013eren)o(t)i(con)o(texts)h(in) d Fd(TLR)p Fo(.)g(First,)h(if)262 2144 y(an)e(eigen)o(v)n(alue)g(has)g(not)h (con)o(v)o(erged)g(in)f(20)g(iterations,)g(then)h(the)g(iteration)f(is)h (assumed)f(to)262 2194 y(b)q(e)i(stuc)o(k)i(in)d(a)h(cycle,)h(and)g(one)f (arbitrary)g(\(random\))f(double-shift)h(is)g(applied.)17 b(Second,)262 2243 y(if)9 b(the)j(LR)e(iteration)g(encoun)o(ters)j(a)d(zero)i(\(small\))c (diagonal)h(elemen)o(t,)h(then)i(the)f(iteration)262 2293 y(breaks)20 b(do)o(wn,)h(and)e(one)h(arbitrary)g(shift)f(is)h(applied)f(to)h(c)o(hange)g (the)g(v)n(alues)g(of)f(the)262 2343 y(diagonal)12 b(elemen)o(ts.)19 b(\(Another)c(ob)o(vious)f(metho)q(d)f(for)h(a)o(v)o(oiding)e(a)i(zero)h (diagonal)e(is)h(to)262 2393 y(piv)o(ot)f(inside)h(LR,)f(but)h(this)g(has)g (the)g(ma)r(jor)f(dra)o(wbac)o(k)h(of)f(destro)o(ying)h(the)h(tridiagonal)262 2443 y(structure)h(of)e(the)h(matrix\).)j(Up)c(to)g(ten)h(consecutiv)o(e)h (arbitrary)e(shifts)h(are)g(tried)f(if)g(the)967 2574 y(3)p eop %%Page: 4 4 3 bop 262 307 a Fo(breakdo)o(wn)17 b(condition)f(p)q(ersists,)j(after)e(whic) o(h)g(the)h(algorithm)d(ab)q(orts)i(with)g(an)g(error)262 357 y(condition.)f(Ho)o(w)o(ev)o(er,)11 b(a)f(single)h(arbitrary)f(shift)g(pro)o (v)o(ed)h(su\016cien)o(t)g(during)g(all)e(our)i(tests.)322 407 y(Because)k(of)e(the)i(p)q(oten)o(tial)e(for)h(breakdo)o(wn)g(and)f(the)i (need)g(to)e(restart)i(an)f(iteration)262 457 y(with)k(a)g(di\013eren)o(t)i (shift,)f(a)g(cop)o(y)f(of)g(the)i(matrix)d(is)h(made)g(b)q(efore)h(the)h (start)f(of)f(eac)o(h)262 506 y(iteration.)g(This)d(requires)g(at)f(most)f(2) p Fl(n)h Fo(storage.)20 b(One)15 b Fl(n)f Fo(v)o(ector)h(m)o(ust)f(b)q(e)g (supplied)h(b)o(y)262 556 y(the)d(user)h(for)f(this)g(purp)q(ose.)18 b(The)12 b(second)i Fl(n)d Fo(v)o(ector)i(initially)c(holds)j(the)h(sup)q (erdiagonal,)262 606 y(but)h(this)g(space)g(is)g(reclaimed)f(after)h(the)h (matrix)d(is)i(scaled.)262 743 y Fp(3)69 b(Impro)n(ving)34 b(the)f(Accuracy)h(of)g(the)g(Eigen)n(v)l(alue)365 818 y(and)23 b(Computing)f(the)h(Eigen)n(v)n(ector)262 909 y Fo(Appro)o(ximations)15 b(to)i(the)h(eigen)o(v)n(alues)g(of)f Fl(A)g Fo(are)h(obtained)g(b)o(y)f (reducing)h(the)g(matrix)262 959 y(to)c(tridiagonal)f(form)g Fl(T)20 b Fo(\(with)15 b Fd(ATOTRI)p Fo(\))e(and)i(then)g(calculating)f(the)h (eigen)o(v)n(alues)g(of)f Fl(T)262 1009 y Fo(\(with)9 b Fd(TLR)p Fo(\).)g(In)g(man)o(y)f(cases,)j(particularly)e(for)g(small)f(matrices,)h (these)i(computed)f(eigen-)262 1058 y(v)n(alues)15 b(closely)g(appro)o (ximate)e(the)j(eigen)o(v)n(alues)f(of)g Fl(A)p Fo(.)22 b(Ho)o(w)o(ev)o(er,) 15 b(for)g(larger)h(matrices,)262 1108 y(or)d(for)g(matrices)g(whose)h(eigen) o(v)n(alues)f(are)h(ill-conditioned,)d(the)k(rounding)d(errors)j(in)o(tro-) 262 1158 y(duced)j(during)e(the)i(reduction)g(of)e Fl(A)i Fo(to)f (tridiagonal)e(form,)g(coupled)j(with)f(the)g(errors)262 1208 y(in)o(tro)q(duced)12 b(b)o(y)f(LR)g(iteration,)h(can)g(induce)g (signi\014can)o(t)f(errors)i(in)e(the)h(computed)g(eigen-)262 1258 y(v)n(alues.)k(Hence,)c(w)o(e)e(regard)h(the)f(reduction)h(to)f (tridiagonal)e(form)g Fl(T)16 b Fo(and)10 b(the)h(subsequen)o(t)262 1307 y(calculation)f(of)h(the)h(eigen)o(v)n(alues)g(of)f Fl(T)17 b Fo(as)12 b(yielding)e(appro)o(ximations)f(to)j(the)g(eigen)o(v)n(alues)262 1357 y(of)h Fl(A)h Fo(that)g(are)g(then)h(impro)o(v)o(ed)d(in)h(a)h (subsequen)o(t)h(phase)g(of)e(the)i(computation.)262 1473 y Fj(3.1)55 b(In)n(v)n(erse)18 b(Iteration)g(with)h(Ra)n(yleigh)e(Quotien)n(ts) 262 1550 y Fo(One)c(standard)h(tec)o(hnique)f(for)g(impro)o(ving)d(the)k (accuracy)g(of)e(an)h(eigen)o(v)n(alue)f(and)h(at)g(the)262 1600 y(same)k(time)g(computing)g(the)i(asso)q(ciated)g(eigen)o(v)o(ector)g (is)f(to)g(apply)g(in)o(v)o(erse)h(iteration)262 1650 y(coupled)f(with)h (calculating)e(the)i(Ra)o(yleigh)e(quotien)o(t.)32 b(If)19 b(only)e(a)i(few)f(eigenpairs)h(are)262 1700 y(desired,)e(then)f(in)o(v)o (erse)h(iteration)f(is)g(fairly)e(attractiv)o(e,)j(since)g(it)e(is)h (accurate)h(and)f(rea-)262 1749 y(sonably)11 b(rapid.)17 b(The)c Fs(EISP)m(A)o(CK)e Fo(routine)i Fd(INVIT)e Fo(p)q(erforms)g(in)o(v)o(erse)i (iteration)f(\(without)262 1799 y(Ra)o(yleigh)g(quotien)o(ts\))j(to)f(a)g (Hessen)o(b)q(erg)i(matrix.)h(Eac)o(h)e(iteration)f(requires)h Fl(O)q Fo(\()p Fl(n)1587 1784 y Fk(2)1606 1799 y Fo(\))f(op-)262 1849 y(erations,)e(since)h(solving)e(a)h(linear)f(system)h(with)g(a)g(new)h (righ)o(t-hand)e(side)i(is)f(required)h(for)262 1899 y(eac)o(h)i(iteration.)k (If)14 b(the)h(complete)f(eigensystem)h(of)f(a)g(dense)i(matrix)c(is)j (required,)g(then)262 1949 y(the)f Fs(EISP)m(A)o(CK)g Fo(routine)h Fd(RG)f Fo(is)g(recommended)f(b)q(ecause)j(it)e(is)g(robust,)h(highly)e (accurate,)262 1998 y(and)g(requires)i(only)e Fl(O)q Fo(\()p Fl(n)663 1983 y Fk(3)682 1998 y Fo(\))h(op)q(erations)g(for)g(the)g(full)f (eigensystem.)322 2048 y(Another)i(alternativ)o(e)f(is)h(to)f(apply)g(in)o(v) o(erse)h(iteration)f(with)g(Ra)o(yleigh)f(quotien)o(ts)i(to)262 2098 y(the)g(tridiagonal)e(matrix)g Fl(T)21 b Fo(obtained)15 b(from)e Fl(A)j Fo(b)o(y)e Fd(ATOTRI)p Fo(.)g(Again,)g(the)h(solution)f(of)h (a)262 2148 y(di\013eren)o(t)e(linear)f(system)g(for)h(eac)o(h)g(iteration)f (is)g(required,)h(but)g(the)g(linear)f(systems)h(no)o(w)262 2198 y(ha)o(v)o(e)i(a)g(tridiagonal)f(co)q(e\016cien)o(t)j(matrix)c(and)j (therefore)h(can)f(b)q(e)g(solv)o(ed)g(in)f(only)g Fl(O)q Fo(\()p Fl(n)p Fo(\))262 2248 y(steps.)21 b(Th)o(us,)15 b(in)o(v)o(erse)g(iteration)g (with)f(Ra)o(yleigh)f(quotien)o(ts)i(applied)f(to)h(the)g(matrix)e Fl(T)262 2297 y Fo(is)j(a)g(v)o(ery)h(fast)f(means)g(of)g(obtaining)f (accurate)i(appro)o(ximations)d(to)j(the)g(eigensystem)262 2347 y(of)f Fl(T)6 b Fo(.)26 b(Unfortunately)m(,)17 b(to)f(obtain)g(the)i (eigen)o(v)o(ectors)g(of)e(the)h(original)e(matrix)g Fl(A)p Fo(,)i(one)262 2397 y(m)o(ust)g(apply)h(the)h(in)o(v)o(erse)g(of)f(the)h (transformations)e(that)i(reduced)h Fl(A)f Fo(to)g(tridiagonal)262 2447 y(form)14 b(to)i(the)h(computed)f(eigen)o(v)o(ectors)i(of)d Fl(T)6 b Fo(,)17 b(and)f(the)h(eigen)o(v)o(ectors)g(of)f Fl(A)g Fo(will)f(su\013er)967 2574 y(4)p eop %%Page: 5 5 4 bop 262 307 a Fo(from)15 b(an)o(y)i(resulting)h(roundo\013)f(error.)30 b(Moreo)o(v)o(er,)18 b(the)g(eigen)o(v)n(alues)f(of)g Fl(T)24 b Fo(ma)o(y)15 b(di\013er)262 357 y(from)10 b(those)j(of)f Fl(A)h Fo(for)f(the)h(same)f(reason.)18 b(The)13 b(results)g(giv)o(en)f(in)g (Section)h(4)g(indicate)f(the)262 407 y(degree)j(of)e(inaccuracy)i(stemming)c (from)h(these)j(roundo\013)f(errors.)322 457 y(In)i(summary)m(,)d(in)o(v)o (erse)k(iteration)f(on)g Fl(T)23 b Fo(can)16 b(giv)o(e)g(a)g(useful)g(rapid)g (initial)f(appro)o(xi-)262 506 y(mation)d(to)j(the)g(eigensystem)g(of)f Fl(A)p Fo(.)21 b(But)16 b(there)g(ma)o(y)d(b)q(e)i(inaccuracies)h(in)o(tro)q (duced)g(b)o(y)262 556 y(rounding)g(error)h(either)h(in)e(calculating)g(the)h (eigen)o(v)n(alues)g(or)g(in)f(obtaining)f(the)j(eigen-)262 606 y(v)o(ectors)d(of)e Fl(A)h Fo(from)e(the)i(eigen)o(v)o(ectors)i(of)d Fl(T)6 b Fo(.)262 721 y Fj(3.2)55 b(Iterativ)n(e)17 b(Re\014nemen)n(t)262 798 y Fo(It)i(has)g(long)f(b)q(een)j(kno)o(wn)d(that)i(Newton's)f(metho)q(d)g (for)f(the)i(solution)f(of)f(nonlinear)262 847 y(systems)c(can)h(b)q(e)g (applied)f(to)g(the)h(problem)e(of)h(calculating)f(the)i(eigensystem)g(of)e (a)h(ma-)262 897 y(trix)g([6].)21 b(Moreo)o(v)o(er,)15 b(in)g([1)o(],)f (Dongarra)g Fc(et)i(al.)22 b Fo(describ)q(e)16 b(an)f(algorithm)e(for)h (impro)o(ving)262 947 y(the)i(accuracy)h(of)e(an)g(eigenpair)h(based)g(on)f (Newton's)h(metho)q(d.)23 b(The)16 b(main)e(dra)o(wbac)o(k)262 997 y(of)e(their)i(approac)o(h)g(is)f(that)g(it)g(is)h(to)q(o)f(costly)m(,)f (in)h(general.)19 b(In)13 b(this)g(section,)h(w)o(e)g(describ)q(e)262 1047 y(a)i(less)i(costly)f(v)n(arian)o(t)e(of)i(the)g(algorithm)d(giv)o(en)j (in)f([1)o(])h(that)f(tak)o(es)i(adv)n(an)o(tage)e(of)g(the)262 1097 y(tridiagonalization)11 b(of)i Fl(A)h Fo(while)f(still)g(obtaining)g(a)g (high)h(degree)h(of)e(accuracy)m(.)322 1146 y(Assume)j(that)h(\()p Fl(\025)p Fo(,)d Fl(x)p Fo(\))j(is)g(an)f(appro)o(ximate)f(eigenpair)i(of)f (the)i(matrix)d Fl(A)i Fo(and)g(that)262 1196 y Fl(\025)9 b Fo(+)g Fl(\016)r(\025)15 b Fo(and)e Fl(x)c Fo(+)h Fl(\016)r(x)j Fo(are)i(a)e(neigh)o(b)q(oring)g(eigenpair)h(suc)o(h)h(that)f(the)g (relationship)702 1280 y Fl(A)p Fo(\()p Fl(x)c Fo(+)f Fl(\016)r(x)p Fo(\))j(=)f(\()p Fl(\025)f Fo(+)f Fl(\016)r(\025)p Fo(\)\()p Fl(x)h Fo(+)f Fl(\016)r(x)p Fo(\))p Fl(;)262 1365 y Fo(is)16 b(exact.)28 b(Th)o(us,)17 b Fl(\016)r(\025)g Fo(and)g Fl(\016)r(x)f 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1913 y(\016)r(\025)1089 1830 y Fi(\023)1131 1889 y Fo(=)1175 1830 y Fi(\022)1227 1863 y Fl(r)1226 1913 y Fo(0)1268 1830 y Fi(\023)1305 1889 y Fl(;)262 2002 y Fo(where)14 b Fl(e)400 1987 y Fh(T)400 2012 y(s)426 2002 y Fl(\016)r(x)d Fo(=)h(0)h(is)g(a)g (normalization)d(applied)i(to)h Fl(x)g Fo(suc)o(h)h(that)f(the)h Fl(s)f Fo(comp)q(onen)o(t)f(of)h Fl(x)262 2051 y Fo(equals)g(one,)h(implying) d Fl(\016)r(x)691 2057 y Fh(s)720 2051 y Fo(=)h(0)h(\(see)j([1)o(])d(for)h (details\).)322 2101 y(When)i(the)h(original)d(appro)o(ximate)h(eigen)o(v)n (alue)h(is)g(found)g(b)o(y)g(using)g(the)h(reduction)262 2151 y(to)c(tridiagonal)f(form,)g(this)i(yields)g(a)f(matrix)f Fl(N)19 b Fo(suc)o(h)14 b(that)855 2235 y Fl(A)d Fo(=)h Fl(N)979 2218 y Fg(\000)p Fk(1)1023 2235 y Fl(T)6 b(N)r(:)262 2319 y Fo(Using)13 b(the)i(transformations)d(from)g(the)j(reduction)f(to)g(tridiagonal)e(form,)g (w)o(e)i(ha)o(v)o(e)400 2368 y Fi(\022)451 2401 y Fl(N)531 2451 y Fo(1)572 2368 y Fi(\023)7 b(\022)661 2401 y Fl(A)i Ff(\000)h Fl(\025I)45 b Ff(\000)p Fl(x)702 2451 y(e)721 2436 y Fh(T)721 2461 y(s)848 2451 y Fo(0)907 2368 y Fi(\023)6 b(\022)996 2401 y Fl(N)1034 2386 y Fg(\000)p Fk(1)1120 2451 y Fo(1)1161 2368 y Fi(\023)h(\022)1250 2401 y Fl(N)1329 2451 y Fo(1)1371 2368 y Fi(\023)f(\022)1460 2401 y Fl(\016)r(x)1460 2451 y(\016)r(\025)1525 2368 y Fi(\023)967 2574 y Fo(5)p eop %%Page: 6 6 5 bop 780 326 a Fo(=)824 267 y Fi(\022)875 301 y Fl(N)954 350 y Fo(1)996 267 y Fi(\023)6 b(\022)1085 301 y Fl(r)1085 350 y Fo(0)1126 267 y Fi(\023)1164 326 y Fl(;)262 415 y Fo(whic)o(h)13 b(can)h(b)q(e)h(rewritten)g(as)619 451 y Fi(\022)671 484 y Fl(T)g Ff(\000)10 b Fl(\025I)46 b Ff(\000)p Fl(N)5 b(x)670 534 y(e)689 519 y Fh(T)689 544 y(s)716 534 y Fl(N)754 519 y Fg(\000)p Fk(1)876 534 y Fo(0)955 451 y Fi(\023)h(\022)p 1044 451 44 2 v 1044 485 a Fl(\016)r(x)1043 535 y(\016)r(\025)1108 451 y Fi(\023)1150 509 y Fo(=)1194 451 y Fi(\022)p 1246 461 20 2 v 1246 484 a Fl(r)1246 534 y Fo(0)1287 451 y Fi(\023)1325 509 y Fl(;)262 619 y Fo(where)p 383 597 V 16 w Fl(r)14 b Fo(=)f Fl(N)5 b(r)16 b Fo(and)p 616 586 44 2 v 15 w Fl(\016)r(x)d Fo(=)g Fl(N)5 b(\016)r(x)p Fo(.)21 b(The)15 b(solution)g(to)f(the)i (resulting)f(linear)f(system)h(will)262 669 y(pro)q(duce)h(appro)o(ximations) c(to)j(the)h(errors)g Fl(\016)r(\025)f Fo(and)g Fl(\016)r(x)p Fo(,)g(yielding)f(new)h(appro)o(ximations)262 719 y(to)h(the)i(eigenpair.)27 b(The)18 b(linear)e(system)h(is)g(easily)f(solv)o(ed)h(b)o(y)g(transforming)e (it)i(in)o(to)f(a)262 769 y(tridiagonal)f(system)h(of)h(equations)g(b)o(y)f (a)h(rank-one)g(mo)q(di\014cation.)25 b(The)17 b(soft)o(w)o(are)g(w)o(e)262 819 y(ha)o(v)o(e)11 b(implem)o(en)o(ted)e(applies)i(the)h(Sherman-Morrison)e (form)o(ula)f([5)o(])i(to)g(solv)o(e)g(the)h(system)262 868 y(of)h(equations.)322 918 y(Giv)o(en)c(the)h(original)e(matrix)h Fl(A)p Fo(,)h(the)g(tridiagonal)e(matrix)h Fl(T)d Fo(,)k(the)g (transformations)f Fl(N)262 968 y Fo(that)16 b(reduced)i Fl(A)f Fo(to)f Fl(T)22 b Fo(\()p Fl(A)16 b Fo(=)g Fl(N)807 953 y Fg(\000)p Fk(1)852 968 y Fl(T)6 b(N)f Fo(\),)16 b(and)g(the)h(appro)o(ximate)e(eigen)o (v)n(alue)h Fl(\025)1589 974 y Fk(0)1607 968 y Fo(,)h(the)262 1018 y(re\014nemen)o(t)d(algorithm)d(can)j(b)q(e)h(describ)q(ed)g(as)f(follo) o(ws.)349 1146 y Fl(v)i Fo(is)d(an)h(initial)e(guess)j(for)e(the)i(eigen)o(v) o(ector;)349 1196 y(p)q(erform)e(one)h(step)h(of)e(in)o(v)o(erse)i (iteration,)e(\()p Fl(T)i Ff(\000)10 b Fl(\025)1156 1202 y Fk(0)1175 1196 y Fl(I)s Fo(\))p Fl(v)1232 1202 y Fk(0)1263 1196 y Fo(=)h Fl(v)q Fo(;)349 1245 y(for)j Fl(i)e Fo(=)f(0)p Fl(;)c Fo(1)p Fl(;)g Fo(2)p Fl(;)g(:)g(:)g(:)418 1295 y(x)442 1301 y Fh(i)467 1295 y Fo(=)12 b Fl(N)5 b(v)569 1301 y Fh(i)583 1295 y Fo(;)418 1345 y Fl(r)437 1351 y Fh(i)462 1345 y Fo(=)12 b Fl(Ax)561 1351 y Fh(i)584 1345 y Ff(\000)e Fl(\025)650 1351 y Fh(i)664 1345 y Fl(x)688 1351 y Fh(i)701 1345 y Fo(;)418 1395 y Fl(g)439 1380 y Fh(T)438 1406 y(i)477 1395 y Fo(=)i(\()p Fl(e)556 1380 y Fh(T)556 1405 y(s)583 1395 y Fl(N)r(;)7 b Ff(\000)p Fo(1\))706 1380 y Fh(T)732 1395 y Fo(;)418 1470 y(solv)o(e)520 1412 y Fi(\022)571 1445 y Fl(T)16 b Ff(\000)9 b Fl(\025)676 1451 y Fh(i)690 1445 y Fl(I)45 b Ff(\000)p Fl(N)823 1430 y Fg(\000)p Fk(1)868 1445 y Fl(x)892 1451 y Fh(i)618 1495 y Fl(g)639 1480 y Fh(T)638 1505 y(i)819 1495 y Fo(0)926 1412 y Fi(\023)7 b(\022)1017 1445 y Fl(y)1037 1451 y Fh(i)1015 1495 y Fl(\026)1040 1501 y Fh(i)1075 1412 y Fi(\023)1117 1470 y Fo(=)1160 1412 y Fi(\022)1212 1445 y Fl(N)1250 1430 y Fg(\000)p Fk(1)1294 1445 y Fl(r)1313 1451 y Fh(i)1259 1495 y Fo(0)1348 1412 y Fi(\023)1378 1470 y Fo(;)418 1544 y Fl(w)448 1550 y Fh(i)473 1544 y Fo(=)12 b Fl(N)5 b(y)575 1550 y Fh(i)589 1544 y Fo(;)418 1594 y Fl(x)442 1600 y Fh(i)p Fk(+1)509 1594 y Fo(=)12 b Fl(x)577 1600 y Fh(i)600 1594 y Fo(+)e Fl(w)672 1600 y Fh(i)685 1594 y Fo(;)418 1644 y Fl(\025)442 1650 y Fh(i)p Fk(+1)510 1644 y Fo(=)i Fl(\025)578 1650 y Fh(i)601 1644 y Fo(+)e Fl(\026)668 1650 y Fh(i)681 1644 y Fo(;)418 1701 y(c)o(hec)o(k)15 b(if)e(con)o(v)o(erged,)777 1681 y Fg(jj)p Fh(Ax)841 1685 y Fb(i)p Fa(+1)890 1681 y Fg(\000)p Fh(\025)936 1685 y Fb(i)p Fa(+1)984 1681 y Fh(x)1003 1685 y Fb(i)p Fa(+1)1051 1681 y Fg(jj)p 777 1692 294 2 v 869 1716 a Fk(10)p Fg(j)n(j)p Fh(A)p Fg(jj)p Fh(\017)1087 1701 y Ff(\024)f Fo(1;)349 1758 y(end.)262 1836 y(Note)e(that)g(the)h(eigenpair)f(is)g (re\014ned)i(relativ)o(e)e(to)g(the)g(original)f(matrix,)f Fc(i.e.)p Fo(,)i(the)h(residual)262 1886 y(is)g(computed)g(with)h(the)g (original)e(data)h Fl(A)h Fo(and)f(the)i(impro)o(v)o(emen)o(t)c(is)i(b)q (eing)h(made)e(to)i(the)262 1936 y(eigen)o(v)o(ector)h(of)e Fl(A)p Fo(.)18 b(The)12 b(tridiagonal)e(matrix)h Fl(T)18 b Fo(and)12 b(the)h(transformations)d Fl(N)17 b Fo(are)c(used)262 1985 y(solely)i(to)g(simplify)e(solving)h(the)j(system)e(of)g(equations.)23 b(Hence)18 b(the)e(con)o(v)o(ergence)h(will)262 2035 y(b)q(e)d(to)g(the)g (eigensystem)g(of)f(the)i(original)d(matrix)g Fl(A)p Fo(,)h(not)h(the)h (tridiagonal)d(matrix)g Fl(T)6 b Fo(.)322 2085 y(Because)12 b(of)f(the)g(relationship)f(with)h(Newton's)g(metho)q(d,)f(con)o(v)o(ergence) j(will)c(b)q(e)j(guar-)262 2135 y(an)o(teed)i(when)h Fl(\014)r(\021)q(\024)d (<)g Fo(1)p Fl(=)p Fo(2,)g(where)700 2246 y Fl(\014)i Fo(=)781 2186 y Fi(\014)781 2211 