*DECK POISTG SUBROUTINE POISTG (NPEROD, N, MPEROD, M, A, B, C, IDIMY, Y, + IERROR, W) C***BEGIN PROLOGUE POISTG C***PURPOSE Solve a block tridiagonal system of linear equations C that results from a staggered grid finite difference C approximation to 2-D elliptic PDE's. C***LIBRARY SLATEC (FISHPACK) C***CATEGORY I2B4B C***TYPE SINGLE PRECISION (POISTG-S) C***KEYWORDS ELLIPTIC, FISHPACK, HELMHOLTZ, PDE, TRIDIAGONAL C***AUTHOR Adams, J., (NCAR) C Swarztrauber, P. N., (NCAR) C Sweet, R., (NCAR) C***DESCRIPTION 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 *Long Description: 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 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 The machine dependent constant PI is defined in 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 C***REFERENCES U. Schumann and R. Sweet, A direct method for the C solution of Poisson's equation with Neumann boundary C conditions on a staggered grid of arbitrary size, C Journal of Computational Physics 20, (1976), C pp. 171-182. C***ROUTINES CALLED POSTG2 C***REVISION HISTORY (YYMMDD) C 801001 DATE WRITTEN C 861211 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE POISTG C C DIMENSION Y(IDIMY,*) DIMENSION W(*) ,B(*) ,A(*) ,C(*) C***FIRST EXECUTABLE STATEMENT POISTG 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