C FISHPK10 FROM PORTLIB 12/30/83 SUBROUTINE HSTSSP (A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD, 1 ELMBDA,F,IDIMF,PERTRB,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 HSTSSP SOLVES THE STANDARD FIVE-POINT FINITE DIFFERENCE C APPROXIMATION ON A STAGGERED GRID TO THE HELMHOLTZ EQUATION IN C SPHERICAL COORDINATES AND ON THE SURFACE OF THE UNIT SPHERE C (RADIUS OF 1) C C (1/SIN(THETA))(D/DTHETA)(SIN(THETA)(DU/DTHETA)) + C C (1/SIN(THETA)**2)(D/DPHI)(DU/DPHI) + LAMBDA*U = F(THETA,PHI) C C WHERE THETA IS COLATITUDE AND PHI IS LONGITUDE. C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C * * * * * * * * PARAMETER DESCRIPTION * * * * * * * * * * C C * * * * * * ON INPUT * * * * * * C C A,B C THE RANGE OF THETA (COLATITUDE), I.E. A .LE. THETA .LE. B. A C MUST BE LESS THAN B AND A MUST BE NON-NEGATIVE. A AND B ARE IN C RADIANS. A = 0 CORRESPONDS TO THE NORTH POLE AND B = PI C CORRESPONDS TO THE SOUTH POLE. C C C * * * IMPORTANT * * * C C IF B IS EQUAL TO PI, THEN B MUST BE COMPUTED USING THE STATEMENT C C B = PIMACH(DUM) C C THIS INSURES THAT B IN THE USER"S PROGRAM IS EQUAL TO PI IN THIS C PROGRAM WHICH PERMITS SEVERAL TESTS OF THE INPUT PARAMETERS THAT C OTHERWISE WOULD NOT BE POSSIBLE. C C * * * * * * * * * * * * C C C C M C THE NUMBER OF GRID POINTS IN THE INTERVAL (A,B). THE GRID POINTS C IN THE THETA-DIRECTION ARE GIVEN BY THETA(I) = A + (I-0.5)DTHETA C FOR I=1,2,...,M WHERE DTHETA =(B-A)/M. M MUST BE GREATER THAN 2. C C MBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT THETA = A AND C THETA = B. C C = 1 IF THE SOLUTION IS SPECIFIED AT THETA = A AND THETA = B. C (SEE NOTE 3 BELOW) C C = 2 IF THE SOLUTION IS SPECIFIED AT THETA = A AND THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO THETA IS SPECIFIED AT C THETA = B (SEE NOTES 2 AND 3 BELOW). C C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO THETA IS C SPECIFIED AT THETA = A (SEE NOTES 1, 2 BELOW) AND THETA = B. C C = 4 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO THETA IS C SPECIFIED AT THETA = A (SEE NOTES 1 AND 2 BELOW) AND THE C SOLUTION IS SPECIFIED AT THETA = B. C C = 5 IF THE SOLUTION IS UNSPECIFIED AT THETA = A = 0 AND THE C SOLUTION IS SPECIFIED AT THETA = B. (SEE NOTE 3 BELOW) C C = 6 IF THE SOLUTION IS UNSPECIFIED AT THETA = A = 0 AND THE C