y(\014)781 2236 y(\014)781 2261 y(\014)795 2186 y(\014)795 2211 y(\014)795 2236 y(\014)795 2261 y(\014)809 2188 y(\022)860 2221 y Fl(A)c Ff(\000)f Fl(\025)966 2227 y Fh(i)980 2221 y Fl(I)45 b Ff(\000)p Fl(x)1099 2227 y Fh(i)908 2271 y Fl(e)927 2256 y Fh(T)927 2281 y(s)1068 2271 y Fo(0)1134 2188 y Fi(\023)1164 2186 y(\014)1164 2211 y(\014)1164 2236 y(\014)1164 2261 y(\014)1178 2186 y(\014)1178 2211 y(\014)1178 2236 y(\014)1178 2261 y(\014)1192 2196 y Fg(\000)p Fk(1)1243 2246 y Fl(;)262 2350 y(\021)16 b Fo(=)g Ff(jj)o Fl(x)394 2356 y Fh(i)p Fk(+1)459 2350 y Ff(\000)10 b Fl(x)525 2356 y Fh(i)538 2350 y Ff(jj)o Fo(,)17 b(and)f Fl(\024)g Fo(=)g(2)g(\()p Fl(\024)h Fo(is)f(a)g(b)q(ound)g(on)h(the)g(second)g(deriv)n(ativ)o(e\).)25 b(As)17 b(can)262 2399 y(b)q(e)e(seen,)i(the)e(con)o(v)o(ergence)i(rate)f (and)f(error)h(b)q(ound)f(dep)q(end)h(on)f(the)h(condition)e(of)h(the)262 2449 y(matrix,)c(\(see)k([)p Fe(?)p Fo(])i(for)d(additional)e(information\).) 967 2574 y(6)p eop %%Page: 7 7 6 bop 322 307 a Fo(The)15 b(approac)o(h)g(describ)q(ed)i(here)f(will)d(not)i (only)f(impro)o(v)o(e)f(the)j(accuracy)g(of)e(the)i(ap-)262 357 y(pro)o(ximate)10 b(eigen)o(v)n(alue)j Fl(\025)g Fo(but)f(will)g(also)g (compute)g(the)h(eigen)o(v)o(ector.)19 b(The)13 b(con)o(v)o(ergence)262 407 y(theorem)g(for)h(this)g(iterativ)o(e)f(pro)q(cedure)j(can)e(b)q(e)h (found)e(in)h([1)o(].)322 457 y(During)9 b(the)i(impro)o(v)o(emen)o(t)d (phase,)j(the)g(subprogram)e(named)h Fd(REFINE)f Fo(is)h(called)g(with)262 506 y(the)i(original)d(data)j(matrix)e Fd(A)p Fo(,)g(the)j(reduced)g (tridiagonal)d(matrix)f Fd(T)p Fo(,)i(the)i(transformation)262 556 y Fd(N)p Fo(,)19 b(and)h(an)g(appro)o(ximate)e(eigen)o(v)n(alue)i Fd(\(WR,WI\))f Fo(as)h(parameters.)37 b(A)20 b(single)g(in)o(v)o(erse)262 606 y(iteration)f(step)i(is)f(p)q(erformed)f(with)h(the)g(tridiagonal)e (matrix)h Fd(T)g Fo(\(using)h Fd(GTINIT)p Fo(\))f(to)262 656 y(obtain)f(an)h(initial)e(appro)o(ximation)f(to)j(the)h(eigen)o(v)o(ector)g (asso)q(ciated)g(with)e(the)i(giv)o(en)262 706 y(eigen)o(v)n(alue.)c(On)10 b(return)h(from)e Fd(REFINE)f Fo(the)j(impro)o(v)o(ed)d(eigen)o(v)n(alue)i (is)g(stored)h(in)e Fd(\(WR,WI\))262 756 y Fo(and)k(the)i(impro)o(v)o(ed)d (eigen)o(v)o(ector)j(in)e Fd(\(XR,XI\))p Fo(.)262 893 y Fp(4)69 b(Examples)22 b(and)h(P)n(erformance)262 984 y Fo(W)m(e)d(presen)o(t)i(t)o(w) o(o)e(test)h(suites)h(to)e(illustrate)g(the)h(sp)q(eed)h(and)f(accuracy)g(of) f(the)h(new)262 1034 y(algorithms.)15 b(The)e(p)q(erformance)f(of)h Fd(RG)f Fo(is)g(included)h(for)f(comparison.)17 b(All)11 b(exp)q(erimen)o(ts) 262 1083 y(w)o(ere)k(executed)h(on)e(an)g(IBM)h(RS/6000)e(mo)q(del)g(530,)g (using)h(the)h(F)m(ortran)f(compiler)f Fd(xlf)262 1133 y Fo(with)g (optimization.)322 1183 y(T)m(ables)j(1)g(and)g(2)g(sho)o(w)g(the)h(results)h (for)e(random)f(matrices)h(with)g(the)h(en)o(tries)g(uni-)262 1233 y(formly)7 b(distributed)k(in)f([)p Ff(\000)p Fo(1)p Fl(:)p Fo(0)p Fl(;)d Fo(1)p Fl(:)p Fo(0])o(.)14 b(T)m(able)c(1)f(sho)o(ws)i(the)f (maxim)o(um)5 b(di\013erence)12 b(b)q(et)o(w)o(een)262 1283 y(the)g(eigen)o(v)n(alues)f(computed)h(b)o(y)f Fd(ATOTRI-TLR)f Fo(and)i(those)g(calculated)g(b)o(y)g Fd(RG)p Fo(.)e(The)j(max-)262 1332 y(im)o(um)d(di\013erence)16 b(of)e(the)h(impro)o(v)o(ed)d(eigen)o(v)n (alues)i(is)g(also)g(giv)o(en.)k(Finally)m(,)12 b(the)j(residual)262 1382 y(is)e(giv)o(en)g(for)h(the)g(results)h(from)d(in)o(v)o(erse)i (iteration,)f(iterativ)o(e)h(re\014nemen)o(t,)g(and)f Fd(RG)p Fo(.)g(The)262 1432 y(in)o(v)o(erse)j(iteration)g(results)h(are)f(obtained)f (b)o(y)h(calling)f(the)h Fs(EISP)m(A)o(CK)f Fo(routines)i Fd(ELMHES)262 1482 y Fo(and)12 b Fd(HQR)p Fo(,)g(follo)o(w)o(ed)f(b)o(y)h Fd(INVIT)g