*DECK DUMACH DOUBLE PRECISION FUNCTION DUMACH () C***BEGIN PROLOGUE DUMACH C***PURPOSE Compute the unit roundoff of the machine. C***CATEGORY R1 C***TYPE DOUBLE PRECISION (RUMACH-S, DUMACH-D) C***KEYWORDS MACHINE CONSTANTS C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C *Usage: C DOUBLE PRECISION A, DUMACH C A = DUMACH() C C *Function Return Values: C A : the unit roundoff of the machine. C C *Description: C The unit roundoff is defined as the smallest positive machine C number u such that 1.0 + u .ne. 1.0. This is computed by DUMACH C in a machine-independent manner. C C***REFERENCES (NONE) C***ROUTINES CALLED DUMSUM C***REVISION HISTORY (YYYYMMDD) C 19930216 DATE WRITTEN C 19930818 Added SLATEC-format prologue. (FNF) C 20030707 Added DUMSUM to force normal storage of COMP. (ACH) C***END PROLOGUE DUMACH C DOUBLE PRECISION U, COMP C***FIRST EXECUTABLE STATEMENT DUMACH U = 1.0D0 10 U = U*0.5D0 CALL DUMSUM(1.0D0, U, COMP) IF (COMP .NE. 1.0D0) GO TO 10 DUMACH = U*2.0D0 RETURN C----------------------- End of Function DUMACH ------------------------ END SUBROUTINE DUMSUM(A,B,C) C Routine to force normal storing of A + B, for DUMACH. DOUBLE PRECISION A, B, C C = A + B RETURN END *DECK DCFODE SUBROUTINE DCFODE (METH, ELCO, TESCO) C***BEGIN PROLOGUE DCFODE C***SUBSIDIARY C***PURPOSE Set ODE integrator coefficients. C***TYPE DOUBLE PRECISION (SCFODE-S, DCFODE-D) C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C C DCFODE is called by the integrator routine to set coefficients C needed there. The coefficients for the current method, as C given by the value of METH, are set for all orders and saved. C The maximum order assumed here is 12 if METH = 1 and 5 if METH = 2. C (A smaller value of the maximum order is also allowed.) C DCFODE is called once at the beginning of the problem, C and is not called again unless and until METH is changed. C C The ELCO array contains the basic method coefficients. C The coefficients el(i), 1 .le. i .le. nq+1, for the method of C order nq are stored in ELCO(i,nq). They are given by a genetrating C polynomial, i.e., C l(x) = el(1) + el(2)*x + ... + el(nq+1)*x**nq. C For the implicit Adams methods, l(x) is given by C dl/dx = (x+1)*(x+2)*...*(x+nq-1)/factorial(nq-1), l(-1) = 0. C For the BDF methods, l(x) is given by C l(x) = (x+1)*(x+2)* ... *(x+nq)/K, C where K = factorial(nq)*(1 + 1/2 + ... + 1/nq). C C The TESCO array contains test constants used for the C local error test and the selection of step size and/or order. C At order nq, TESCO(k,nq) is used for the selection of step C size at order nq - 1 if k = 1, at order nq if k = 2, and at order C nq + 1 if k = 3. C C***SEE ALSO DLSODE C***ROUTINES CALLED (NONE) C***REVISION HISTORY (YYMMDD) C 791129 DATE WRITTEN C 890501 Modified prologue to SLATEC/LDOC format. (FNF) C 890503 Minor cosmetic changes. (FNF) C 930809 Renamed to allow single/double precision versions. (ACH) C***END PROLOGUE DCFODE C**End INTEGER METH INTEGER I, IB, NQ, NQM1, NQP1 DOUBLE PRECISION ELCO, TESCO DOUBLE PRECISION AGAMQ, FNQ, FNQM1, PC, PINT, RAGQ, 1 RQFAC, RQ1FAC, TSIGN, XPIN DIMENSION ELCO(13,12), TESCO(3,12) DIMENSION PC(12) C C***FIRST EXECUTABLE STATEMENT DCFODE GO TO (100, 200), METH C 100 ELCO(1,1) = 1.0D0 ELCO(2,1) = 1.0D0 TESCO(1,1) = 0.0D0 TESCO(2,1) = 2.0D0 TESCO(1,2) = 1.0D0 TESCO(3,12) = 0.0D0 PC(1) = 1.0D0 RQFAC = 1.0D0 DO 140 NQ = 2,12 C----------------------------------------------------------------------- C The PC array will contain the coefficients of the polynomial C p(x) = (x+1)*(x+2)*...*(x+nq-1). C Initially, p(x) = 1. C----------------------------------------------------------------------- RQ1FAC = RQFAC RQFAC = RQFAC/NQ NQM1 = NQ - 1 FNQM1 = NQM1 NQP1 = NQ + 1 C Form coefficients of p(x)*(x+nq-1). ---------------------------------- PC(NQ) = 0.0D0 DO 110 IB = 1,NQM1 I = NQP1 - IB 110 PC(I) = PC(I-1) + FNQM1*PC(I) PC(1) = FNQM1*PC(1) C Compute integral, -1 to 0, of p(x) and x*p(x). ----------------------- PINT = PC(1) XPIN = PC(1)/2.0D0 TSIGN = 1.0D0 DO 120 I = 2,NQ TSIGN = -TSIGN PINT = PINT + TSIGN*PC(I)/I 120 XPIN = XPIN + TSIGN*PC(I)/(I+1) C Store coefficients in ELCO and TESCO. -------------------------------- ELCO(1,NQ) = PINT*RQ1FAC ELCO(2,NQ) = 1.0D0 DO 130 I = 2,NQ 130 ELCO(I+1,NQ) = RQ1FAC*PC(I)/I AGAMQ = RQFAC*XPIN RAGQ = 1.0D0/AGAMQ TESCO(2,NQ) = RAGQ IF (NQ .LT. 12) TESCO(1,NQP1) = RAGQ*RQFAC/NQP1 TESCO(3,NQM1) = RAGQ 140 CONTINUE RETURN C 200 PC(1) = 1.0D0 RQ1FAC = 1.0D0 DO 230 NQ = 1,5 C----------------------------------------------------------------------- C The PC array will contain the coefficients of the polynomial C p(x) = (x+1)*(x+2)*...*(x+nq). C Initially, p(x) = 1. C----------------------------------------------------------------------- FNQ = NQ NQP1 = NQ + 1 C Form coefficients of p(x)*(x+nq). ------------------------------------ PC(NQP1) = 0.0D0 DO 210 IB = 1,NQ I = NQ + 2 - IB 210 PC(I) = PC(I-1) + FNQ*PC(I) PC(1) = FNQ*PC(1) C Store coefficients in ELCO and TESCO. -------------------------------- DO 220 I = 1,NQP1 220 ELCO(I,NQ) = PC(I)/PC(2) ELCO(2,NQ) = 1.0D0 TESCO(1,NQ) = RQ1FAC TESCO(2,NQ) = NQP1/ELCO(1,NQ) TESCO(3,NQ) = (NQ+2)/ELCO(1,NQ) RQ1FAC = RQ1FAC/FNQ 230 CONTINUE RETURN C----------------------- END OF SUBROUTINE DCFODE ---------------------- END *DECK DINTDY SUBROUTINE DINTDY (T, K, YH, NYH, DKY, IFLAG) C***BEGIN PROLOGUE DINTDY C***SUBSIDIARY C***PURPOSE Interpolate solution derivatives. C***TYPE DOUBLE PRECISION (SINTDY-S, DINTDY-D) C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C C DINTDY computes interpolated values of the K-th derivative of the C dependent variable vector y, and stores it in DKY. This routine C is called within the package with K = 0 and T = TOUT, but may C also be called by the user for any K up to the current order. C (See detailed instructions in the usage documentation.) C C The computed values in DKY are gotten by interpolation using the C Nordsieck history array YH. This array corresponds uniquely to a C vector-valued polynomial of degree NQCUR or less, and DKY is set C to the K-th derivative of this polynomial at T. C The formula for DKY is: C q C DKY(i) = sum c(j,K) * (T - tn)**(j-K) * h**(-j) * YH(i,j+1) C j=K C where c(j,K) = j*(j-1)*...*(j-K+1), q = NQCUR, tn = TCUR, h = HCUR. C The quantities nq = NQCUR, l = nq+1, N = NEQ, tn, and h are C communicated by COMMON. The above sum is done in reverse order. C IFLAG is returned negative if either K or T is out of bounds. C C***SEE ALSO DLSODE C***ROUTINES CALLED XERRWD C***COMMON BLOCKS DLS001 C***REVISION HISTORY (YYMMDD) C 791129 DATE WRITTEN C 890501 Modified prologue to SLATEC/LDOC format. (FNF) C 890503 Minor cosmetic changes. (FNF) C 930809 Renamed to allow single/double precision versions. (ACH) C 010418 Reduced size of Common block /DLS001/. (ACH) C 031105 Restored 'own' variables to Common block /DLS001/, to C enable interrupt/restart feature. (ACH) C 050427 Corrected roundoff decrement in TP. (ACH) C***END PROLOGUE DINTDY C**End INTEGER K, NYH, IFLAG DOUBLE PRECISION T, YH, DKY DIMENSION YH(NYH,*), DKY(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER I, IC, J, JB, JB2, JJ, JJ1, JP1 DOUBLE PRECISION C, R, S, TP CHARACTER*80 MSG C C***FIRST EXECUTABLE STATEMENT DINTDY IFLAG = 0 IF (K .LT. 0 .OR. K .GT. NQ) GO TO 80 TP = TN - HU - 100.0D0*UROUND*SIGN(ABS(TN) + ABS(HU), HU) IF ((T-TP)*(T-TN) .GT. 0.0D0) GO TO 90 C S = (T - TN)/H IC = 1 IF (K .EQ. 0) GO TO 15 JJ1 = L - K DO 10 JJ = JJ1,NQ 10 IC = IC*JJ 15 C = IC DO 20 I = 1,N 20 DKY(I) = C*YH(I,L) IF (K .EQ. NQ) GO TO 55 JB2 = NQ - K DO 50 JB = 1,JB2 J = NQ - JB JP1 = J + 1 IC = 1 IF (K .EQ. 0) GO TO 35 JJ1 = JP1 - K DO 30 JJ = JJ1,J 30 IC = IC*JJ 35 C = IC DO 40 I = 1,N 40 DKY(I) = C*YH(I,JP1) + S*DKY(I) 50 CONTINUE IF (K .EQ. 0) RETURN 55 R = H**(-K) DO 60 I = 1,N 60 DKY(I) = R*DKY(I) RETURN C 80 MSG = 'DINTDY- K (=I1) illegal ' CALL XERRWD (MSG, 30, 51, 0, 1, K, 0, 0, 0.0D0, 0.0D0) IFLAG = -1 RETURN 90 MSG = 'DINTDY- T (=R1) illegal ' CALL XERRWD (MSG, 30, 52, 0, 0, 0, 0, 1, T, 0.0D0) MSG=' T not in interval TCUR - HU (= R1) to TCUR (=R2) ' CALL XERRWD (MSG, 60, 52, 0, 0, 0, 0, 2, TP, TN) IFLAG = -2 RETURN C----------------------- END OF SUBROUTINE DINTDY ---------------------- END *DECK DPREPJ SUBROUTINE DPREPJ (NEQ, Y, YH, NYH, EWT, FTEM, SAVF, WM, IWM, 1 F, JAC) C***BEGIN PROLOGUE DPREPJ C***SUBSIDIARY C***PURPOSE Compute and process Newton iteration matrix. C***TYPE DOUBLE PRECISION (SPREPJ-S, DPREPJ-D) C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C C DPREPJ is called by DSTODE to compute and process the matrix C P = I - h*el(1)*J , where J is an approximation to the Jacobian. C Here J is computed by the user-supplied routine JAC if C MITER = 1 or 4, or by finite differencing if MITER = 2, 3, or 5. C If MITER = 3, a diagonal approximation to J is used. C J is stored in WM and replaced by P. If MITER .ne. 3, P is then C subjected to LU decomposition in preparation for later solution C of linear systems with P as coefficient matrix. This is done C by DGEFA if MITER = 1 or 2, and by DGBFA if MITER = 4 or 5. C C In addition to variables described in DSTODE and DLSODE prologues, C communication with DPREPJ uses the following: C Y = array containing predicted values on entry. C FTEM = work array of length N (ACOR in DSTODE). C SAVF = array containing f evaluated at predicted y. C WM = real work space for matrices. On output it contains the C inverse diagonal matrix if MITER = 3 and the LU decomposition C of P if MITER is 1, 2 , 4, or 5. C Storage of matrix elements starts at WM(3). C WM also contains the following matrix-related data: C WM(1) = SQRT(UROUND), used in numerical Jacobian increments. C WM(2) = H*EL0, saved for later use if MITER = 3. C IWM = integer work space containing pivot information, starting at C IWM(21), if MITER is 1, 2, 4, or 5. IWM also contains band C parameters ML = IWM(1) and MU = IWM(2) if MITER is 4 or 5. C EL0 = EL(1) (input). C IERPJ = output error flag, = 0 if no trouble, .gt. 0 if C P matrix found to be singular. C JCUR = output flag = 1 to indicate that the Jacobian matrix C (or approximation) is now current. C