DERIVATIVE OF THE SOLUTION WITH RESPECT TO THETA IS C SPECIFIED AT THETA = B (SEE NOTE 2 BELOW). C C = 7 IF THE SOLUTION IS SPECIFIED AT THETA = A AND THE C SOLUTION IS UNSPECIFIED AT THETA = B = PI. (SEE NOTE 3 BELOW) C C = 8 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO C THETA IS SPECIFIED AT THETA = A (SEE NOTE 1 BELOW) C AND THE SOLUTION IS UNSPECIFIED AT THETA = B = PI. C C = 9 IF THE SOLUTION IS UNSPECIFIED AT THETA = A = 0 AND C THETA = B = PI. C C NOTES: 1. IF A = 0, DO NOT USE MBDCND = 3, 4, OR 8, C BUT INSTEAD USE MBDCND = 5, 6, OR 9. C C 2. IF B = PI, DO NOT USE MBDCND = 2, 3, OR 6, C BUT INSTEAD USE MBDCND = 7, 8, OR 9. C C 3. WHEN THE SOLUTION IS SPECIFIED AT THETA = 0 AND/OR C THETA = PI AND THE OTHER BOUNDARY CONDITIONS ARE C COMBINATIONS OF UNSPECIFIED, NORMAL DERIVATIVE, OR C PERIODICITY A SINGULAR SYSTEM RESULTS. THE UNIQUE C SOLUTION IS DETERMINED BY EXTRAPOLATION TO THE C SPECIFICATION OF THE SOLUTION AT EITHER THETA = 0 OR C THETA = PI. BUT IN THESE CASES THE RIGHT SIDE OF THE C SYSTEM WILL BE PERTURBED BY THE CONSTANT PERTRB. C C BDA C A ONE-DIMENSIONAL ARRAY OF LENGTH N THAT SPECIFIES THE BOUNDARY C VALUES (IF ANY) OF THE SOLUTION AT THETA = A. WHEN C MBDCND = 1, 2, OR 7, C C BDA(J) = U(A,PHI(J)) , J=1,2,...,N. C C WHEN MBDCND = 3, 4, OR 8, C C BDA(J) = (D/DTHETA)U(A,PHI(J)) , J=1,2,...,N. C C WHEN MBDCND HAS ANY OTHER VALUE, BDA IS A DUMMY VARIABLE. C C BDB C A ONE-DIMENSIONAL ARRAY OF LENGTH N THAT SPECIFIES THE BOUNDARY C VALUES OF THE SOLUTION AT THETA = B. WHEN MBDCND = 1,4, OR 5, C C BDB(J) = U(B,PHI(J)) , J=1,2,...,N. C C WHEN MBDCND = 2,3, OR 6, C C BDB(J) = (D/DTHETA)U(B,PHI(J)) , J=1,2,...,N. C C WHEN MBDCND HAS ANY OTHER VALUE, BDB IS A DUMMY VARIABLE. C C C,D C THE RANGE OF PHI (LONGITUDE), I.E. C .LE. PHI .LE. D. C C MUST BE LESS THAN D. IF D-C = 2*PI, PERIODIC BOUNDARY C CONDITIONS ARE USUALLY PRESCRIBED. C C N C THE NUMBER OF UNKNOWNS IN THE INTERVAL (C,D). THE UNKNOWNS IN C THE PHI-DIRECTION ARE GIVEN BY PHI(J) = C + (J-0.5)DPHI, C J=1,2,...,N, WHERE DPHI = (D-C)/N. N MUST BE GREATER THAN 2. C C NBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT PHI = C C AND PHI = D. C C = 0 IF THE SOLUTION IS PERIODIC IN PHI, I.E. C U(I,J) = U(I,N+J). C C = 1 IF THE SOLUTION IS SPECIFIED AT PHI = C AND PHI = D C (SEE NOTE BELOW). C C = 2 IF THE SOLUTION IS SPECIFIED AT PHI = C AND THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO PHI IS SPECIFIED AT C PHI = D (SEE NOTE BELOW). C C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO PHI IS C SPECIFIED AT PHI = C AND PHI = D. C C = 4 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO PHI IS C SPECIFIED AT PHI = C AND THE SOLUTION IS SPECIFIED AT C PHI = D (SEE NOTE BELOW). C C NOTE: WHEN NBDCND = 1, 2, OR 4, DO NOT USE MBDCND = 5, 6, 7, 8, C OR 9 (THE FORMER INDICATES THAT THE SOLUTION IS SPECIFIED AT C A POLE; THE LATTER INDICATES THE SOLUTION IS UNSPECIFIED). USE C INSTEAD MBDCND = 1 OR 2. C C BDC C A ONE DIMENSIONAL ARRAY OF LENGTH M THAT SPECIFIES THE BOUNDARY C VALUES OF THE SOLUTION AT PHI = C. WHEN NBDCND = 1 OR 2, C C BDC(I) = U(THETA(I),C) , I=1,2,...,M. C C WHEN NBDCND = 3 OR 4, C C BDC(I) = (D/DPHI)U(THETA(I),C), I=1,2,...,M. C C WHEN NBDCND = 0, BDC IS A DUMMY VARIABLE. C C BDD C A ONE-DIMENSIONAL ARRAY OF LENGTH M THAT SPECIFIES THE BOUNDARY C VALUES OF THE SOLUTION AT PHI = D. WHEN NBDCND = 1 OR 4, C C BDD(I) = U(THETA(I),D) , I=1,2,...,M. C C WHEN NBDCND = 2 OR 3, C C BDD(I) = (D/DPHI)U(THETA(I),D) , I=1,2,...,M. C C WHEN NBDCND = 0, BDD IS A DUMMY VARIABLE. C C ELMBDA C THE CONSTANT LAMBDA IN THE HELMHOLTZ EQUATION. IF LAMBDA IS C GREATER THAN 0, A SOLUTION MAY NOT EXIST. HOWEVER, HSTSSP WILL C ATTEMPT TO FIND A SOLUTION. C C F C A TWO-DIMENSIONAL ARRAY THAT SPECIFIES THE VALUES OF THE RIGHT C SIDE OF THE HELMHOLTZ EQUATION. FOR I=1,2,...,M AND J=1,2,...,N C C F(I,J) = F(THETA(I),PHI(J)) . C C F MUST BE DIMENSIONED AT LEAST M X N. C C IDIMF C THE ROW (OR FIRST) DIMENSION OF THE ARRAY F AS IT APPEARS IN THE C PROGRAM CALLING HSTSSP. THIS PARAMETER IS USED TO SPECIFY THE C VARIABLE DIMENSION OF F. IDIMF MUST BE AT LEAST M. C C W C A ONE-DIMENSIONAL ARRAY THAT MUST BE PROVIDED BY THE USER FOR C WORK SPACE. W MAY REQUIRE UP TO 13M + 4N + M*INT(LOG2(N)) C LOCATIONS. THE ACTUAL NUMBER OF LOCATIONS USED IS COMPUTED BY C HSTSSP AND IS RETURNED IN THE LOCATION W(1). C C C * * * * * * ON OUTPUT * * * * * * C C F C CONTAINS THE SOLUTION U(I,J) OF THE FINITE DIFFERENCE C APPROXIMATION FOR THE GRID POINT (THETA(I),PHI(J)) FOR C I=1,2,...,M, J=1,2,...,N. C C PERTRB C IF A COMBINATION OF