SUBROUTINE POISTG (NPEROD,N,MPEROD,M,A,B,C,IDIMY,Y,IERROR,W) C C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C * * C * F I S H P A K * C * * C * * C * A PACKAGE OF FORTRAN SUBPROGRAMS FOR THE SOLUTION OF * C * * C * SEPARABLE ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS * C * * C * (VERSION 3.1 , OCTOBER 1980) * C * * C * BY * C * * C * JOHN ADAMS, PAUL SWARZTRAUBER AND ROLAND SWEET * C * * C * OF * C * * C * THE NATIONAL CENTER FOR ATMOSPHERIC RESEARCH * C * * C * BOULDER, COLORADO (80307) U.S.A. * C * * C * WHICH IS SPONSORED BY * C * * C * THE NATIONAL SCIENCE FOUNDATION * C * * C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C C * * * * * * * * * PURPOSE * * * * * * * * * * * * * * * * * * C C C SUBROUTINE POISTG SOLVES THE LINEAR SYSTEM OF EQUATIONS C C A(I)*X(I-1,J) + B(I)*X(I,J) + C(I)*X(I+1,J) C + X(I,J-1) - 2.*X(I,J) + X(I,J+1) = Y(I,J) C C FOR I=1,2,...,M AND J=1,2,...,N. C C THE INDICES I+1 AND I-1 ARE EVALUATED MODULO M, I.E. C X(0,J) = X(M,J) AND X(M+1,J) = X(1,J), AND X(I,0) MAY BE EQUAL TO C X(I,1) OR -X(I,1) AND X(I,N+1) MAY BE EQUAL TO X(I,N) OR -X(I,N) C DEPENDING ON AN INPUT PARAMETER. C C C * * * * * * * * PARAMETER DESCRIPTION * * * * * * * * * * C C * * * * * * ON INPUT * * * * * * C C NPEROD C INDICATES THE VALUES WHICH X(I,0) AND X(I,N+1) ARE ASSUMED C TO HAVE. C = 1 IF X(I,0) = -X(I,1) AND X(I,N+1) = -X(I,N) C = 2 IF X(I,0) = -X(I,1) AND X(I,N+1) = X(I,N) C = 3 IF X(I,0) = X(I,1) AND X(I,N+1) = X(I,N) C = 4 IF X(I,0) = X(I,1) AND X(I,N+1) = -X(I,N) C C N C THE NUMBER OF UNKNOWNS IN THE J-DIRECTION. N MUST C BE GREATER THAN 2. C C MPEROD C = 0 IF A(1) AND C(M) ARE NOT ZERO C = 1 IF A(1) = C(M) = 0 C C M C THE NUMBER OF UNKNOWNS IN THE I-DIRECTION. M MUST C BE GREATER THAN 2. C C A,B,C C ONE-DIMENSIONAL ARRAYS OF LENGTH M THAT SPECIFY THE COEFFICIENTS C IN THE LINEAR EQUATIONS GIVEN ABOVE. IF MPEROD = 0 THE ARRAY C ELEMENTS MUST NOT DEPEND ON THE INDEX I, BUT MUST BE CONSTANT. C SPECIFICALLY, THE SUBROUTINE CHECKS THE FOLLOWING CONDITION C C A(I) = C(1) C B(I) = B(1) C C(I) = C(1) C C FOR I = 1, 2, ..., M. C C IDIMY C THE ROW (OR FIRST) DIMENSION OF THE TWO-DIMENSIONAL ARRAY Y AS C IT APPEARS IN THE PROGRAM CALLING POISTG. THIS PARAMETER IS C USED TO SPECIFY THE VARIABLE DIMENSION OF Y. IDIMY MUST BE AT C LEAST M. C C Y C A TWO-DIMENSIONAL ARRAY THAT SPECIFIES THE VALUES OF THE C RIGHT SIDE OF THE LINEAR SYSTEM OF EQUATIONS GIVEN ABOVE. C Y MUST BE DIMENSIONED AT LEAST M X N. C C W C A ONE-DIMENSIONAL WORK ARRAY THAT MUST BE PROVIDED BY THE USER C FOR WORK SPACE. W MAY REQUIRE UP TO 9M + 4N + M(INT(LOG2(N))) C LOCATIONS. THE ACTUAL NUMBER OF LOCATIONS USED IS COMPUTED BY C POISTG AND RETURNED IN LOCATION