Fo(for)g(ev)o(ery)h(eigenpair.)18 b(The)13 b(maxim)n(um)8 b(residual)13 b(o)o(v)o(er)g(all)262 1532 y(the)j(eigenpairs)h(is)e(rep)q (orted)j(in)e(the)g(table.)25 b(Similarly)l(,)13 b Fd(REFINE)i Fo(w)o(as)h(called)g(for)g(ev)o(ery)262 1581 y(eigenpair,)h(and)h(the)g (maxim)n(um)c(residual)j(is)h(rep)q(orted.)31 b(In)18 b(ev)o(ery)g(case)h (the)f(smallest)262 1631 y(maxim)n(um)9 b(residual)14 b(w)o(as)g(generated)h (with)f(the)g(new)g(iterativ)o(e)g(re\014nemen)o(t)h(routines.)262 1759 y(T)m(able)8 b(1:)16 b(Comparison)8 b(of)h(the)h(accuracy)h(of)e(the)h (new)g(routines)g(to)g(the)g Fs(EISP)m(A)o(CK)f Fo(routine)262 1809 y Fd(RG)p Fo(.)k(Residual)h(is)h(max)5 b Ff(jj)o Fl(Ax)k Ff(\000)h Fl(\025x)p Ff(j)o(j)828 1822 y Fg(1)878 1809 y Fo(and)k Fl(e)978 1815 y Fh(\025)1015 1809 y Fo(is)h(max)5 b Ff(j)p Fl(\025)1178 1815 y Fh(i)1201 1809 y Ff(\000)p 1243 1775 39 2 v 10 w Fl(\025)1267 1815 y Fh(i)1281 1809 y Ff(j)p Fo(.)20 b(\()p Fl(\025)1365 1815 y Fh(i)1394 1809 y Fo(is)15 b(the)g(eigen)o(v)n (alue)262 1859 y(obtained)e(from)f Fd(RG)p Fo(,)h(and)p 680 1825 V 14 w Fl(\025)704 1865 y Fh(i)732 1859 y Fo(is)h(the)g(computed)f (eigen)o(v)n(alue.)p 298 1873 1346 2 v 297 1923 2 50 v 776 1908 a(Accuracy)i(of)e(Routines)p 1643 1923 V 297 1973 V 733 1958 a(on)g(dense)j(random)c(matrices)p 1643 1973 V 298 1974 1346 2 v 297 2024 2 50 v 494 2024 V 494 2024 V 520 2009 a Fd(ATOTRI-TLR)p 762 2024 V 762 2024 V 127 w(RG)p 1014 2024 V 1014 2024 V 175 w(INVIT)p 1265 2024 V 1265 2024 V 194 w(REFINE)p 1643 2024 V 297 2074 V 323 2059 a Fo(problem)p 494 2074 V 494 2074 V 762 2074 V 762 2074 V 316 w(\()p Fs(EISP)m(A)o(CK)p Fo(\))p 1014 2074 V 1014 2074 V 50 w(\()p Fs(EISP)m(A)o(CK)p Fo(\))p 1265 2074 V 1265 2074 V 1643 2074 V 297 2124 V 364 2109 a(size)p 494 2124 V 494 2124 V 181 w Fl(e)628 2115 y Fh(\025)p 762 2124 V 762 2124 V 818 2109 a Fo(residual)p 1014 2124 V 1014 2124 V 111 w(residual)p 1265 2124 V 1265 2124 V 128 w Fl(e)1358 2115 y Fh(\025)p 1452 2124 V 1452 2124 V 1478 2109 a Fo(residual)p 1643 2124 V 298 2125 1346 2 v 297 2175 2 50 v 429 2160 a(10)p 494 2175 V 90 w(8.7E-14)p 762 2175 V 122 w(1.8E-14)p 1014 2175 V 109 w(2.0E-14)p 1257 2175 V 1266 2175 V 85 w(4.4E-15)p 1452 2175 V 51 w(3.7E-16)p 1643 2175 V 297 2225 V 408 2210 a(100)p 494 2225 V 90 w(7.2E-06)p 762 2225 V 122 w(5.3E-12)p 1014 2225 V 109 w(2.9E-12)p 1257 2225 V 1266 2225 V 85 w(2.7E-13)p 1452 2225 V 51 w(5.1E-13)p 1643 2225 V 297 2275 V 408 2260 a(500)p 494 2275 V 90 w(1.2E-02)p 762 2275 V 122 w(2.5E-09)p 1014 2275 V 109 w(3.0E-10)p 1257 2275 V 1266 2275 V 85 w(4.3E-12)p 1452 2275 V 51 w(2.3E-12)p 1643 2275 V 298 2276 1346 2 v 322 2368 a(T)m(able)g(2)g(compares)h(the)g(execution)h(times)e(of)g(three)i(metho)q (ds)e(of)h(calculating)f(eigen-)262 2418 y(pairs)19 b(for)g(a)g(nonsymmetric) e(matrix.)33 b(F)m(or)19 b(reference)j(the)e(time)e(required)i(for)f Fd(RG)g Fo(to)967 2574 y(7)p eop %%Page: 8 8 7 bop 262 336 a Fo(T)m(able)14 b(2:)21 b(Comparison)14 b(of)g(execution)j (times)d(in)h(seconds)i(of)e(the)h(new)g(routines)g(to)f(the)262 386 y Fs(EISP)m(A)o(CK)e Fo(routine)h Fd(RG)p Fo(.)f(Time)f(for)i Fd(INVIT)f Fo(and)g Fd(REFINE)g Fo(are)h(p)q(er)h(eigenpair.)p 367 400 1208 2 v 366 450 2 50 v 747 435 a(P)o(erformance)e(of)h(Routines)p 1573 450 V 366 499 V 733 484 a(on)f(dense)j(random)c(matrices)p 1573 499 V 367 501 1208 2 v 366 551 2 50 v 392 536 a(problem)p 564 551 V 564 551 V 105 w Fd(RG)p 770 551 V 770 551 V 106 w(ATOTRI)p 950 551 V 950 551 V 48 w(ELMHES)p 1131 551 V 1131 551 V 80 w(INVIT)p 1352 551 V 1352 551 V 100 w(REFINE)p 1573 551 V 366 601 V 433 586 a Fo(size)p 564 601 V 564 601 V 92 w(all)h(\()p Fl(\025;)7 b(x)p Fo(\))p 770 601 V 770 601 V 82 w Fd(TLR)p 950 601 V 950 601 V 115 w(HQR)p 