This routine also uses the COMMON variables EL0, H, TN, UROUND, C MITER, N, NFE, and NJE. C C***SEE ALSO DLSODE C***ROUTINES CALLED DGBFA, DGEFA, DVNORM C***COMMON BLOCKS DLS001 C***REVISION HISTORY (YYMMDD) C 791129 DATE WRITTEN C 890501 Modified prologue to SLATEC/LDOC format. (FNF) C 890504 Minor cosmetic changes. (FNF) C 930809 Renamed to allow single/double precision versions. (ACH) C 010418 Reduced size of Common block /DLS001/. (ACH) C 031105 Restored 'own' variables to Common block /DLS001/, to C enable interrupt/restart feature. (ACH) C***END PROLOGUE DPREPJ C**End EXTERNAL F, JAC INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, EWT, FTEM, SAVF, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), FTEM(*), SAVF(*), 1 WM(*), IWM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER I, I1, I2, IER, II, J, J1, JJ, LENP, 1 MBA, MBAND, MEB1, MEBAND, ML, ML3, MU, NP1 DOUBLE PRECISION CON, DI, FAC, HL0, R, R0, SRUR, YI, YJ, YJJ, 1 DVNORM C C***FIRST EXECUTABLE STATEMENT DPREPJ NJE = NJE + 1 IERPJ = 0 JCUR = 1 HL0 = H*EL0 GO TO (100, 200, 300, 400, 500), MITER C If MITER = 1, call JAC and multiply by scalar. ----------------------- 100 LENP = N*N DO 110 I = 1,LENP 110 WM(I+2) = 0.0D0 CALL JAC (NEQ, TN, Y, 0, 0, WM(3), N) CON = -HL0 DO 120 I = 1,LENP 120 WM(I+2) = WM(I+2)*CON GO TO 240 C If MITER = 2, make N calls to F to approximate J. -------------------- 200 FAC = DVNORM (N, SAVF, EWT) R0 = 1000.0D0*ABS(H)*UROUND*N*FAC IF (R0 .EQ. 0.0D0) R0 = 1.0D0 SRUR = WM(1) J1 = 2 DO 230 J = 1,N YJ = Y(J) R = MAX(SRUR*ABS(YJ),R0/EWT(J)) Y(J) = Y(J) + R FAC = -HL0/R CALL F (NEQ, TN, Y, FTEM) DO 220 I = 1,N 220 WM(I+J1) = (FTEM(I) - SAVF(I))*FAC Y(J) = YJ J1 = J1 + N 230 CONTINUE NFE = NFE + N C Add identity matrix. ------------------------------------------------- 240 J = 3 NP1 = N + 1 DO 250 I = 1,N WM(J) = WM(J) + 1.0D0 250 J = J + NP1 C Do LU decomposition on P. -------------------------------------------- CALL DGEFA (WM(3), N, N, IWM(21), IER) IF (IER .NE. 0) IERPJ = 1 RETURN C If MITER = 3, construct a diagonal approximation to J and P. --------- 300 WM(2) = HL0 R = EL0*0.1D0 DO 310 I = 1,N 310 Y(I) = Y(I) + R*(H*SAVF(I) - YH(I,2)) CALL F (NEQ, TN, Y, WM(3)) NFE = NFE + 1 DO 320 I = 1,N R0 = H*SAVF(I) - YH(I,2) DI = 0.1D0*R0 - H*(WM(I+2) - SAVF(I)) WM(I+2) = 1.0D0 IF (ABS(R0) .LT. UROUND/EWT(I)) GO TO 320 IF (ABS(DI) .EQ. 0.0D0) GO TO 330 WM(I+2) = 0.1D0*R0/DI 320 CONTINUE RETURN 330 IERPJ = 1 RETURN C If MITER = 4, call JAC and multiply by scalar. ----------------------- 400 ML = IWM(1) MU = IWM(2) ML3 = ML + 3 MBAND = ML + MU + 1 MEBAND = MBAND + ML LENP = MEBAND*N DO 410 I = 1,LENP 410 WM(I+2) = 0.0D0 CALL JAC (NEQ, TN, Y, ML, MU, WM(ML3), MEBAND) CON = -HL0 DO 420 I = 1,LENP 420 WM(I+2) = WM(I+2)*CON GO TO 570 C If MITER = 5, make MBAND calls to F to approximate J. ---------------- 500 ML = IWM(1) MU = IWM(2) MBAND = ML + MU + 1 MBA = MIN(MBAND,N) MEBAND = MBAND + ML MEB1 = MEBAND - 1 SRUR = WM(1) FAC = DVNORM (N, SAVF, EWT) R0 = 1000.0D0*ABS(H)*UROUND*N*FAC IF (R0 .EQ. 0.0D0) R0 = 1.0D0 DO 560 J = 1,MBA DO 530 I = J,N,MBAND YI = Y(I) R = MAX(SRUR*ABS(YI),R0/EWT(I)) 530 Y(I) = Y(I) + R CALL F (NEQ, TN, Y, FTEM) DO 550 JJ = J,N,MBAND Y(JJ) = YH(JJ,1) YJJ = Y(JJ) R = MAX(SRUR*ABS(YJJ),R0/EWT(JJ)) FAC = -HL0/R I1 = MAX(JJ-MU,1) I2 = MIN(JJ+ML,N) II = JJ*MEB1 - ML + 2 DO 540 I = I1,I2 540 WM(II+I) = (FTEM(I) - SAVF(I))*FAC 550 CONTINUE 560 CONTINUE NFE = NFE + MBA C Add identity matrix. ------------------------------------------------- 570 II = MBAND + 2 DO 580 I = 1,N WM(II) = WM(II) + 1.0D0 580 II = II + MEBAND C Do LU decomposition of P. -------------------------------------------- CALL DGBFA (WM(3), MEBAND, N, ML, MU, IWM(21), IER) IF (IER .NE. 0) IERPJ = 1 RETURN C----------------------- END OF SUBROUTINE DPREPJ ---------------------- END *DECK DSOLSY SUBROUTINE DSOLSY (WM, IWM, X, TEM) C***BEGIN PROLOGUE DSOLSY C***SUBSIDIARY C***PURPOSE ODEPACK linear system solver. C***TYPE DOUBLE PRECISION (SSOLSY-S, DSOLSY-D) C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C C This routine manages the solution of the linear system arising from C a chord iteration. It is called if MITER .ne. 0. C If MITER is 1 or 2, it calls DGESL to accomplish this. C If MITER = 3 it updates the coefficient h*EL0 in the diagonal C matrix, and then computes the solution. C If MITER is 4 or 5, it calls DGBSL. C Communication with DSOLSY uses the following variables: C WM = real work space containing the inverse diagonal matrix if C MITER = 3 and the LU decomposition of the matrix otherwise. C Storage of matrix elements starts at WM(3). C WM also contains the following matrix-related data: C WM(1) = SQRT(UROUND) (not used here), C WM(2) = HL0, the previous value of h*EL0, used if MITER = 3. C IWM = integer work space containing pivot information, starting at C IWM(21), if MITER is 1, 2, 4, or 5. IWM also contains band C parameters ML = IWM(1) and MU = IWM(2) if MITER is 4 or 5. C X = the right-hand side vector on input, and the solution vector C on output, of length N. C TEM = vector of work space of length N, not used in this version. C IERSL = output flag (in COMMON). IERSL = 0 if no trouble occurred. C IERSL = 1 if a singular matrix arose with MITER = 3. C This routine also uses the COMMON variables EL0, H, MITER, and N. C C***SEE ALSO DLSODE C***ROUTINES CALLED DGBSL, DGESL C***COMMON BLOCKS DLS001 C***REVISION HISTORY (YYMMDD) C 791129 DATE WRITTEN C 890501 Modified prologue to SLATEC/LDOC format. (FNF) C 890503 Minor cosmetic changes. (FNF) C 930809 Renamed to allow single/double precision versions. (ACH) C 010418 Reduced size of Common block /DLS001/. (ACH) C 031105 Restored 'own' variables to Common block /DLS001/, to C enable interrupt/restart feature. (ACH) C***END PROLOGUE DSOLSY C**End INTEGER IWM DOUBLE PRECISION WM, X, TEM DIMENSION WM(*), IWM(*), X(*), TEM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER I, MEBAND, ML, MU DOUBLE PRECISION DI, HL0, PHL0, R C C***FIRST EXECUTABLE STATEMENT DSOLSY IERSL = 0 GO TO (100, 100, 300, 400, 400), MITER 100 CALL DGESL (WM(3), N, N, IWM(21), X, 0) RETURN C 300 PHL0 = WM(2) HL0 = H*EL0 WM(2) = HL0 IF (HL0 .EQ. PHL0) GO TO 330 R = HL0/PHL0 DO 320 I = 1,N DI = 1.0D0 - R*(1.0D0 - 1.0D0/WM(I+2)) IF (ABS(DI) .EQ. 0.0D0) GO TO 390 320 WM(I+2) = 1.0D0/DI 330 DO 340 I = 1,N 340 X(I) = WM(I+2)*X(I) RETURN 390 IERSL = 1 RETURN C 400 ML = IWM(1) MU = IWM(2) MEBAND = 2*ML + MU + 1 CALL DGBSL (WM(3), MEBAND, N, ML, MU, IWM(21), X, 0) RETURN C----------------------- END OF SUBROUTINE DSOLSY ---------------------- END *DECK DSRCOM SUBROUTINE DSRCOM (RSAV, ISAV, JOB) C***BEGIN PROLOGUE DSRCOM C***SUBSIDIARY C***PURPOSE Save/restore ODEPACK COMMON blocks. C***TYPE DOUBLE PRECISION (SSRCOM-S, DSRCOM-D) C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C C This routine saves or restores (depending on JOB) the contents of C the COMMON block DLS001, which is used internally C by one or more ODEPACK solvers. C C RSAV = real array of length 218 or more. C ISAV = integer array of length 37 or more. C JOB = flag indicating to save or restore the COMMON blocks: C JOB = 1 if COMMON is to be saved (written to RSAV/ISAV) C JOB = 2 if COMMON is to be restored (read from RSAV/ISAV) C A call with JOB = 2 presumes a prior call with JOB = 1. C C***SEE ALSO DLSODE C***ROUTINES CALLED (NONE) C***COMMON BLOCKS DLS001 C***REVISION HISTORY (YYMMDD) C 791129 DATE WRITTEN C 890501 Modified prologue to SLATEC/LDOC format. (FNF) C 890503 Minor cosmetic changes. (FNF) C 921116 Deleted treatment of block /EH0001/. (ACH) C 930801 Reduced Common block length by 2. (ACH) C 930809 Renamed to allow single/double precision versions. (ACH) C 010418 Reduced Common block length by 209+12. (ACH) C 031105 Restored 'own' variables to Common block /DLS001/, to C enable interrupt/restart feature. (ACH) C 031112 Added SAVE statement for data-loaded constants. C***END PROLOGUE DSRCOM C**End INTEGER ISAV, JOB INTEGER ILS INTEGER I, LENILS, LENRLS DOUBLE PRECISION RSAV, RLS DIMENSION RSAV(*), ISAV(*) SAVE LENRLS, LENILS COMMON /DLS001/ RLS(218), ILS(37) DATA LENRLS/218/, LENILS/37/ C C***FIRST EXECUTABLE STATEMENT DSRCOM IF (JOB .EQ. 2) GO TO 100 C DO 10 I = 1,LENRLS 10 RSAV(I) = RLS(I) DO 20 I = 1,LENILS 20 ISAV(I) = ILS(I) RETURN C 100 CONTINUE DO 110 I = 1,LENRLS 110 RLS(I) = RSAV(I) DO 120 I = 1,LENILS 120 ILS(I) = ISAV(I) RETURN C----------------------- END OF SUBROUTINE DSRCOM ---------------------- END *DECK DSTODE SUBROUTINE DSTODE (NEQ, Y, YH, NYH, YH1, EWT, SAVF, ACOR, 1 WM, IWM, F, JAC, PJAC, SLVS) C***BEGIN PROLOGUE DSTODE C***SUBSIDIARY C***PURPOSE Performs one step of an ODEPACK integration. C***TYPE DOUBLE PRECISION (SSTODE-S, DSTODE-D) C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C C DSTODE performs one step of the integration of an initial value C problem for a system of ordinary differential equations. C Note: DSTODE is independent of the value of the iteration method C indicator MITER, when this is .ne. 0, and hence is independent C of the type of chord method used, or the Jacobian structure. C Communication with DSTODE is done with the following variables: C C NEQ = integer array containing problem size in NEQ(1), and C passed as the NEQ argument in all calls to F and JAC. C Y = an array of length .ge. N used as the Y argument in C all calls to F and JAC. C YH = an NYH by LMAX array containing the dependent variables C and their approximate scaled derivatives, where C LMAX = MAXORD + 1. YH(i,j+1) contains the approximate C j-th derivative of y(i), scaled by h**j/factorial(j) C (j = 0,1,...,NQ). on entry for the first step, the first C two columns of YH must be set from the initial values. C NYH = a constant integer .ge. N, the first dimension of YH. C YH1 = a one-dimensional array occupying the same space as YH. C EWT = an array of length N containing multiplicative weights C for local error measurements. Local errors in Y(i) are C compared to 1.0/EWT(i) in various error tests. C SAVF = an array of working storage, of length N. C Also used for input of YH(*,MAXORD+2) when JSTART = -1 C and MAXORD .lt. the current order NQ. C ACOR = a work array of length N, used for the accumulated C corrections. On a successful return, ACOR(i) contains C the estimated one-step local error in Y(i). C WM,IWM = real and integer work arrays associated with matrix C operations in chord iteration (MITER .ne. 0). C PJAC = name of routine to evaluate and preprocess Jacobian matrix C and P = I - h*el0*JAC, if a chord method is being used. C SLVS = name of routine to solve linear system in chord iteration. C CCMAX = maximum relative change in h*el0 before PJAC is called. C H = the step size to be attempted on the next step. C H is altered by the error control algorithm during the C problem. H can be either positive or negative, but its C sign must remain constant throughout the problem. C HMIN = the minimum absolute value of the step size h to be used. C HMXI = inverse of the maximum absolute value of h to be used. C HMXI = 0.0 is allowed and corresponds to an infinite hmax. C HMIN and HMXI may be changed at any time, but will not C take effect until the next change of h is considered. C TN = the independent variable. TN is updated on each step taken. C JSTART = an integer used for input only, with the following C values and meanings: C 0 perform the first step. C .gt.0 take a new step continuing from the last. C -1 take the next step with a new value of H, MAXORD, C N, METH, MITER, and/or matrix parameters. C -2 take the next step with a new value of H, C but with other inputs unchanged. C On return, JSTART is set to 1 to facilitate continuation. C KFLAG = a completion code with the following meanings: C 0 the step was succesful. C -1 the requested error could not be achieved. C -2 corrector convergence could not be achieved. C -3 fatal error in PJAC or SLVS. C A return with KFLAG = -1 or -2 means either C abs(H) = HMIN or 10 consecutive failures occurred. C On a return with KFLAG negative, the values of TN and C the YH array are as of the beginning of the last C step, and H is the last step size attempted. C MAXORD = the maximum order of integration method to be allowed. C MAXCOR = the maximum number of corrector iterations allowed. C MSBP = maximum number of steps between PJAC calls (MITER .gt. 0). C MXNCF = maximum number of convergence failures allowed. C METH/MITER = the method flags. See description in driver. C N = the number of first-order differential equations. C The values of CCMAX, H, HMIN, HMXI, TN, JSTART, KFLAG, MAXORD, C MAXCOR, MSBP, MXNCF, METH, MITER, and N are communicated via COMMON. C C***SEE ALSO DLSODE C***ROUTINES CALLED DCFODE, DVNORM C***COMMON BLOCKS DLS001 C***REVISION HISTORY (YYMMDD) C 791129 DATE WRITTEN C 890501 Modified prologue to SLATEC/LDOC format. (FNF) C 890503 Minor cosmetic changes. (FNF) C 930809 Renamed to allow single/double precision versions. (ACH) C 010418 Reduced size of Common block /DLS001/. (ACH) C 031105 Restored 'own' variables to Common block /DLS001/, to C enable interrupt/restart feature. (ACH) C***END PROLOGUE DSTODE C**End EXTERNAL F, JAC, PJAC, SLVS INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, YH1, EWT, SAVF, ACOR, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*), 1 ACOR(*), WM(*), IWM(*) INTEGER IOWND, IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER I, I1, IREDO, IRET, J, JB, M, NCF, NEWQ DOUBLE PRECISION CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO, 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION DCON, DDN, DEL, DELP, DSM, DUP, EXDN, EXSM, EXUP, 1 R, RH, RHDN, RHSM, RHUP, TOLD, DVNORM COMMON /DLS001/ CONIT, CRATE, EL(13), ELCO(13,12), 1 HOLD, RMAX, TESCO(3,12), 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 3 IOWND(6), IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU C C***FIRST EXECUTABLE STATEMENT DSTODE KFLAG = 0 TOLD = TN NCF = 0 IERPJ = 0 IERSL = 0 JCUR = 0 ICF = 0 DELP = 0.0D0 IF (JSTART .GT. 0) GO TO 200 IF (JSTART .EQ. -1) GO TO 100 IF (JSTART .EQ. -2) GO TO 160 C----------------------------------------------------------------------- C On the first call, the order is set to 1, and other variables are C initialized. RMAX is the maximum ratio by which H can be increased C in a single step. It is initially 1.E4 to compensate for the small C initial H, but then is normally equal to 10. If a failure C occurs (in corrector convergence or error test), RMAX is set to 2 C for the next increase. C----------------------------------------------------------------------- LMAX = MAXORD + 1 NQ = 1 L = 2 IALTH = 2 RMAX = 10000.0D0 RC = 0.0D0 EL0 = 1.0D0 CRATE = 0.7D0 HOLD = H MEO = METH NSLP = 0 IPUP = MITER IRET = 3 GO TO 140 C----------------------------------------------------------------------- C The following block handles preliminaries needed when JSTART = -1. C IPUP is set to MITER to force a matrix update. C If an order increase is about to be considered (IALTH = 1), C IALTH is reset to 2 to postpone consideration one more step. C If the caller has changed METH, DCFODE is called to reset C the coefficients of the method. C If the caller has changed MAXORD to a value less than the current C order NQ, NQ is reduced to MAXORD, and a new H chosen accordingly. C If H is to be changed, YH must be rescaled. C If H or METH is being changed, IALTH is reset to L = NQ + 1 C to prevent further changes in H for that many steps. C----------------------------------------------------------------------- 100 IPUP = MITER LMAX = MAXORD + 1 IF (IALTH .EQ. 1) IALTH = 2 IF (METH .EQ. MEO) GO TO 110 CALL DCFODE (METH, ELCO, TESCO) MEO = METH IF (NQ .GT. MAXORD) GO TO 120 IALTH = L IRET = 1 GO TO 150 110 IF (NQ .LE. MAXORD) GO TO 160 120 NQ = MAXORD L = LMAX DO 125 I = 1,L 125 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) DDN = DVNORM (N, SAVF, EWT)/TESCO(1,L) EXDN = 1.0D0/L RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) RH = MIN(RHDN,1.0D0) IREDO = 3 IF (H .EQ. HOLD) GO TO 170 RH = MIN(RH,ABS(H/HOLD)) H = HOLD GO TO 175 C----------------------------------------------------------------------- C DCFODE is called to get all the integration coefficients for the C current METH. Then the EL vector and related constants are reset C whenever the order NQ is changed, or at the start of the problem. C----------------------------------------------------------------------- 140 CALL DCFODE (METH, ELCO, TESCO) 150 DO 155 I = 1,L 155 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) GO TO (160, 170, 200), IRET C----------------------------------------------------------------------- C If H is being changed, the H ratio RH is checked against C RMAX, HMIN, and HMXI, and the YH array rescaled. IALTH is set to C L = NQ + 1 to prevent a change of H for that many steps, unless C forced by a convergence or error test failure. C----------------------------------------------------------------------- 160 IF (H .EQ. HOLD) GO TO 200 RH = H/HOLD H = HOLD IREDO = 3 GO TO 175 170 RH = MAX(RH,HMIN/ABS(H)) 175 RH = MIN(RH,RMAX) RH = RH/MAX(1.0D0,ABS(H)*HMXI*RH) R = 1.0D0 DO 180 J = 2,L R = R*RH DO 180 I = 1,N 180 YH(I,J) = YH(I,J)*R H = H*RH RC = RC*RH IALTH = L IF (IREDO .EQ. 0) GO TO 690 C----------------------------------------------------------------------- C This section computes the predicted values by effectively C multiplying the YH array by the Pascal Triangle matrix. C RC is the ratio of new to old values of the coefficient H*EL(1). C When RC differs from 1 by more than CCMAX, IPUP is set to MITER C to force PJAC to be called, if a Jacobian is involved. C In any case, PJAC is called at least every MSBP steps. C----------------------------------------------------------------------- 200 IF (ABS(RC-1.0D0) .GT. CCMAX) IPUP = MITER IF (NST .GE. NSLP+MSBP) IPUP = MITER TN = TN + H I1 = NQNYH + 1 DO 215 JB = 1,NQ I1 = I1 - NYH Cdir\$ ivdep DO 210 I = I1,NQNYH 210 YH1(I) = YH1(I) + YH1(I+NYH) 215 CONTINUE C----------------------------------------------------------------------- C Up to MAXCOR corrector iterations are taken. A convergence test is C made on the R.M.S. norm of each correction, weighted by the error C weight vector EWT. The sum of the corrections is accumulated in the C vector ACOR(i). The YH array is not altered in the corrector loop. C----------------------------------------------------------------------- 220 M = 0 DO 230 I = 1,N 230 Y(I) = YH(I,1) CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 IF (IPUP .LE. 0) GO TO 250 C----------------------------------------------------------------------- C If indicated, the matrix P = I - h*el(1)*J is reevaluated and C preprocessed before starting the corrector iteration. IPUP is set C to 0 as an indicator that this has been done. C----------------------------------------------------------------------- CALL PJAC (NEQ, Y, YH, NYH, EWT, ACOR, SAVF, WM, IWM, F, JAC) IPUP = 0 RC = 1.0D0 NSLP = NST CRATE = 0.7D0 IF (IERPJ .NE. 0) GO TO 430 250 DO 260 I = 1,N 260 ACOR(I) = 0.0D0 270 IF (MITER .NE. 0) GO TO 350 C----------------------------------------------------------------------- C In the case of functional iteration, update Y directly from C the result of the last function evaluation. C----------------------------------------------------------------------- DO 290 I = 1,N SAVF(I) = H*SAVF(I) - YH(I,2) 290 Y(I) = SAVF(I) - ACOR(I) DEL = DVNORM (N, Y, EWT) DO 300 I = 1,N Y(I) = YH(I,1) + EL(1)*SAVF(I) 300 ACOR(I) = SAVF(I) GO TO 400 C----------------------------------------------------------------------- C In the case of the chord method, compute the corrector error, C and solve the linear system with that as right-hand side and C P as coefficient matrix. C----------------------------------------------------------------------- 350 DO 360 I = 1,N 360 Y(I) = H*SAVF(I) - (YH(I,2) + ACOR(I)) CALL SLVS (WM, IWM, Y, SAVF) IF (IERSL .LT. 0) GO TO 430 IF (IERSL .GT. 0) GO TO 410 DEL = DVNORM (N, Y, EWT) DO 380 I = 1,N ACOR(I) = ACOR(I) + Y(I) 380 Y(I) = YH(I,1) + EL(1)*ACOR(I) C----------------------------------------------------------------------- C Test for convergence. If M.gt.0, an estimate of the convergence C rate constant is stored in CRATE, and this is used in the