PERIODIC, DERIVATIVE, OR UNSPECIFIED C BOUNDARY CONDITIONS IS SPECIFIED FOR A POISSON EQUATION C (LAMBDA = 0), A SOLUTION MAY NOT EXIST. PERTRB IS A CON- C STANT, CALCULATED AND SUBTRACTED FROM F, WHICH ENSURES C THAT A SOLUTION EXISTS. HSTSSP THEN COMPUTES THIS C SOLUTION, WHICH IS A LEAST SQUARES SOLUTION TO THE C ORIGINAL APPROXIMATION. THIS SOLUTION PLUS ANY CONSTANT IS ALSO C A SOLUTION; HENCE, THE SOLUTION IS NOT UNIQUE. THE VALUE OF C PERTRB SHOULD BE SMALL COMPARED TO THE RIGHT SIDE F. C OTHERWISE, A SOLUTION IS OBTAINED TO AN ESSENTIALLY DIFFERENT C PROBLEM. THIS COMPARISON SHOULD ALWAYS BE MADE TO INSURE THAT C A MEANINGFUL SOLUTION HAS BEEN OBTAINED. C C IERROR C AN ERROR FLAG THAT INDICATES INVALID INPUT PARAMETERS. C EXCEPT TO NUMBERS 0 AND 14, A SOLUTION IS NOT ATTEMPTED. C C = 0 NO ERROR C C = 1 A .LT. 0 OR B .GT. PI C C = 2 A .GE. B C C = 3 MBDCND .LT. 1 OR MBDCND .GT. 9 C C = 4 C .GE. D C C = 5 N .LE. 2 C C = 6 NBDCND .LT. 0 OR NBDCND .GT. 4 C C = 7 A .GT. 0 AND MBDCND = 5, 6, OR 9 C C = 8 A = 0 AND MBDCND = 3, 4, OR 8 C C = 9 B .LT. PI AND MBDCND .GE. 7 C C = 10 B = PI AND MBDCND = 2,3, OR 6 C C = 11 MBDCND .GE. 5 AND NDBCND = 1, 2, OR 4 C C = 12 IDIMF .LT. M C C = 13 M .LE. 2 C C = 14 LAMBDA .GT. 0 C C SINCE THIS IS THE ONLY MEANS OF INDICATING A POSSIBLY C INCORRECT CALL TO HSTSSP, THE USER SHOULD TEST IERROR AFTER C THE CALL. C C W C W(1) CONTAINS THE REQUIRED LENGTH OF W. C C C * * * * * * * PROGRAM SPECIFICATIONS * * * * * * * * * * * * C C DIMENSION OF BDA(N),BDB(N),BDC(M),BDD(M),F(IDIMF,N), C ARGUMENTS W(SEE ARGUMENT LIST) C C LATEST JUNE 1, 1977 C REVISION C C SUBPROGRAMS HSTSSP,POISTG,POSTG2,GENBUN,POISD2,POISN2,POISP2, C REQUIRED COSGEN,MERGE,TRIX,TRI3,PIMACH 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 AT NCAR IN APRIL, 1977 C C ALGORITHM THIS SUBROUTINE DEFINES THE FINITE-DIFFERENCE C EQUATIONS, INCORPORATES BOUNDARY DATA, ADJUSTS THE C RIGHT SIDE WHEN THE SYSTEM IS SINGULAR AND CALLS C EITHER POISTG OR GENBUN WHICH SOLVES THE LINEAR C SYSTEM OF EQUATIONS. C C SPACE 8427(DECIMAL) = 20353(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 HSTSSP IS ROUGHLY PROPORTIONAL C TO M*N*LOG2(N). SOME TYPICAL VALUES ARE LISTED IN C THE TABLE BELOW. C THE SOLUTION PROCESS EMPLOYED RESULTS IN A LOSS C OF NO MORE THAN FOUR SIGNIFICANT