1131 601 V 1131 601 V 82 w Fo(\(p)q(er)15 b Fl(\025;)7 b(x)p Fo(\))p 1352 601 V 1352 601 V 49 w(\(p)q(er)15 b Fl(\025;)7 b(x)p Fo(\))p 1573 601 V 367 602 1208 2 v 366 652 2 50 v 498 637 a(10)p 564 652 V 91 w(.024)p 770 652 V 118 w(.004)p 950 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y(reduce)k(the)g(matrix)d(to)h(Hessen)o (b)q(erg)k(form)16 b(and)i(calculate)g(all)e(its)i(eigen)o(v)n(alues.)30 b(In)18 b(a)262 1125 y(separate)c(column,)d(the)i(a)o(v)o(erage)g(time)f(to)g (calculate)h(an)g(eigenpair)g(is)f(giv)o(en.)17 b(\(The)d(time)262 1175 y(for)e Fd(ELMBAK)g Fo(is)h(divided)g(among)e(the)j Fl(n)f Fo(eigenpairs)g(calculated\).)18 b(The)c(table)f(also)f(sho)o(ws)262 1225 y(the)i(time)e(to)h(reduce)i(the)f(matrix)e(to)h(tridiagonal)e(form)h (and)h(calculate)h(its)f(eigen)o(v)n(alues.)262 1275 y(This)j(op)q(eration)h (is)g(amazingly)d(fast)j(on)g(a)g(cac)o(he-based)h(mac)o(hine)e(lik)o(e)g (the)i(RS/6000.)262 1325 y(The)12 b(a)o(v)o(erage)f(time)g(p)q(er)h (eigenpair)f(for)h(impro)o(ving)d(the)j(eigen)o(v)n(alue)f(and)g(calculating) g(the)262 1374 y(corresp)q(onding)19 b(eigen)o(v)o(ector)g(with)f(iterativ)o (e)g(re\014nemen)o(t)g(is)g(ab)q(out)g(four)g(times)g(more)262 1424 y(than)f(using)h Fd(INVIT)p Fo(.)e(But)i(b)q(ecause)i(the)e(routines)h Fd(ATOTRI)d Fo(and)i Fd(TLR)f Fo(are)h(so)g(fast,)g(the)262 1474 y(total)11 b(time)g(for)h(calculating)g(up)g(to)h(20\045)e(of)h(the)h (eigenpairs)g(is)f(smaller)f(\(and)h(the)h(results)262 1524 y(more)f(accurate\))k(using)d(the)i(new)f(routines.)322 1574 y(The)g(results)h(of)e(running)h(the)g Fs(EISP)m(A)o(CK)g Fo(general)g (matrix)e(test)j(suite)g([8)o(])e(are)i(sho)o(wn)262 1624 y(in)10 b(T)m(able)h(3.)17 b(This)11 b(test)i(suite)f(consists)g(of)f(35)g(small)e (matrices)i(\(none)h(exceeding)h(20)t Ff(\002)t Fo(20\))262 1673 y(that)e(are)i(designed)f(to)g(b)q(e)h(pathological)c(with)j(resp)q(ect) i(to)e(their)g(eigen)o(v)n(alues)g(and)g(eigen-)262 1723 y(v)o(ectors.)22 b(Most)15 b(of)g(the)h(matrices)e(are)i(ill-conditioned,)d(some)h(are)h (defectiv)o(e,)h(some)e(are)262 1773 y(derogatory)m(,)j(and)g(some)f(are)i (all)e(three.)30 b(The)18 b(accuracy)g(and)f(robustness)i(of)e(the)h(new)262 1823 y(algorithms)9 b(are)k(displa)o(y)o(ed)f(b)o(y)g(this)g(test,)h(where)h 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a(19)p 616 1603 V 93 w(8E-07)p 858 1603 V 102 w(9.0E-15)p 1062 1603 V 85 w(8.8E-09)p 1302 1603 V 91 w(2.7E-08)p 1522 1603 V 418 1653 V 550 1638 a(20)p 616 1653 V 93 w(1E-16)p 858 1653 V 102 w(1.4E-14)p 1062 1653 V 85 w(1.0E-15)p 1302 1653 V 91 w(9.7E-15)p 1522 1653 V 418 1702 V 550 1687 a(21)p 616 1702 V 93 w(1E-16)p 858 1702 V 102 w(6.3E-15)p 1062 1702 V 85 w(2.2E-16)p 1302 1702 V 91 w(6.0E-15)p 1522 1702 V 418 1752 V 550 1737 a(22)p 616 1752 V 93 w(5E-14)p 858 1752 V 102 w(1.0E-13)p 1062 1752 V 85 w(7.1E-16)p 1302 1752 V 91 w(2.1E-14)p 1522 1752 V 418 1802 V 550 1787 a(23)p 616 1802 V 93 w(6E-14)p 858 1802 V 102 w(2.0E-10)p 1062 1802 V 85 w(3.2E-17)p 1302 1802 V 91 w(2.9E-14)p 1522 1802 V 418 1852 V 550 1837 a(24)p 616 1852 V 93 w(1E-10)p 858 1852 V 102 w(2.5E-06)p 1062 1852 V 85 w(6.2E-09)p 1302 1852 V 91 w(1.1E-02)p 1522 1852 V 418 1902 V 550 1887 a(25)p 616 1902 V 93 w(1E-16)p 858 1902 V 102 w(8.7E-07)p 1062 1902 V 85 w(2.2E-15)p 1302 1902 V 91 w(6.0E-14)p 1522 1902 V 418 1951 V 550 1936 a(26)p 616 1951 V 93 w(1E-16)p 858 1951 V 102 w(4.3E-13)p 1062 1951 V 85 w(3.6E-14)p 1302 1951 V 91 w(2.2E-15)p 1522 1951 V 418 2001 V 550 1986 a(27)p 616 2001 V 84 w(6E+07)p 858 2001 V 93 w(3.6E-01)p 1062 2001 V 85 w(9.0E-10)p 1302 2001 V 91 w(2.6E-06)p 1522 2001 V 418 2051 V 550 2036 a(28)p 616 2051 V 93 w(1E-16)p 858 2051 V 102 w(4.8E-14)p 1062 2051 V 85 w(1.2E-14)p 1302 