test. C----------------------------------------------------------------------- 400 IF (M .NE. 0) CRATE = MAX(0.2D0*CRATE,DEL/DELP) DCON = DEL*MIN(1.0D0,1.5D0*CRATE)/(TESCO(2,NQ)*CONIT) IF (DCON .LE. 1.0D0) GO TO 450 M = M + 1 IF (M .EQ. MAXCOR) GO TO 410 IF (M .GE. 2 .AND. DEL .GT. 2.0D0*DELP) GO TO 410 DELP = DEL CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 GO TO 270 C----------------------------------------------------------------------- C The corrector iteration failed to converge. C If MITER .ne. 0 and the Jacobian is out of date, PJAC is called for C the next try. Otherwise the YH array is retracted to its values C before prediction, and H is reduced, if possible. If H cannot be C reduced or MXNCF failures have occurred, exit with KFLAG = -2. C----------------------------------------------------------------------- 410 IF (MITER .EQ. 0 .OR. JCUR .EQ. 1) GO TO 430 ICF = 1 IPUP = MITER GO TO 220 430 ICF = 2 NCF = NCF + 1 RMAX = 2.0D0 TN = TOLD I1 = NQNYH + 1 DO 445 JB = 1,NQ I1 = I1 - NYH Cdir\$ ivdep DO 440 I = I1,NQNYH 440 YH1(I) = YH1(I) - YH1(I+NYH) 445 CONTINUE IF (IERPJ .LT. 0 .OR. IERSL .LT. 0) GO TO 680 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 670 IF (NCF .EQ. MXNCF) GO TO 670 RH = 0.25D0 IPUP = MITER IREDO = 1 GO TO 170 C----------------------------------------------------------------------- C The corrector has converged. JCUR is set to 0 C to signal that the Jacobian involved may need updating later. C The local error test is made and control passes to statement 500 C if it fails. C----------------------------------------------------------------------- 450 JCUR = 0 IF (M .EQ. 0) DSM = DEL/TESCO(2,NQ) IF (M .GT. 0) DSM = DVNORM (N, ACOR, EWT)/TESCO(2,NQ) IF (DSM .GT. 1.0D0) GO TO 500 C----------------------------------------------------------------------- C After a successful step, update the YH array. C Consider changing H if IALTH = 1. Otherwise decrease IALTH by 1. C If IALTH is then 1 and NQ .lt. MAXORD, then ACOR is saved for C use in a possible order increase on the next step. C If a change in H is considered, an increase or decrease in order C by one is considered also. A change in H is made only if it is by a C factor of at least 1.1. If not, IALTH is set to 3 to prevent C testing for that many steps. C----------------------------------------------------------------------- KFLAG = 0 IREDO = 0 NST = NST + 1 HU = H NQU = NQ DO 470 J = 1,L DO 470 I = 1,N 470 YH(I,J) = YH(I,J) + EL(J)*ACOR(I) IALTH = IALTH - 1 IF (IALTH .EQ. 0) GO TO 520 IF (IALTH .GT. 1) GO TO 700 IF (L .EQ. LMAX) GO TO 700 DO 490 I = 1,N 490 YH(I,LMAX) = ACOR(I) GO TO 700 C----------------------------------------------------------------------- C The error test failed. KFLAG keeps track of multiple failures. C Restore TN and the YH array to their previous values, and prepare C to try the step again. Compute the optimum step size for this or C one lower order. After 2 or more failures, H is forced to decrease C by a factor of 0.2 or less. C----------------------------------------------------------------------- 500 KFLAG = KFLAG - 1 TN = TOLD I1 = NQNYH + 1 DO 515 JB = 1,NQ I1 = I1 - NYH Cdir\$ ivdep DO 510 I = I1,NQNYH 510 YH1(I) = YH1(I) - YH1(I+NYH) 515 CONTINUE RMAX = 2.0D0 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 660 IF (KFLAG .LE. -3) GO TO 640 IREDO = 2 RHUP = 0.0D0 GO TO 540 C----------------------------------------------------------------------- C Regardless of the success or failure of the step, factors C RHDN, RHSM, and RHUP are computed, by which H could be multiplied C at order NQ - 1, order NQ, or order NQ + 1, respectively. C In the case of failure, RHUP = 0.0 to avoid an order increase. C The largest of these is determined and the new order chosen C accordingly. If the order is to be increased, we compute one C additional scaled derivative. C----------------------------------------------------------------------- 520 RHUP = 0.0D0 IF (L .EQ. LMAX) GO TO 540 DO 530 I = 1,N 530 SAVF(I) = ACOR(I) - YH(I,LMAX) DUP = DVNORM (N, SAVF, EWT)/TESCO(3,NQ) EXUP = 1.0D0/(L+1) RHUP = 1.0D0/(1.4D0*DUP**EXUP + 0.0000014D0) 540 EXSM = 1.0D0/L RHSM = 1.0D0/(1.2D0*DSM**EXSM + 0.0000012D0) RHDN = 0.0D0 IF (NQ .EQ. 1) GO TO 560 DDN = DVNORM (N, YH(1,L), EWT)/TESCO(1,NQ) EXDN = 1.0D0/NQ RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) 560 IF (RHSM .GE. RHUP) GO TO 570 IF (RHUP .GT. RHDN) GO TO 590 GO TO 580 570 IF (RHSM .LT. RHDN) GO TO 580 NEWQ = NQ RH = RHSM GO TO 620 580 NEWQ = NQ - 1 RH = RHDN IF (KFLAG .LT. 0 .AND. RH .GT. 1.0D0) RH = 1.0D0 GO TO 620 590 NEWQ = L RH = RHUP IF (RH .LT. 1.1D0) GO TO 610 R = EL(L)/L DO 600 I = 1,N 600 YH(I,NEWQ+1) = ACOR(I)*R GO TO 630 610 IALTH = 3 GO TO 700 620 IF ((KFLAG .EQ. 0) .AND. (RH .LT. 1.1D0)) GO TO 610 IF (KFLAG .LE. -2) RH = MIN(RH,0.2D0) C----------------------------------------------------------------------- C If there is a change of order, reset NQ, l, and the coefficients. C In any case H is reset according to RH and the YH array is rescaled. C Then exit from 690 if the step was OK, or redo the step otherwise. C----------------------------------------------------------------------- IF (NEWQ .EQ. NQ) GO TO 170 630 NQ = NEWQ L = NQ + 1 IRET = 2 GO TO 150 C----------------------------------------------------------------------- C Control reaches this section if 3 or more failures have occured. C If 10 failures have occurred, exit with KFLAG = -1. C It is assumed that the derivatives that have accumulated in the C YH array have errors of the wrong order. Hence the first C derivative is recomputed, and the order is set to 1. Then C H is reduced by a factor of 10, and the step is retried, C until it succeeds or H reaches HMIN. C----------------------------------------------------------------------- 640 IF (KFLAG .EQ. -10) GO TO 660 RH = 0.1D0 RH = MAX(HMIN/ABS(H),RH) H = H*RH DO 645 I = 1,N 645 Y(I) = YH(I,1) CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 DO 650 I = 1,N 650 YH(I,2) = H*SAVF(I) IPUP = MITER IALTH = 5 IF (NQ .EQ. 1) GO TO 200 NQ = 1 L = 2 IRET = 3 GO TO 150 C----------------------------------------------------------------------- C All returns are made through this section. H is saved in HOLD C to allow the caller to change H on the next step. C----------------------------------------------------------------------- 660 KFLAG = -1 GO TO 720 670 KFLAG = -2 GO TO 720 680 KFLAG = -3 GO TO 720 690 RMAX = 10.0D0 700 R = 1.0D0/TESCO(2,NQU) DO 710 I = 1,N 710 ACOR(I) = ACOR(I)*R 720 HOLD = H JSTART = 1 RETURN C----------------------- END OF SUBROUTINE DSTODE ---------------------- END *DECK DEWSET SUBROUTINE DEWSET (N, ITOL, RTOL, ATOL, YCUR, EWT) C***BEGIN PROLOGUE DEWSET C***SUBSIDIARY C***PURPOSE Set error weight vector. C***TYPE DOUBLE PRECISION (SEWSET-S, DEWSET-D) C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C C This subroutine sets the error weight vector EWT according to C EWT(i) = RTOL(i)*ABS(YCUR(i)) + ATOL(i), i = 1,...,N, C with the subscript on RTOL and/or ATOL possibly replaced by 1 above, C depending on the value of ITOL. C C***SEE ALSO DLSODE C***ROUTINES CALLED (NONE) C***REVISION HISTORY (YYMMDD) C 791129 DATE WRITTEN C 890501 Modified prologue to SLATEC/LDOC format. (FNF) C 890503 Minor cosmetic changes. (FNF) C 930809 Renamed to allow single/double precision versions. (ACH) C***END PROLOGUE DEWSET C**End INTEGER N, ITOL INTEGER I DOUBLE PRECISION RTOL, ATOL, YCUR, EWT DIMENSION RTOL(*), ATOL(*), YCUR(N), EWT(N) C C***FIRST EXECUTABLE STATEMENT DEWSET GO TO (10, 20, 30, 40), ITOL 10 CONTINUE DO 15 I = 1,N 15 EWT(I) = RTOL(1)*ABS(YCUR(I)) + ATOL(1) RETURN 20 CONTINUE DO 25 I = 1,N 25 EWT(I) = RTOL(1)*ABS(YCUR(I)) + ATOL(I) RETURN 30 CONTINUE DO 35 I = 1,N 35 EWT(I) = RTOL(I)*ABS(YCUR(I)) + ATOL(1) RETURN 40 CONTINUE DO 45 I = 1,N 45 EWT(I) = RTOL(I)*ABS(YCUR(I)) + ATOL(I) RETURN C----------------------- END OF SUBROUTINE DEWSET ---------------------- END *DECK DVNORM DOUBLE PRECISION FUNCTION DVNORM (N, V, W) C***BEGIN PROLOGUE DVNORM C***SUBSIDIARY C***PURPOSE Weighted root-mean-square vector norm. C***TYPE DOUBLE PRECISION (SVNORM-S, DVNORM-D) C***AUTHOR Hindmarsh, Alan C., (LLNL) C***DESCRIPTION C C This function routine computes the weighted root-mean-square norm C of the vector of length N contained in the array V, with weights C contained in the array W of length N: C DVNORM = SQRT( (1/N) * SUM( V(i)*W(i) )**2 ) C C***SEE ALSO DLSODE C***ROUTINES CALLED (NONE) C***REVISION HISTORY (YYMMDD) C 791129 DATE WRITTEN C 890501 Modified prologue to SLATEC/LDOC format. (FNF) C 890503 Minor cosmetic changes. (FNF) C 930809 Renamed to allow single/double precision versions. (ACH) C***END PROLOGUE DVNORM C**End INTEGER N, I DOUBLE PRECISION V, W, SUM DIMENSION V(N), W(N) C C***FIRST EXECUTABLE STATEMENT DVNORM SUM = 0.0D0 DO 10 I = 1,N 10 SUM = SUM + (V(I)*W(I))**2 DVNORM = SQRT(SUM/N) RETURN C----------------------- END OF FUNCTION DVNORM ------------------------ END *DECK DIPREP SUBROUTINE DIPREP (NEQ, Y, RWORK, IA, JA, IPFLAG, F, JAC) EXTERNAL F, JAC INTEGER NEQ, IA, JA, IPFLAG DOUBLE PRECISION Y, RWORK DIMENSION NEQ(*), Y(*), RWORK(*), IA(*), JA(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION RLSS COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSS01/ RLSS(6), 1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU INTEGER I, IMAX, LEWTN, LYHD, LYHN C----------------------------------------------------------------------- C This routine serves as an interface between the driver and C Subroutine DPREP. It is called only if MITER is 1 or 2. C Tasks performed here are: C * call DPREP, C * reset the required WM segment length LENWK, C * move YH back to its final location (following WM in RWORK), C * reset pointers for YH, SAVF, EWT, and ACOR, and C * move EWT to its new position if ISTATE = 1. C IPFLAG is an output error indication flag. IPFLAG = 0 if there was C no trouble, and IPFLAG is the value of the DPREP error flag IPPER C if there was trouble in Subroutine DPREP. C----------------------------------------------------------------------- IPFLAG = 0 C Call DPREP to do matrix preprocessing operations. -------------------- CALL