W(1). C C C * * * * * * ON OUTPUT * * * * * * C C Y C CONTAINS THE SOLUTION X. C C IERROR C AN ERROR FLAG THAT INDICATES INVALID INPUT PARAMETERS. EXCEPT C FOR NUMBER ZERO, A SOLUTION IS NOT ATTEMPTED. C = 0 NO ERROR C = 1 IF M .LE. 2 C = 2 IF N .LE. 2 C = 3 IDIMY .LT. M C = 4 IF NPEROD .LT. 1 OR NPEROD .GT. 4 C = 5 IF MPEROD .LT. 0 OR MPEROD .GT. 1 C = 6 IF MPEROD = 0 AND C A(I) .NE. C(1) OR B(I) .NE. B(1) OR C(I) .NE. C(1) C FOR SOME I = 1, 2, ..., M. C = 7 IF MPEROD .EQ. 1 .AND. (A(1).NE.0 .OR. C(M).NE.0) C C W C W(1) CONTAINS THE REQUIRED LENGTH OF W. C C C * * * * * * * PROGRAM SPECIFICATIONS * * * * * * * * * * * * C C DIMENSION OF A(M),B(M),C(M),Y(IDIMY,N), C ARGUMENTS W(SEE ARGUMENT LIST) C C LATEST JUNE 1, 1977 C REVISION C C SUBPROGRAMS POISTG,POSTG2,COSGEN,MERGE,TRIX,TRI3,PIMACH C REQUIRED C C SPECIAL NONE C CONDITIONS C C COMMON NONE C BLOCKS C C I/O NONE C C PRECISION SINGLE C C SPECIALIST ROLAND SWEET C C LANGUAGE FORTRAN C C HISTORY WRITTEN BY ROLAND SWEET IN 1973 C REVISED BY ROLAND SWEET IN 1977 C C ALGORITHM THIS SUBROUTINE IS AN IMPLEMENTATION OF THE C ALGORITHM PRESENTED IN THE REFERENCE. C C SPACE 3297(DECIMAL) = 6341(OCTAL) LOCATIONS ON THE C REQUIRED NCAR CONTROL DATA 7600 C C TIMING AND THE EXECUTION TIME T ON THE NCAR CONTROL DATA C ACCURACY 7600 FOR SUBROUTINE POISTG IS ROUGHLY PROPORTIONAL C TO M*N*LOG2(N). SOME TYPICAL VALUES ARE LISTED C IN THE TABLE BELOW. MORE COMPREHENSIVE TIMING C CHARTS MAY BE FOUND IN THE REFERENCE. C TO MEASURE THE ACCURACY OF THE ALGORITHM A C UNIFORM RANDOM NUMBER GENERATOR WAS USED TO CREATE C A SOLUTION ARRAY X FOR THE SYSTEM GIVEN IN THE C 'PURPOSE' WITH C C A(I) = C(I) = -0.5*B(I) = 1, I=1,2,...,M C C AND, WHEN MPEROD = 1 C C A(1) = C(M) = 0 C B(1) = B(M) =-1. C C THE SOLUTION X WAS SUBSTITUTED INTO THE GIVEN SYS- C TEM AND, USING DOUBLE PRECISION, A RIGHT SIDE Y WAS C COMPUTED. USING THIS ARRAY Y SUBROUTINE POISTG WAS C CALLED TO PRODUCE AN APPROXIMATE SOLUTION Z. THEN C THE RELATIVE ERROR, DEFINED AS C C E = MAX(ABS(Z(I,J)-X(I,J)))/MAX(ABS(X(I,J))) C C WHERE THE TWO MAXIMA ARE TAKEN OVER ALL I=1,2,...,M C AND J=1,2,...,N, WAS COMPUTED. THE VALUE OF E IS C GIVEN IN THE TABLE BELOW FOR SOME TYPICAL VALUES OF C M AND N. C C C M (=N) MPEROD NPEROD T(MSECS) E C ------ ------ ------ -------- ------ C C 31 0-1 1-4 45 9.E-13 C 31 1 1 21 4.E-13 C 31 1 3 41 3.E-13 C 32 0-1 1-4 51 3.E-12 C 32 1 1 32 3.E-13 C 32 1 3 48 1.E-13 C 33 0-1 1-4 42 1.E-12 C 33 1 1 30 4.E-13 C 33 1 3 34 1.E-13 C 63 0-1 1-4 186 3.E-12 C 63 1 1 91 1.E-12 C 63 1 3 173 2.E-13 C 64 0-1 1-4 209 4.E-12 C 64 1 1 128 1.E-12 C 64 1 3 199 6.E-13 C 65 0-1 1-4 143 2.E-13 C 65 1 1 160 1.E-11 C 65 1 3 138 4.E-13 C C PORTABILITY AMERICAN