DIGITS FOR N AND M C AS LARGE AS 64. MORE DETAILED INFORMATION ABOUT C ACCURACY CAN BE FOUND IN THE DOCUMENTATION FOR C SUBROUTINE POISTG WHICH IS THE ROUTINE THAT C ACTUALLY SOLVES THE FINITE DIFFERENCE EQUATIONS. C C C M(=N) MBDCND NBDCND T(MSECS) C ----- ------ ------ -------- C C 32 1-9 1-4 56 C 64 1-9 1-4 230 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 F(IDIMF,1) ,BDA(1) ,BDB(1) ,BDC(1) , 1 BDD(1) ,W(1) IERROR = 0 PI = PIMACH(DUM) IF (A.LT.0. .OR. B.GT.PI) IERROR = 1 IF (A .GE. B) IERROR = 2 IF (MBDCND.LE.0 .OR. MBDCND.GT.9) IERROR = 3 IF (C .GE. D) IERROR = 4 IF (N .LE. 2) IERROR = 5 IF (NBDCND.LT.0 .OR. NBDCND.GE.5) IERROR = 6 IF (A.GT.0. .AND. (MBDCND.EQ.5 .OR. MBDCND.EQ.6 .OR. MBDCND.EQ.9)) 1 IERROR = 7 IF (A.EQ.0. .AND. (MBDCND.EQ.3 .OR. MBDCND.EQ.4 .OR. MBDCND.EQ.8)) 1 IERROR = 8 IF (B.LT.PI .AND. MBDCND.GE.7) IERROR = 9 IF (B.EQ.PI .AND. (MBDCND.EQ.2 .OR. MBDCND.EQ.3 .OR. MBDCND.EQ.6)) 1 IERROR = 10 IF (MBDCND.GE.5 .AND. 1 (NBDCND.EQ.1 .OR. NBDCND.EQ.2 .OR. NBDCND.EQ.4)) IERROR = 11 IF (IDIMF .LT. M) IERROR = 12 IF (M .LE. 2) IERROR = 13 IF (IERROR .NE. 0) RETURN DELTAR = (B-A)/FLOAT(M) DLRSQ = DELTAR**2 DELTHT = (D-C)/FLOAT(N) DLTHSQ = DELTHT**2 NP = NBDCND+1 ISW = 1 JSW = 1 MB = MBDCND IF (ELMBDA .NE. 0.) GO TO 105 GO TO (101,102,105,103,101,105,101,105,105),MBDCND 101 IF (A.NE.0. .OR. B.NE.PI) GO TO 105 MB = 9 GO TO 104 102 IF (A .NE. 0.) GO TO 105 MB = 6 GO TO 104 103 IF (B .NE. PI) GO TO 105 MB = 8 104 JSW = 2 105 CONTINUE C C DEFINE A,B,C COEFFICIENTS IN W-ARRAY. C IWB = M IWC = IWB+M IWR = IWC+M IWS = IWR+M DO 106 I=1,M J = IWR+I W(J) = SIN(A+(FLOAT(I)-0.5)*DELTAR) W(I) = SIN((A+FLOAT(I-1)*DELTAR))/DLRSQ 106 CONTINUE MM1 = M-1 DO 107 I=1,MM1 K = IWC+I W(K) = W(I+1) J = IWR+I K = IWB+I W(K) = ELMBDA*W(J)-(W(I)+W(I+1)) 107 CONTINUE W(IWR) = SIN(B)/DLRSQ W(IWC) = ELMBDA*W(IWS)-(W(M)+W(IWR)) DO 109 I=1,M J = IWR+I A1 = W(J) DO 108 J=1,N F(I,J) = A1*F(I,J) 108 CONTINUE 109 CONTINUE C C ENTER BOUNDARY DATA FOR THETA-BOUNDARIES. C GO TO (110,110,112,112,114,114,110,112,114),MB 110 A1 = 2.*W(1) W(IWB+1) = W(IWB+1)-W(1) DO 111 J=1,N F(1,J) = F(1,J)-A1*BDA(J) 111 CONTINUE GO TO 114 112 A1 = DELTAR*W(1) W(IWB+1) = W(IWB+1)+W(1) DO 113 J=1,N F(1,J) = F(1,J)+A1*BDA(J) 113 CONTINUE 114 GO TO (115,117,117,115,115,117,119,119,119),MB 115 A1 = 2.