DPREP (NEQ, Y, RWORK(LYH), RWORK(LSAVF), RWORK(LEWT), 1 RWORK(LACOR), IA, JA, RWORK(LWM), RWORK(LWM), IPFLAG, F, JAC) LENWK = MAX(LREQ,LWMIN) IF (IPFLAG .LT. 0) RETURN C If DPREP was successful, move YH to end of required space for WM. ---- LYHN = LWM + LENWK IF (LYHN .GT. LYH) RETURN LYHD = LYH - LYHN IF (LYHD .EQ. 0) GO TO 20 IMAX = LYHN - 1 + LENYHM DO 10 I = LYHN,IMAX 10 RWORK(I) = RWORK(I+LYHD) LYH = LYHN C Reset pointers for SAVF, EWT, and ACOR. ------------------------------ 20 LSAVF = LYH + LENYH LEWTN = LSAVF + N LACOR = LEWTN + N IF (ISTATC .EQ. 3) GO TO 40 C If ISTATE = 1, move EWT (left) to its new position. ------------------ IF (LEWTN .GT. LEWT) RETURN DO 30 I = 1,N 30 RWORK(I+LEWTN-1) = RWORK(I+LEWT-1) 40 LEWT = LEWTN RETURN C----------------------- End of Subroutine DIPREP ---------------------- END *DECK DPREP SUBROUTINE DPREP (NEQ, Y, YH, SAVF, EWT, FTEM, IA, JA, 1 WK, IWK, IPPER, F, JAC) EXTERNAL F,JAC INTEGER NEQ, IA, JA, IWK, IPPER DOUBLE PRECISION Y, YH, SAVF, EWT, FTEM, WK DIMENSION NEQ(*), Y(*), YH(*), SAVF(*), EWT(*), FTEM(*), 1 IA(*), JA(*), WK(*), IWK(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSS01/ CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH, 1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU INTEGER I, IBR, IER, IPIL, IPIU, IPTT1, IPTT2, J, JFOUND, K, 1 KNEW, KMAX, KMIN, LDIF, LENIGP, LIWK, MAXG, NP1, NZSUT DOUBLE PRECISION DQ, DYJ, ERWT, FAC, YJ C----------------------------------------------------------------------- C This routine performs preprocessing related to the sparse linear C systems that must be solved if MITER = 1 or 2. C The operations that are performed here are: C * compute sparseness structure of Jacobian according to MOSS, C * compute grouping of column indices (MITER = 2), C * compute a new ordering of rows and columns of the matrix, C * reorder JA corresponding to the new ordering, C * perform a symbolic LU factorization of the matrix, and C * set pointers for segments of the IWK/WK array. C In addition to variables described previously, DPREP uses the C following for communication: C YH = the history array. Only the first column, containing the C current Y vector, is used. Used only if MOSS .ne. 0. C SAVF = a work array of length NEQ, used only if MOSS .ne. 0. C EWT = array of length NEQ containing (inverted) error weights. C Used only if MOSS = 2 or if ISTATE = MOSS = 1. C FTEM = a work array of length NEQ, identical to ACOR in the driver, C used only if MOSS = 2. C WK = a real work array of length LENWK, identical to WM in C the driver. C IWK = integer work array, assumed to occupy the same space as WK. C LENWK = the length of the work arrays WK and IWK. C ISTATC = a copy of the driver input argument ISTATE (= 1 on the C first call, = 3 on a continuation call). C IYS = flag value from ODRV or CDRV. C IPPER = output error flag with the following values and meanings: C 0 no error. C -1 insufficient storage for internal structure pointers. C -2 insufficient storage for JGROUP. C -3 insufficient storage for ODRV. C -4 other error flag from ODRV (should never occur). C -5 insufficient storage for CDRV. C -6 other error flag from CDRV. C----------------------------------------------------------------------- IBIAN = LRAT*2 IPIAN = IBIAN + 1 NP1 = N + 1 IPJAN = IPIAN + NP1 IBJAN = IPJAN - 1 LIWK = LENWK*LRAT IF (IPJAN+N-1 .GT. LIWK) GO TO 210 IF (MOSS .EQ. 0) GO TO 30 C IF (ISTATC .EQ. 3) GO TO 20 C ISTATE = 1 and MOSS .ne. 0. Perturb Y for structure determination. -- DO 10 I = 1,N ERWT = 1.0D0/EWT(I) FAC = 1.0D0 + 1.0D0/(I + 1.0D0) Y(I) = Y(I) + FAC*SIGN(ERWT,Y(I)) 10 CONTINUE GO TO (70, 100), MOSS C 20 CONTINUE C ISTATE = 3 and MOSS .ne. 0. Load Y from YH(*,1). -------------------- DO 25 I = 1,N 25 Y(I) = YH(I) GO TO (70, 100), MOSS C C MOSS = 0. Process user's IA,JA. Add diagonal entries if necessary. - 30 KNEW = IPJAN KMIN = IA(1) IWK(IPIAN) = 1 DO 60 J = 1,N JFOUND = 0 KMAX = IA(J+1) - 1 IF (KMIN .GT. KMAX) GO TO 45 DO 40 K = KMIN,KMAX I = JA(K) IF (I .EQ. J) JFOUND = 1 IF (KNEW .GT. LIWK) GO TO 210 IWK(KNEW) = I KNEW = KNEW + 1 40 CONTINUE IF (JFOUND .EQ. 1) GO TO 50 45 IF (KNEW .GT. LIWK) GO TO 210 IWK(KNEW) = J KNEW = KNEW + 1 50 IWK(IPIAN+J) = KNEW + 1 - IPJAN KMIN = KMAX + 1 60 CONTINUE GO TO 140 C C MOSS = 1. Compute structure from user-supplied Jacobian routine JAC. 70 CONTINUE C A dummy call to F allows user to create temporaries for use in JAC. -- CALL F (NEQ, TN, Y, SAVF) K = IPJAN IWK(IPIAN) = 1 DO 90 J = 1,N IF (K .GT. LIWK) GO TO 210 IWK(K) = J K = K + 1 DO 75 I = 1,N 75 SAVF(I) = 0.0D0 CALL JAC (NEQ, TN, Y, J, IWK(IPIAN), IWK(IPJAN), SAVF) DO 80 I = 1,N IF (ABS(SAVF(I)) .LE. SETH) GO TO 80 IF (I .EQ. J) GO TO 80 IF (K .GT. LIWK) GO TO 210 IWK(K) = I K = K + 1 80 CONTINUE IWK(IPIAN+J) = K + 1 - IPJAN 90 CONTINUE GO TO 140 C C MOSS = 2. Compute structure from results of N + 1 calls to F. ------- 100 K = IPJAN IWK(IPIAN) = 1 CALL F (NEQ, TN, Y, SAVF) DO 120 J = 1,N IF (K .GT. LIWK) GO TO 210 IWK(K) = J K = K + 1 YJ = Y(J) ERWT = 1.0D0/EWT(J) DYJ = SIGN(ERWT,YJ) Y(J) = YJ + DYJ CALL F (NEQ, TN, Y, FTEM) Y(J) = YJ DO 110 I = 1,N DQ = (FTEM(I) - SAVF(I))/DYJ IF (ABS(DQ) .LE. SETH) GO TO 110 IF (I .EQ. J) GO TO 110 IF (K .GT. LIWK) GO TO 210 IWK(K) = I K = K + 1 110 CONTINUE IWK(IPIAN+J) = K + 1 - IPJAN 120 CONTINUE C 140 CONTINUE IF (MOSS .EQ. 0 .OR. ISTATC .NE. 1) GO TO 150 C If ISTATE = 1 and MOSS .ne. 0, restore Y from YH. -------------------- DO 145 I = 1,N 145 Y(I) = YH(I) 150 NNZ = IWK(IPIAN+N) - 1 LENIGP = 0 IPIGP = IPJAN + NNZ IF (MITER .NE. 2) GO TO 160 C C Compute grouping of column indices (MITER = 2). ---------------------- MAXG = NP1 IPJGP = IPJAN + NNZ IBJGP = IPJGP - 1 IPIGP = IPJGP + N IPTT1 = IPIGP + NP1 IPTT2 = IPTT1 + N LREQ = IPTT2 + N - 1 IF (LREQ .GT. LIWK) GO TO 220 CALL JGROUP (N, IWK(IPIAN), IWK(IPJAN), MAXG, NGP, IWK(IPIGP), 1 IWK(IPJGP), IWK(IPTT1), IWK(IPTT2), IER) IF (IER .NE. 0) GO TO 220 LENIGP = NGP + 1 C C Compute new ordering of rows/columns of Jacobian. -------------------- 160 IPR = IPIGP + LENIGP IPC = IPR IPIC = IPC + N IPISP = IPIC + N IPRSP = (IPISP - 2)/LRAT + 2 IESP = LENWK + 1 - IPRSP IF (IESP .LT. 0) GO TO 230 IBR = IPR - 1 DO 170 I = 1,N 170 IWK(IBR+I) = I NSP = LIWK + 1 - IPISP CALL ODRV (N, IWK(IPIAN), IWK(IPJAN), WK, IWK(IPR), IWK(IPIC), 1 NSP, IWK(IPISP), 1, IYS) IF (IYS .EQ. 11*N+1) GO TO 240 IF (IYS .NE. 0) GO TO 230 C C Reorder JAN and do symbolic LU factorization of matrix. -------------- IPA = LENWK + 1 - NNZ NSP = IPA - IPRSP LREQ = MAX(12*N/LRAT, 6*N/LRAT+2*N+NNZ) + 3 LREQ = LREQ + IPRSP - 1 + NNZ IF (LREQ .GT. LENWK) GO TO 250 IBA = IPA - 1 DO 180 I = 1,NNZ 180 WK(IBA+I) = 0.0D0 IPISP = LRAT*(IPRSP - 1) + 1 CALL CDRV (N,IWK(IPR),IWK(IPC),IWK(IPIC),IWK(IPIAN),IWK(IPJAN), 1 WK(IPA),WK(IPA),WK(IPA),NSP,IWK(IPISP),WK(IPRSP),IESP,5,IYS) LREQ = LENWK - IESP IF (IYS .EQ. 10*N+1) GO TO 250 IF (IYS .NE. 0) GO TO 260 IPIL = IPISP IPIU = IPIL + 2*N + 1 NZU = IWK(IPIL+N) - IWK(IPIL) NZL = IWK(IPIU+N) - IWK(IPIU) IF (LRAT .GT. 1) GO TO 190 CALL ADJLR (N, IWK(IPISP), LDIF) LREQ = LREQ + LDIF 190 CONTINUE IF (LRAT .EQ. 2 .AND. NNZ .EQ. N) LREQ = LREQ + 1 NSP = NSP + LREQ - LENWK IPA = LREQ + 1 - NNZ IBA = IPA - 1 IPPER = 0 RETURN C 210 IPPER = -1 LREQ = 2 + (2*N + 1)/LRAT LREQ = MAX(LENWK+1,LREQ) RETURN C 220 IPPER = -2 LREQ = (LREQ - 1)/LRAT + 1 RETURN C 230 IPPER = -3 CALL CNTNZU (N, IWK(IPIAN), IWK(IPJAN), NZSUT) LREQ = LENWK - IESP + (3*N + 4*NZSUT - 1)/LRAT + 1 RETURN C 240 IPPER = -4 RETURN C 250 IPPER = -5 RETURN C 260 IPPER = -6 LREQ = LENWK RETURN C----------------------- End of Subroutine DPREP ----------------------- END *DECK JGROUP SUBROUTINE JGROUP (N,IA,JA,MAXG,NGRP,IGP,JGP,INCL,JDONE,IER) INTEGER N, IA, JA, MAXG, NGRP, IGP, JGP, INCL, JDONE, IER DIMENSION IA(*), JA(*), IGP(*), JGP(*), INCL(*), JDONE(*) C----------------------------------------------------------------------- C This subroutine constructs groupings of the column indices of C the Jacobian matrix, used in the numerical evaluation of the C Jacobian by finite differences. C C Input: C N = the order of the matrix. C IA,JA = sparse structure descriptors of the matrix by rows. C MAXG = length of available storage in the IGP array. C C Output: C NGRP = number of groups. C JGP = array of length N containing the column indices by groups. C IGP = pointer array of length NGRP + 1 to the locations in JGP C of the beginning of each group. C IER = error indicator. IER = 0 if no error occurred, or 1 if C MAXG was insufficient. C C INCL and JDONE are working arrays of length N. C----------------------------------------------------------------------- INTEGER I, J, K, KMIN, KMAX, NCOL, NG C IER = 0 DO 10 J = 1,N 10 JDONE(J) = 0 NCOL = 1 DO 60 NG = 1,MAXG IGP(NG) = NCOL DO 20 I = 1,N 20 INCL(I) = 0 DO 50 J = 1,N C Reject column J if it is already in a group.-------------------------- IF (JDONE(J) .EQ. 1) GO TO 50 KMIN = IA(J) KMAX = IA(J+1) - 1 DO 30 K = KMIN,KMAX C Reject column J if it overlaps any column already in this group.------ I = JA(K) IF (INCL(I) .EQ. 1) GO TO 50 30 CONTINUE C Accept column J into group NG.---------------------------------------- JGP(NCOL) = J NCOL = NCOL + 1 JDONE(J) = 1 DO 40 K = KMIN,KMAX I = JA(K) 40 INCL(I) = 1 50 CONTINUE C Stop if this group is empty (grouping is complete).------------------- IF (NCOL .EQ. IGP(NG)) GO TO 70 60 CONTINUE C Error return if not all columns were chosen (MAXG too small).