NATIONAL STANDARDS INSTITUTE FORTRAN. C ALL MACHINE DEPENDENT CONSTANTS ARE LOCATED IN THE C FUNCTION PIMACH. C C REQUIRED COS C RESIDENT C ROUTINES C C REFERENCE SCHUMANN, U. AND R. SWEET,"A DIRECT METHOD FOR C THE SOLUTION OF POISSON"S EQUATION WITH NEUMANN C BOUNDARY CONDITIONS ON A STAGGERED GRID OF C ARBITRARY SIZE," J. COMP. PHYS. 20(1976), C PP. 171-182. C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C DIMENSION Y(IDIMY,1) DIMENSION W(1) ,B(1) ,A(1) ,C(1) IERROR = 0 IF (M .LE. 2) IERROR = 1 IF (N .LE. 2) IERROR = 2 IF (IDIMY .LT. M) IERROR = 3 IF (NPEROD.LT.1 .OR. NPEROD.GT.4) IERROR = 4 IF (MPEROD.LT.0 .OR. MPEROD.GT.1) IERROR = 5 IF (MPEROD .EQ. 1) GO TO 103 DO 101 I=1,M IF (A(I) .NE. C(1)) GO TO 102 IF (C(I) .NE. C(1)) GO TO 102 IF (B(I) .NE. B(1)) GO TO 102 101 CONTINUE GO TO 104 102 IERROR = 6 RETURN 103 IF (A(1).NE.0. .OR. C(M).NE.0.) IERROR = 7 104 IF (IERROR .NE. 0) RETURN IWBA = M+1 IWBB = IWBA+M IWBC = IWBB+M IWB2 = IWBC+M IWB3 = IWB2+M IWW1 = IWB3+M IWW2 = IWW1+M IWW3 = IWW2+M IWD = IWW3+M IWTCOS = IWD+M IWP = IWTCOS+4*N DO 106 I=1,M K = IWBA+I-1 W(K) = -A(I) K = IWBC+I-1 W(K) = -C(I) K = IWBB+I-1 W(K) = 2.-B(I) DO 105 J=1,N Y(I,J) = -Y(I,J) 105 CONTINUE 106 CONTINUE NP = NPEROD MP = MPEROD+1 GO TO (110,107),MP 107 CONTINUE GO TO (108,108,108,119),NPEROD 108 CONTINUE CALL POSTG2 (NP,N,M,W(IWBA),W(IWBB),W(IWBC),IDIMY,Y,W,W(IWB2), 1 W(IWB3),W(IWW1),W(IWW2),W(IWW3),W(IWD),W(IWTCOS), 2 W(IWP)) IPSTOR = W(IWW1) IREV = 2 IF (NPEROD .EQ. 4) GO TO 120 109 CONTINUE GO TO (123,129),MP 110 CONTINUE C C REORDER UNKNOWNS WHEN MP =0 C MH = (M+1)/2 MHM1 = MH-1 MODD = 1 IF (MH*2 .EQ. M) MODD = 2 DO 115 J=1,N DO 111 I=1,MHM1 MHPI = MH+I MHMI = MH-I W(I) = Y(MHMI,J)-Y(MHPI,J) W(MHPI) = Y(MHMI,J)+Y(MHPI,J) 111 CONTINUE W(MH) = 2.*Y(MH,J) GO TO (113,112),MODD 112 W(M) = 2.*Y(M,J) 113 CONTINUE DO 114 I=1,M Y(I,J) = W(I) 114 CONTINUE 115 CONTINUE K = IWBC+MHM1-1 I = IWBA+MHM1 W(K) = 0. W(I) = 0. W(K+1) = 2.*W(K+1) GO TO (116,117),MODD 116 CONTINUE K = IWBB+MHM1-1 W(K) = W(K)-W(I-1) W(IWBC-1) = W(IWBC-1)+W(IWBB-1) GO TO 118 117 W(IWBB-1) = W(K+1) 118 CONTINUE GO TO 107 119 CONTINUE C C REVERSE COLUMNS WHEN NPEROD = 4. C IREV = 1 NBY2 = N/2 NP = 2 120 DO 122 J=1,NBY2 MSKIP = N+1-J DO 121 I=1,M A1 = Y(I,J) Y(I,J) = Y(I,MSKIP) Y(I,MSKIP) = A1 121 CONTINUE 122 CONTINUE GO TO (108,109),IREV 123 CONTINUE DO 128 J=1,N DO 124 I=1,MHM1 MHMI = MH-I MHPI = MH+I W(MHMI) = .5*(Y(MHPI,J)+Y(I,J)) W(MHPI) = .5*(Y(MHPI,J)-Y(I,J)) 124 CONTINUE W(MH) = .5*Y(MH,J) GO TO (126,125),MODD 125 W(M) = .5*Y(M,J) 126 CONTINUE DO 127 I=1,M Y(I,J) = W(I) 127 CONTINUE 128 CONTINUE 129 CONTINUE C C RETURN STORAGE REQUIREMENTS FOR W ARRAY. C W(1) = IPSTOR+IWP-1 RETURN END .