*W(IWR) W(IWC) = W(IWC)-W(IWR) DO 116 J=1,N F(M,J) = F(M,J)-A1*BDB(J) 116 CONTINUE GO TO 119 117 A1 = DELTAR*W(IWR) W(IWC) = W(IWC)+W(IWR) DO 118 J=1,N F(M,J) = F(M,J)-A1*BDB(J) 118 CONTINUE C C ENTER BOUNDARY DATA FOR PHI-BOUNDARIES. C 119 A1 = 2./DLTHSQ GO TO (129,120,120,122,122),NP 120 DO 121 I=1,M J = IWR+I F(I,1) = F(I,1)-A1*BDC(I)/W(J) 121 CONTINUE GO TO 124 122 A1 = 1./DELTHT DO 123 I=1,M J = IWR+I F(I,1) = F(I,1)+A1*BDC(I)/W(J) 123 CONTINUE 124 A1 = 2./DLTHSQ GO TO (129,125,127,127,125),NP 125 DO 126 I=1,M J = IWR+I F(I,N) = F(I,N)-A1*BDD(I)/W(J) 126 CONTINUE GO TO 129 127 A1 = 1./DELTHT DO 128 I=1,M J = IWR+I F(I,N) = F(I,N)-A1*BDD(I)/W(J) 128 CONTINUE 129 CONTINUE C C ADJUST RIGHT SIDE OF SINGULAR PROBLEMS TO INSURE EXISTENCE OF A C SOLUTION. C PERTRB = 0. IF (ELMBDA) 139,131,130 130 IERROR = 14 GO TO 139 131 GO TO (139,139,132,139,139,132,139,132,132),MB 132 GO TO (133,139,139,133,139),NP 133 CONTINUE ISW = 2 DO 135 J=1,N DO 134 I=1,M PERTRB = PERTRB+F(I,J) 134 CONTINUE 135 CONTINUE A1 = FLOAT(N)*(COS(A)-COS(B))/(2.*SIN(0.5*DELTAR)) PERTRB = PERTRB/A1 DO 137 I=1,M J = IWR+I A1 = PERTRB*W(J) DO 136 J=1,N F(I,J) = F(I,J)-A1 136 CONTINUE 137 CONTINUE A2 = 0. A3 = 0. DO 138 J=1,N A2 = A2+F(1,J) A3 = A3+F(M,J) 138 CONTINUE A2 = A2/W(IWR+1) A3 = A3/W(IWS) 139 CONTINUE C C MULTIPLY I-TH EQUATION THROUGH BY R(I)*DELTHT**2 C DO 141 I=1,M J = IWR+I A1 = DLTHSQ*W(J) W(I) = A1*W(I) J = IWC+I W(J) = A1*W(J) J = IWB+I W(J) = A1*W(J) DO 140 J=1,N F(I,J) = A1*F(I,J) 140 CONTINUE 141 CONTINUE LP = NBDCND W(1) = 0. W(IWR) = 0. C C CALL POISTG OR GENBUN TO SOLVE THE SYSTEM OF EQUATIONS. C IF (NBDCND .EQ. 0) GO TO 142 CALL POISTG (LP,N,1,M,W,W(IWB+1),W(IWC+1),IDIMF,F,IERR1,W(IWR+1)) GO TO 143 142 CALL GENBUN (LP,N,1,M,W,W(IWB+1),W(IWC+1),IDIMF,F,IERR1,W(IWR+1)) 143 CONTINUE W(1) = W(IWR+1)+3.*FLOAT(M) IF (ISW.NE.2 .OR. JSW.NE.2) GO TO 150 IF (MB .NE. 8) GO TO 145 A1 = 0. DO 144 J=1,N A1 = A1+F(M,J) 144 CONTINUE A1 = (A1-DLRSQ*A3/16.)/FLOAT(N) IF (NBDCND .EQ. 3) A1 = A1+(BDD(M)-BDC(M))/(D-C) A1 = BDB(1)-A1 GO TO 147 145 A1 = 0. DO 146 J=1,N A1 = A1+F(1,J) 146 CONTINUE A1 = (A1-DLRSQ*A2/16.)/FLOAT(N) IF (NBDCND .EQ. 3) A1 = A1+(BDD(1)-BDC(1))/(D-C) A1 = BDA(1)-A1 147 DO 149 I=1,M DO 148 J=1,N F(I,J) = F(I,J)+A1 148 CONTINUE 149 CONTINUE 150 CONTINUE RETURN END