--------- IF (NCOL .LE. N) GO TO 80 NG = MAXG 70 NGRP = NG - 1 RETURN 80 IER = 1 RETURN C----------------------- End of Subroutine JGROUP ---------------------- END *DECK ADJLR SUBROUTINE ADJLR (N, ISP, LDIF) INTEGER N, ISP, LDIF DIMENSION ISP(*) C----------------------------------------------------------------------- C This routine computes an adjustment, LDIF, to the required C integer storage space in IWK (sparse matrix work space). C It is called only if the word length ratio is LRAT = 1. C This is to account for the possibility that the symbolic LU phase C may require more storage than the numerical LU and solution phases. C----------------------------------------------------------------------- INTEGER IP, JLMAX, JUMAX, LNFC, LSFC, NZLU C IP = 2*N + 1 C Get JLMAX = IJL(N) and JUMAX = IJU(N) (sizes of JL and JU). ---------- JLMAX = ISP(IP) JUMAX = ISP(IP+IP) C NZLU = (size of L) + (size of U) = (IL(N+1)-IL(1)) + (IU(N+1)-IU(1)). NZLU = ISP(N+1) - ISP(1) + ISP(IP+N+1) - ISP(IP+1) LSFC = 12*N + 3 + 2*MAX(JLMAX,JUMAX) LNFC = 9*N + 2 + JLMAX + JUMAX + NZLU LDIF = MAX(0, LSFC - LNFC) RETURN C----------------------- End of Subroutine ADJLR ----------------------- END *DECK CNTNZU SUBROUTINE CNTNZU (N, IA, JA, NZSUT) INTEGER N, IA, JA, NZSUT DIMENSION IA(*), JA(*) C----------------------------------------------------------------------- C This routine counts the number of nonzero elements in the strict C upper triangle of the matrix M + M(transpose), where the sparsity C structure of M is given by pointer arrays IA and JA. C This is needed to compute the storage requirements for the C sparse matrix reordering operation in ODRV. C----------------------------------------------------------------------- INTEGER II, JJ, J, JMIN, JMAX, K, KMIN, KMAX, NUM C NUM = 0 DO 50 II = 1,N JMIN = IA(II) JMAX = IA(II+1) - 1 IF (JMIN .GT. JMAX) GO TO 50 DO 40 J = JMIN,JMAX IF (JA(J) - II) 10, 40, 30 10 JJ =JA(J) KMIN = IA(JJ) KMAX = IA(JJ+1) - 1 IF (KMIN .GT. KMAX) GO TO 30 DO 20 K = KMIN,KMAX IF (JA(K) .EQ. II) GO TO 40 20 CONTINUE 30 NUM = NUM + 1 40 CONTINUE 50 CONTINUE NZSUT = NUM RETURN C----------------------- End of Subroutine CNTNZU ---------------------- END *DECK DPRJS SUBROUTINE DPRJS (NEQ,Y,YH,NYH,EWT,FTEM,SAVF,WK,IWK,F,JAC) EXTERNAL F,JAC INTEGER NEQ, NYH, IWK DOUBLE PRECISION Y, YH, EWT, FTEM, SAVF, WK DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), FTEM(*), SAVF(*), 1 WK(*), IWK(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSS01/ CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH, 1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU INTEGER I, IMUL, J, JJ, JOK, JMAX, JMIN, K, KMAX, KMIN, NG DOUBLE PRECISION CON, DI, FAC, HL0, PIJ, R, R0, RCON, RCONT, 1 SRUR, DVNORM C----------------------------------------------------------------------- C DPRJS is called to compute and process the matrix C P = I - H*EL(1)*J , where J is an approximation to the Jacobian. C J is computed by columns, either by the user-supplied routine JAC C if MITER = 1, or by finite differencing if MITER = 2. C if MITER = 3, a diagonal approximation to J is used. C if MITER = 1 or 2, and if the existing value of the Jacobian C (as contained in P) is considered acceptable, then a new value of C P is reconstructed from the old value. In any case, when MITER C is 1 or 2, the P matrix is subjected to LU decomposition in CDRV. C P and its LU decomposition are stored (separately) in WK. C C In addition to variables described previously, communication C with DPRJS uses the following: C Y = array containing predicted values on entry. C FTEM = work array of length N (ACOR in DSTODE). C SAVF = array containing f evaluated at predicted y. C WK = real work space for matrices. On output it contains the C inverse diagonal matrix if MITER = 3, and P and its sparse C LU decomposition if MITER is 1 or 2. C Storage of matrix elements starts at WK(3). C WK also contains the following matrix-related data: C WK(1) = SQRT(UROUND), used in numerical Jacobian increments. C WK(2) = H*EL0, saved for later use if MITER = 3. C IWK = integer work space for matrix-related data, assumed to C be equivalenced to WK. In addition, WK(IPRSP) and IWK(IPISP) C are assumed to have identical locations. C EL0 = EL(1) (input). C IERPJ = output error flag (in Common). C = 0 if no error. C = 1 if zero pivot found in CDRV. C = 2 if a singular matrix arose with MITER = 3. C = -1 if insufficient storage for CDRV (should not occur here). C = -2 if other error found in CDRV (should not occur here). C JCUR = output flag showing status of (approximate) Jacobian matrix: C = 1 to indicate that the Jacobian is now current, or C = 0 to indicate that a saved value was used. C This routine also uses other variables in Common. C----------------------------------------------------------------------- HL0 = H*EL0 CON = -HL0 IF (MITER .EQ. 3) GO TO 300 C See whether J should be reevaluated (JOK = 0) or not (JOK = 1). ------ JOK = 1 IF (NST .EQ. 0 .OR. NST .GE. NSLJ+MSBJ) JOK = 0 IF (ICF .EQ. 1 .AND. ABS(RC - 1.0D0) .LT. CCMXJ) JOK = 0 IF (ICF .EQ. 2) JOK = 0 IF (JOK .EQ. 1) GO TO 250 C C MITER = 1 or 2, and the Jacobian is to be reevaluated. --------------- 20 JCUR = 1 NJE = NJE + 1 NSLJ = NST IPLOST = 0 CONMIN = ABS(CON) GO TO (100, 200), MITER C C If MITER = 1, call JAC, multiply by scalar, and add identity. -------- 100 CONTINUE KMIN = IWK(IPIAN) DO 130 J = 1, N KMAX = IWK(IPIAN+J) - 1 DO 110 I = 1,N 110 FTEM(I) = 0.0D0 CALL JAC (NEQ, TN, Y, J, IWK(IPIAN), IWK(IPJAN), FTEM) DO 120 K = KMIN, KMAX I = IWK(IBJAN+K) WK(IBA+K) = FTEM(I)*CON IF (I .EQ. J) WK(IBA+K) = WK(IBA+K) + 1.0D0 120 CONTINUE KMIN = KMAX + 1 130 CONTINUE GO TO 290 C C If MITER = 2, make NGP calls to F to approximate J and P. ------------ 200 CONTINUE FAC = DVNORM(N, SAVF, EWT) R0 = 1000.0D0 * ABS(H) * UROUND * N * FAC IF (R0 .EQ. 0.0D0) R0 = 1.0D0 SRUR = WK(1) JMIN = IWK(IPIGP) DO 240 NG = 1,NGP JMAX = IWK(IPIGP+NG) - 1 DO 210 J = JMIN,JMAX JJ = IWK(IBJGP+J) R = MAX(SRUR*ABS(Y(JJ)),R0/EWT(JJ)) 210 Y(JJ) = Y(JJ) + R CALL F (NEQ, TN, Y, FTEM) DO 230 J = JMIN,JMAX JJ = IWK(IBJGP+J) Y(JJ) = YH(JJ,1) R = MAX(SRUR*ABS(Y(JJ)),R0/EWT(JJ)) FAC = -HL0/R KMIN =IWK(IBIAN+JJ) KMAX =IWK(IBIAN+JJ+1) - 1 DO 220 K = KMIN,KMAX I = IWK(IBJAN+K) WK(IBA+K) = (FTEM(I) - SAVF(I))*FAC IF (I .EQ. JJ) WK(IBA+K) = WK(IBA+K) + 1.0D0 220 CONTINUE 230 CONTINUE JMIN = JMAX + 1 240 CONTINUE NFE = NFE + NGP GO TO 290 C C If JOK = 1, reconstruct new P from old P. ---------------------------- 250 JCUR = 0 RCON = CON/CON0 RCONT = ABS(CON)/CONMIN IF (RCONT .GT. RBIG .AND. IPLOST .EQ. 1) GO TO 20 KMIN = IWK(IPIAN) DO 275 J = 1,N KMAX = IWK(IPIAN+J) - 1 DO 270 K = KMIN,KMAX I = IWK(IBJAN+K) PIJ = WK(IBA+K) IF (I .NE. J) GO TO 260 PIJ = PIJ - 1.0D0 IF (ABS(PIJ) .GE. PSMALL) GO TO 260 IPLOST = 1 CONMIN = MIN(ABS(CON0),CONMIN) 260 PIJ = PIJ*RCON IF (I .EQ. J) PIJ = PIJ + 1.0D0 WK(IBA+K) = PIJ 270 CONTINUE KMIN = KMAX + 1 275 CONTINUE C C Do numerical factorization of P matrix. ------------------------------ 290 NLU = NLU + 1 CON0 = CON IERPJ = 0 DO 295 I = 1,N 295 FTEM(I) = 0.0D0 CALL CDRV (N,IWK(IPR),IWK(IPC),IWK(IPIC),IWK(IPIAN),IWK(IPJAN), 1 WK(IPA),FTEM,FTEM,NSP,IWK(IPISP),WK(IPRSP),IESP,2,IYS) IF (IYS .EQ. 0) RETURN IMUL = (IYS - 1)/N IERPJ = -2 IF (IMUL .EQ. 8) IERPJ = 1 IF (IMUL .EQ. 10) IERPJ = -1 RETURN C C If MITER = 3, construct a diagonal approximation to J and P. --------- 300 CONTINUE JCUR = 1 NJE = NJE + 1 WK(2) = HL0 IERPJ = 0 R = EL0*0.1D0 DO 310 I = 1,N 310 Y(I) = Y(I) + R*(H*SAVF(I) - YH(I,2)) CALL F (NEQ, TN, Y, WK(3)) NFE = NFE + 1 DO 320 I = 1,N R0 = H*SAVF(I) - YH(I,2) DI = 0.1D0*R0 - H*(WK(I+2) - SAVF(I)) WK(I+2) = 1.0D0 IF (ABS(R0) .LT. UROUND/EWT(I)) GO TO 320 IF (ABS(DI) .EQ. 0.0D0) GO TO 330 WK(I+2) = 0.1D0*R0/DI 320 CONTINUE RETURN 330 IERPJ = 2 RETURN C----------------------- End of Subroutine DPRJS ----------------------- END *DECK DSOLSS SUBROUTINE DSOLSS (WK, IWK, X, TEM) INTEGER IWK DOUBLE PRECISION WK, X, TEM DIMENSION WK(*), IWK(*), X(*), TEM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION RLSS COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSS01/ RLSS(6), 1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU INTEGER I DOUBLE PRECISION DI, HL0, PHL0, R C----------------------------------------------------------------------- C This routine manages the solution of the linear system arising from C a chord iteration. It is called if MITER .ne. 0. C If MITER is 1 or 2, it calls CDRV to accomplish this. C If MITER = 3 it updates the coefficient H*EL0 in the diagonal C matrix, and then computes the solution. C communication with DSOLSS uses the following variables: C WK = real work space containing the inverse diagonal matrix if C MITER = 3 and the LU decomposition of the matrix otherwise. C Storage of matrix elements starts at WK(3). C WK also contains the following matrix-related data: C WK(1) = SQRT(UROUND) (not used here), C WK(2) = HL0, the previous value of H*EL0, used if MITER = 3. C IWK = integer work space for matrix-related data, assumed to C be equivalenced to WK. In addition, WK(IPRSP) and IWK(IPISP) C are assumed to have identical locations. C X = the right-hand side vector on input, and the solution vector C on output, of length N. C TEM = vector of work space of length N, not used in this version. C IERSL = output flag (in Common). C IERSL = 0 if no trouble occurred. C IERSL = -1 if CDRV returned an error flag (MITER = 1 or 2). C This should never occur and is considered fatal. C IERSL = 1 if a singular matrix arose with MITER = 3. C This routine also uses other variables in Common. C----------------------------------------------------------------------- IERSL = 0 GO TO (100, 100, 300), MITER 100 CALL CDRV (N,IWK(IPR),IWK(IPC),IWK(IPIC),IWK(IPIAN),IWK(IPJAN), 1 WK(IPA),X,X,NSP,IWK(IPISP),WK(IPRSP),IESP,4,IERSL) IF (IERSL .NE. 0) IERSL = -1 RETURN C 300 PHL0 = WK(2) HL0 = H*EL0 WK(2) = HL0 IF (HL0 .EQ. PHL0) GO TO 330 R = HL0/PHL0 DO 320 I = 1,N DI = 1.0D0 - R*(1.0D0 - 1.0D0/WK(I+2)) IF (ABS(DI) .EQ. 0.0D0) GO TO 390 320 WK(I+2) = 1.0D0/DI 330 DO 340 I = 1,N 340 X(I) = WK(I+2)*X(I) RETURN 390 IERSL = 1 RETURN C C----------------------- End of Subroutine DSOLSS ---------------------- END *DECK DSRCMS SUBROUTINE DSRCMS (RSAV, ISAV, JOB) C----------------------------------------------------------------------- C This routine saves or restores (depending on JOB) the contents of C the Common blocks DLS001, DLSS01, which are used C internally by one or more ODEPACK solvers. C C RSAV = real array of length 224 or more. C ISAV = integer array of length 71 or more. C JOB = flag indicating to save or restore the Common blocks: C JOB = 1 if Common is to be saved (written to RSAV/ISAV) C JOB = 2 if Common is to be restored (read from RSAV/ISAV) C A call with JOB = 2 presumes a prior call with JOB = 