2051 V 91 w(5.7E-14)p 1522 2051 V 418 2101 V 550 2086 a(29)p 616 2101 V 93 w(1E-16)p 858 2101 V 102 w(2.4E-14)p 1062 2101 V 85 w(2.8E-14)p 1302 2101 V 91 w(4.0E-12)p 1522 2101 V 418 2151 V 550 2136 a(30)p 616 2151 V 93 w(1E-13)p 858 2151 V 102 w(5.2E-14)p 1062 2151 V 85 w(2.3E-13)p 1302 2151 V 91 w(4.2E-13)p 1522 2151 V 418 2200 V 550 2186 a(31)p 616 2200 V 93 w(2E-12)p 858 2200 V 102 w(5.7E-14)p 1062 2200 V 85 w(1.8E-15)p 1302 2200 V 91 w(5.6E-14)p 1522 2200 V 418 2250 V 550 2235 a(32)p 616 2250 V 93 w(1E-03)p 858 2250 V 102 w(1.4E-14)p 1062 2250 V 85 w(1.4E-05)p 1302 2250 V 91 w(4.4E-07)p 1522 2250 V 418 2300 V 550 2285 a(33)p 616 2300 V 93 w(1E-16)p 858 2300 V 102 w(5.4E-01)p 1062 2300 V 85 w(1.9E-04)p 1302 2300 V 91 w(1.1E-08)p 1522 2300 V 418 2350 V 550 2335 a(34)p 616 2350 V 93 w(1E-16)p 858 2350 V 102 w(4.4E-02)p 1062 2350 V 85 w(9.1E-14)p 1302 2350 V 91 w(1.5E-08)p 1522 2350 V 418 2400 V 550 2385 a(35)p 616 2400 V 93 w(1E-01)p 858 2400 V 102 w(1.8E-12)p 1062 2400 V 85 w(1.8E-05)p 1302 2400 V 91 w(2.7E-13)p 1522 2400 V 419 2401 1104 2 v 967 2574 a(9)p eop %%Page: 10 10 9 bop 262 307 a Fo(The)15 b(demand)f(for)h(routines)g(to)g(solv)o(e)g(just)g (suc)o(h)h(problems)e(is)h(gro)o(wing)f(rapidly)g(in)h(the)262 357 y(areas)f(of)f(quan)o(tum)f(c)o(hemistry)i(and)g(materials)e(science.)322 407 y(It)h(is)f(also)h(clear)g(from)e(our)i(tests)i(that)e(if)f(all)g(the)h (eigenpairs)g(are)h(required,)f(then)h(the)262 457 y(EISP)m(A)o(CK)g(routine) g Fd(RG)f Fo(is)h(the)g(recommended)f(approac)o(h.)957 2574 y(10)p eop %%Page: 11 11 10 bop 262 307 a Fp(6)69 b(Description)33 b(of)i(the)g(Soft)n(w)n(are)h(and)g (Program-)365 382 y(ming)21 b(Details)262 473 y Fo(In)13 b(this)h(section)h (w)o(e)f(sho)o(w)g(the)g(soft)o(w)o(are)g(implemen)o(tation)d(of)i(the)h(new) h(algorithms.)392 606 y Fd(SUBROUTINE)20 b(ATOTRI\()g(LDA,)h(A,)h(N,)f (PIVOTS,)f(INFO)h(\))262 656 y(c)43 b(Purpose:)262 706 y(c)86 b(This)21 b(subroutine)f(reduces)h(an)g(n-by-n)f(real)h(general)g(matrix)f(A) i(to)262 756 y(c)86 b(tridiagonal)20 b(form)h(using)g(elementary)e (similarity)h(transformations.)262 805 y(c)262 855 y(c)86 b(At)22 b(each)f(step)g(k)g(the)h(permutation)d(that)i(minimizes)f(the)h(maximum)f (entry)262 905 y(c)86 b(in)22 b(the)f(transformation)e(matrix)h(which)h (reduces)g(column)f(k)i(then)f(row)g(k)262 955 y(c)86 b(is)22 b(applied.)262 1005 y(c)262 1054 y(c)43 b(Arguments:)262 1104 y(c)65 b(LDA)130 b(-integer)262 1154 y(c)283 b(leading)20 b(dimension)g(of)h (A)262 1204 y(c)262 1254 y(c)65 b(A)174 b(-double)20 b(precision)g(array)h (of)g(dimension)f(\(LDA,N\))262 1303 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1652 y(c)86 b(the)22 b(eigenvalues.)262 1702 y(c)262 1752 y(c)43 b(Arguments:)262 1802 y(c)65 b(N)108 b(-integer)262 1851 y(c)217 b(N)22 b(specifies)e(the)h(order)g(of)g(the)g(tridiagonal)f (matrix.)262 1901 y(c)217 b(not)21 b(modified)262 1951 y(c)262 2001 y(c)65 b(DIAG)42 b(-double)21 b(precision)f(array)g(of)i(dimension)e(N) 262 2051 y(c)217 b(On)22 b(entry)e(DIAG)h(contains)f(the)i(diagonal)e(of)h (the)g(tridiagonal)262 2100 y(c)217 b(matrix.)262 2150 y(c)g(On)22 b(exit)f(DIAG)g(contains)f(the)h(real)g(part)g(of)g(the)g(eigenvalues.)262 2200 y(c)262 2250 y(c)65 b(SUB)f(-double)21 b(precision)f(array)g(of)i (dimension)e(N)262 2300 y(c)217 b(On)22 b(entry)e(SUB)i(contains)e(the)h (sub-diagonal)e(of)j(the)262 2350 y(c)217 b(tridiagonal)20 b(matrix.)262 2399 y(c)217 b(On)22 b(exit)f(SUB)g(contains)f(the)h(imaginary) f(part)h(of)g(the)h(eigenvalues.)262 2449 y(c)957 2574 y Fo(12)p eop %%Page: 13 13 12 bop 262 307 a Fd(c)65 b(SUP)f(-double)21 b(precision)f(array)g(of)i (dimension)e(N)262 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