1. C----------------------------------------------------------------------- INTEGER ISAV, JOB INTEGER ILS, ILSS INTEGER I, LENILS, LENISS, LENRLS, LENRSS DOUBLE PRECISION RSAV, RLS, RLSS DIMENSION RSAV(*), ISAV(*) SAVE LENRLS, LENILS, LENRSS, LENISS COMMON /DLS001/ RLS(218), ILS(37) COMMON /DLSS01/ RLSS(6), ILSS(34) DATA LENRLS/218/, LENILS/37/, LENRSS/6/, LENISS/34/ C IF (JOB .EQ. 2) GO TO 100 DO 10 I = 1,LENRLS 10 RSAV(I) = RLS(I) DO 15 I = 1,LENRSS 15 RSAV(LENRLS+I) = RLSS(I) C DO 20 I = 1,LENILS 20 ISAV(I) = ILS(I) DO 25 I = 1,LENISS 25 ISAV(LENILS+I) = ILSS(I) C RETURN C 100 CONTINUE DO 110 I = 1,LENRLS 110 RLS(I) = RSAV(I) DO 115 I = 1,LENRSS 115 RLSS(I) = RSAV(LENRLS+I) C DO 120 I = 1,LENILS 120 ILS(I) = ISAV(I) DO 125 I = 1,LENISS 125 ILSS(I) = ISAV(LENILS+I) C RETURN C----------------------- End of Subroutine DSRCMS ---------------------- END *DECK ODRV subroutine odrv * (n, ia,ja,a, p,ip, nsp,isp, path, flag) c 5/2/83 c*********************************************************************** c odrv -- driver for sparse matrix reordering routines c*********************************************************************** c c description c c odrv finds a minimum degree ordering of the rows and columns c of a matrix m stored in (ia,ja,a) format (see below). for the c reordered matrix, the work and storage required to perform c gaussian elimination is (usually) significantly less. c c note.. odrv and its subordinate routines have been modified to c compute orderings for general matrices, not necessarily having any c symmetry. the miminum degree ordering is computed for the c structure of the symmetric matrix m + m-transpose. c modifications to the original odrv module have been made in c the coding in subroutine mdi, and in the initial comments in c subroutines odrv and md. c c if only the nonzero entries in the upper triangle of m are being c stored, then odrv symmetrically reorders (ia,ja,a), (optionally) c with the diagonal entries placed first in each row. this is to c ensure that if m(i,j) will be in the upper triangle of m with c respect to the new ordering, then m(i,j) is stored in row i (and c thus m(j,i) is not stored), whereas if m(i,j) will be in the c strict lower triangle of m, then m(j,i) is stored in row j (and c thus m(i,j) is not stored). c c c storage of sparse matrices c c the nonzero entries of the matrix m are stored row-by-row in the c array a. to identify the individual nonzero entries in each row, c we need to know in which column each entry lies. these column c indices are stored in the array ja. i.e., if a(k) = m(i,j), then c ja(k) = j. to identify the individual rows, we need to know where c each row starts. these row pointers are stored in the array ia. c i.e., if m(i,j) is the first nonzero entry (stored) in the i-th row c and a(k) = m(i,j), then ia(i) = k. moreover, ia(n+1) points to c the first location following the last element in the last row. c thus, the number of entries in the i-th row is ia(i+1) - ia(i), c the nonzero entries in the i-th row are stored consecutively in c c a(ia(i)), a(ia(i)+1), ..., a(ia(i+1)-1), c c and the corresponding column indices are stored consecutively in c c ja(ia(i)), ja(ia(i)+1), ..., ja(ia(i+1)-1). c c when the coefficient matrix is symmetric, only the nonzero entries c in the upper triangle need be stored. for example, the matrix c c ( 1 0 2 3 0 ) c ( 0 4 0 0 0 ) c m = ( 2 0 5 6 0 ) c ( 3 0 6 7 8 ) c ( 0 0 0 8 9 ) c c could be stored as c c - 1 2 3 4 5 6 7 8 9 10 11 12 13 c ---+-------------------------------------- c ia - 1 4 5 8 12 14 c ja - 1 3 4 2 1 3 4 1 3 4 5 4 5 c a - 1 2 3 4 2 5 6 3 6 7 8 8 9 c c or (symmetrically) as c c - 1 2 3 4 5 6 7 8 9 c ---+-------------------------- c ia - 1 4 5 7 9 10 c ja - 1 3 4 2 3 4 4 5 5 c a - 1 2 3 4 5 6 7 8 9 . c c c parameters c c n - order of the matrix c c ia - integer one-dimensional array containing pointers to delimit c rows in ja and a. dimension = n+1 c c ja - integer one-dimensional array containing the column indices c corresponding to the elements of a. dimension = number of c nonzero entries in (the upper triangle of) m c c a - real one-dimensional array containing the nonzero entries in c (the upper triangle of) m, stored by rows. dimension = c number of nonzero entries in (the upper triangle of) m c c p - integer one-dimensional array used to return the permutation c of the rows and columns of m corresponding to the minimum c degree ordering. dimension = n c c ip - integer one-dimensional array used to return the inverse of c the permutation returned in p. dimension = n c c nsp - declared dimension of the one-dimensional array isp. nsp c must be at least 3n+4k, where k is the number of nonzeroes c in the strict upper triangle of m c c isp - integer one-dimensional array used for working storage. c dimension = nsp c c path - integer path specification. values and their meanings are - c 1 find minimum degree ordering only c 2 find minimum degree ordering and reorder symmetrically c stored matrix (used when only the nonzero entries in c the upper triangle of m are being stored) c 3 reorder symmetrically stored matrix as specified by c input permutation (used when an ordering has already c been determined and only the nonzero entries in the c upper triangle of m are being stored) c 4 same as 2 but put diagonal entries at start of each row c 5 same as 3 but put diagonal entries at start of each row c c flag - integer error flag. values and their meanings are - c 0 no errors detected c 9n+k insufficient storage in md c 10n+1 insufficient storage in odrv c 11n+1 illegal path specification c c c conversion from real to double precision c c change the real declarations in odrv and sro to double precision c declarations. c c----------------------------------------------------------------------- c integer ia(*), ja(*), p(*), ip(*), isp(*), path, flag, * v, l, head, tmp, q c... real a(*) double precision a(*) logical dflag c c----initialize error flag and validate path specification flag = 0 if (path.lt.1 .or. 5.lt.path) go to 111 c c----allocate storage and find minimum degree ordering if ((path-1) * (path-2) * (path-4) .ne. 0) go to 1 max = (nsp-n)/2 v = 1 l = v + max head = l + max next = head + n if (max.lt.n) go to 110 c call md * (n, ia,ja, max,isp(v),isp(l), isp(head),p,ip, isp(v), flag) if (flag.ne.0) go to 100 c c----allocate storage and symmetrically reorder matrix 1 if ((path-2) * (path-3) * (path-4) * (path-5) .ne. 0) go to 2 tmp = (nsp+1) - n q = tmp - (ia(n+1)-1) if (q.lt.1) go to 110 c dflag = path.eq.4 .or. path.eq.5 call sro * (n, ip, ia, ja, a, isp(tmp), isp(q), dflag) c 2 return c c ** error -- error detected in md 100 return c ** error -- insufficient storage 110 flag = 10*n + 1 return c ** error -- illegal path specified 111 flag = 11*n + 1 return end subroutine md * (n, ia,ja, max, v,l, head,last,next, mark, flag) c*********************************************************************** c md -- minimum degree algorithm (based on element model) c*********************************************************************** c c description c c md finds a minimum degree ordering of the rows and columns of a c general sparse matrix m stored in (ia,ja,a) format. c when the structure of m is nonsymmetric, the ordering is that c obtained for the symmetric matrix m + m-transpose. c c c additional parameters c c max - declared dimension of the one-dimensional arrays v and l. c max must be at least n+2k, where k is the number of c nonzeroes in the strict upper triangle of m + m-transpose c c v - integer one-dimensional work array. dimension = max c c l - integer one-dimensional work array. dimension = max c c head - integer one-dimensional work array. dimension = n c c last - integer one-dimensional array used to return the permutation c of the rows and columns of m corresponding to the minimum c degree ordering. dimension = n c c next - integer one-dimensional array used to return the inverse of c the permutation returned in last. dimension = n c c mark - integer one-dimensional work array (may be the same as v). c dimension = n c c flag - integer error flag. values and their meanings are - c 0 no errors detected c 9n+k insufficient storage in md c c c definitions of internal parameters c c ---------+--------------------------------------------------------- c v(s) - value field of list entry c ---------+--------------------------------------------------------- c l(s) - link field of list entry (0 =) end of list) c ---------+--------------------------------------------------------- c l(vi) - pointer to element list of uneliminated vertex vi c ---------+--------------------------------------------------------- c l(ej) - pointer to boundary list of active element ej c ---------+--------------------------------------------------------- c head(d) - vj =) vj head of d-list d c - 0 =) no vertex in d-list d c c c - vi uneliminated vertex c - vi in ek - vi not in ek c ---------+-----------------------------+--------------------------- c next(vi) - undefined but nonnegative - vj =) vj next in d-list c - - 0 =) vi tail of d-list c ---------+-----------------------------+--------------------------- c last(vi) - (not set until mdp) - -d =) vi head of d-list d c --vk =) compute degree - vj =) vj last in d-list c - ej =) vi prototype of ej - 0 =) vi not in any d-list c - 0 =) do not compute degree - c ---------+-----------------------------+--------------------------- c mark(vi) - mark(vk) - nonneg. tag .lt. mark(vk) c c c - vi eliminated vertex c - ei active element - otherwise c ---------+-----------------------------+--------------------------- c next(vi) - -j =) vi was j-th vertex - -j =) vi was j-th vertex c - to be eliminated - to be eliminated c ---------+-----------------------------+--------------------------- c last(vi) - m =) size of ei = m - undefined c ---------+-----------------------------+--------------------------- c mark(vi) - -m =) overlap count of ei - undefined c - with ek = m - c - otherwise nonnegative tag - c - .lt. mark(vk) - c c----------------------------------------------------------------------- c integer ia(*), ja(*), v(*), l(*), head(*), last(*), next(*), * mark(*), flag, tag, dmin, vk,ek, tail equivalence (vk,ek) c c----initialization tag = 0 call mdi * (n, ia,ja, max,v,l, head,last,next, mark,tag, flag) if (flag.ne.0) return c k = 0 dmin = 1 c c----while k .lt. n do 1 if (k.ge.n) go to 4 c c------search for vertex of minimum degree 2 if (head(dmin).gt.0) go to 3 dmin = dmin + 1 go to 2 c c------remove vertex vk of minimum degree from degree list 3 vk = head(dmin) head(dmin) = next(vk) if (head(dmin).gt.0) last(head(dmin)) = -dmin c c------number vertex vk, adjust tag, and tag vk k = k+1 next(vk) = -k last(ek) = dmin - 1 tag = tag + last(ek) mark(vk) = tag c c------form element ek from uneliminated neighbors of vk call mdm * (vk,tail, v,l, last,next, mark) c c------purge inactive elements and do mass elimination call mdp * (k,ek,tail, v,l, head,last,next, mark) c c------update degrees of uneliminated vertices in ek call mdu * (ek,dmin, v,l, head,last,next, mark) c go to 1 c c----generate inverse permutation from permutation 4 do 5 k=1,n next(k) = -next(k) 5 last(next(k)) = k c return end subroutine mdi * (n, ia,ja, max,v,l, head,last,next, mark,tag, flag) c*********************************************************************** c mdi -- initialization c*********************************************************************** integer ia(*), ja(*), v(*), l(*), head(*), last(*), next(*), * mark(*), tag, flag, sfs, vi,dvi, vj c c----initialize degrees, element lists, and degree lists do 1 vi=1,n mark(vi) = 1 l(vi) = 0 1 head(vi) = 0 sfs = n+1 c c----create nonzero structure c----for each nonzero entry a(vi,vj) do 6 vi=1,n jmin = ia(vi) jmax = ia(vi+1) - 1 if (jmin.gt.jmax) go to 6 do 5 j=jmin,jmax vj = ja(j) if (vj-vi) 2, 5, 4 c c------if a(vi,vj) is in strict lower triangle c------check for previous occurrence of a(vj,vi) 2 lvk = vi kmax = mark(vi) - 1 if (kmax .eq. 0) go to 4 do 3 k=1,kmax lvk = l(lvk) if (v(lvk).eq.vj) go to 5 3 continue c----for unentered entries a(vi,vj) 4 if (sfs.ge.max) go to 101 c c------enter vj in element list for vi mark(vi) = mark(vi) + 1 v(sfs) = vj l(sfs) = l(vi) l(vi) = sfs sfs = sfs+1 c c------enter vi in element list for vj mark(vj) = mark(vj) + 1 v(sfs) = vi l(sfs) = l(vj) l(vj) = sfs sfs = sfs+1 5 continue 6 continue c c----create degree lists and initialize mark vector do 7 vi=1,n dvi = mark(vi) next(vi) = head(dvi) head(dvi) = vi last(vi) = -dvi nextvi = next(vi) if (nextvi.gt.0) last(nextvi) = vi 7 mark(vi) = tag c return c c ** error- insufficient storage 101 flag = 9*n + vi return end subroutine mdm * (vk,tail, v,l, last,next, mark) c*********************************************************************** c mdm -- form element from uneliminated neighbors of vk c*********************************************************************** integer vk, tail, v(*), l(*), last(*), next(*), mark(*), * tag, s,ls,vs,es, b,lb,vb, blp,blpmax equivalence (vs, es) c c----initialize tag and list of uneliminated neighbors tag = mark(vk) tail = vk c c----for each vertex/element vs/es in element list of vk ls = l(vk) 1 s = ls if (s.eq.0) go to 5 ls = l(s) vs = v(s) if (next(vs).lt.0) go to 2 c c------if vs is uneliminated vertex, then tag and append to list of c------uneliminated neighbors mark(vs) = tag l(tail) = s tail = s go to 4 c c------if es is active element, then ... c--------for each vertex vb in boundary list of element es 2 lb = l(es) blpmax = last(es) do 3 blp=1,blpmax b = lb lb = l(b) vb = v(b) c c----------if vb is untagged vertex, then tag and append to list of c----------uneliminated neighbors if (mark(vb).ge.tag) go to 3 mark(vb) = tag l(tail) = b tail = b 3 continue c c--------mark es inactive mark(es) = tag c 4 go to 1 c c----terminate list of uneliminated neighbors 5 l(tail) = 0 c return end subroutine mdp * (k,ek,tail, v,l, head,last,next, mark) c*********************************************************************** c mdp -- purge inactive elements and do mass elimination c*********************************************************************** integer ek, tail, v(*), l(*), head(*), last(*), next(*), * mark(*), tag, free, li,vi,lvi,evi, s,ls,es, ilp,ilpmax c c----initialize tag tag = mark(ek) c c----for each vertex vi in ek li = ek ilpmax = last(ek) if (ilpmax.le.0) go to 12 do 11 ilp=1,ilpmax i = li li = l(i) vi = v(li) c c------remove vi from degree list if (last(vi).eq.0) go to 3 if (last(vi).gt.0) go to 1 head(-last(vi)) = next(vi) go to 2 1 next(last(vi)) = next(vi) 2 if (next(vi).gt.0) last(next(vi)) = last(vi) c c------remove inactive items from element list of vi 3 ls = vi 4 s = ls ls = l(s) if (ls.eq.0) go to 6 es = v(ls) if (mark(es).lt.tag) go to 5 free = ls l(s) = l(ls) ls = s 5 go to 4 c c------if vi is interior vertex, then remove from list and eliminate 6 lvi = l(vi) if (lvi.ne.0) go to 7 l(i) = l(li) li = i c k = k+1 next(vi) = -k last(ek) = last(ek) - 1 go to 11 c c------else ... c--------classify vertex vi 7 if (l(lvi).ne.0) go to 9 evi = v(lvi) if (next(evi).ge.0) go to 9 if (mark(evi).lt.0) go to 8 c c----------if vi is prototype vertex, then mark as such, initialize c----------overlap count for corresponding element, and move vi to end c----------of boundary list last(vi) = evi mark(evi) = -1 l(tail) = li tail = li l(i) = l(li) li = i go to 10 c c----------else if vi is duplicate vertex, then mark as such and adjust c----------overlap count for corresponding element 8 last(vi) = 0 mark(evi) = mark(evi) - 1 go to 10 c c----------else mark vi to compute degree 9 last(vi) = -ek c c--------insert ek in element list of vi 10 v(free) = ek l(free) = l(vi) l(vi) = free 11 continue c c----terminate boundary list 12 l(tail) = 0 c return end subroutine mdu * (ek,dmin, v,l, head,last,next, mark) c*********************************************************************** c mdu -- update degrees of uneliminated vertices in ek c*********************************************************************** integer ek, dmin, v(*), l(*), head(*), last(*), next(*), * mark(*), tag, vi,evi,dvi, s,vs,es, b,vb, ilp,ilpmax, * blp,blpmax equivalence (vs, es) c c----initialize tag tag = mark(ek) - last(ek) c c----for each vertex vi in ek i = ek ilpmax = last(ek) if (ilpmax.le.0) go to 11 do 10 ilp=1,ilpmax i = l(i) vi = v(i) if (last(vi)) 1, 10, 8 c c------if vi neither prototype nor duplicate vertex, then merge elements c------to compute degree 1 tag = tag + 1 dvi = last(ek) c c--------for each vertex/element vs/es in element list of vi s = l(vi) 2 s = l(s) if (s.eq.0) go to 9 vs = v(s) if (next(vs).lt.0) go to 3 c c----------if vs is uneliminated vertex, then tag and adjust degree mark(vs) = tag dvi = dvi + 1 go to 5 c c----------if es is active element, then expand c------------check for outmatched vertex 3 if (mark(es).lt.0) go to 6 c c------------for each vertex vb in es b = es blpmax = last(es) do 4 blp=1,blpmax b = l(b) vb = v(b) c c--------------if vb is untagged, then tag and adjust degree if (mark(vb).ge.tag) go to 4 mark(vb) = tag dvi = dvi + 1 4 continue c 5 go to 2 c c------else if vi is outmatched vertex, then adjust overlaps but do not c------compute degree 6 last(vi) = 0 mark(es) = mark(es) - 1 7 s = l(s) if (s.eq.0) go to 10 es = v(s) if (mark(es).lt.0) mark(es) = mark(es) - 1 go to 7 c c------else if vi is prototype vertex, then calculate degree by c------inclusion/exclusion and reset overlap count 8 evi = last(vi) dvi = last(ek) + last(evi) + mark(evi) mark(evi) = 0 c c------insert vi in appropriate degree list 9 next(vi) = head(dvi) head(dvi) = vi last(vi) = -dvi if (next(vi).gt.0) last(next(vi)) = vi if (dvi.lt.dmin) dmin = dvi c 10 continue c 11 return end subroutine sro * (n, ip, ia,ja,a, q, r, dflag) c*********************************************************************** c sro -- symmetric reordering of sparse symmetric matrix c*********************************************************************** c c description c c the nonzero entries of the matrix m are assumed to be stored c symmetrically in (ia,ja,a) format (i.e., not both m(i,j) and m(j,i) c are stored if i ne j). c c sro does not rearrange the order of the rows, but does move c nonzeroes from one row to another to ensure that if m(i,j) will be c in the upper triangle of m with respect to the new ordering, then c m(i,j) is stored in row i (and thus m(j,i) is not stored), whereas c if m(i,j) will be in the strict lower triangle of m, then m(j,i) is c stored in row j (and thus m(i,j) is not stored). c c c additional parameters c c q - integer one-dimensional work array. dimension = n c c r - integer one-dimensional work array. dimension = number of c nonzero entries in the upper triangle of m c c dflag - logical variable. if dflag = .true., then store nonzero c diagonal elements at the beginning of the row c c----------------------------------------------------------------------- c integer ip(*), ia(*), ja(*), q(*), r(*) c... real a(*), ak double precision a(*), ak logical dflag c c c--phase 1 -- find row in which to store each nonzero c----initialize count of nonzeroes to be stored in each row do 1 i=1,n 1 q(i) = 0 c c----for each nonzero element a(j) do 3 i=1,n jmin = ia(i) jmax = ia(i+1) - 1 if (jmin.gt.jmax) go to 3 do 2 j=jmin,jmax c c--------find row (=r(j)) and column (=ja(j)) in which to store a(j) ... k = ja(j) if (ip(k).lt.ip(i)) ja(j) = i if (ip(k).ge.ip(i)) k = i r(j) = k c c--------... and increment count of nonzeroes (=q(r(j)) in that row 2 q(k) = q(k) + 1 3 continue c c c--phase 2 -- find new ia and permutation to apply to (ja,a) c----determine pointers to delimit rows in permuted (ja,a) do 4 i=1,n ia(i+1) = ia(i) + q(i) 4 q(i) = ia(i+1) c c----determine where each (ja(j),a(j)) is stored in permuted (ja,a) c----for each nonzero element (in reverse order) ilast = 0 jmin = ia(1) jmax = ia(n+1) - 1 j = jmax do 6 jdummy=jmin,jmax i = r(j) if (.not.dflag .or. ja(j).ne.i .or. i.eq.ilast) go to 5 c c------if dflag, then put diagonal nonzero at beginning of row r(j) = ia(i) ilast = i go to 6 c c------put (off-diagonal) nonzero in last unused location in row 5 q(i) = q(i) - 1 r(j) = q(i) c 6 j = j-1 c c c--phase 3 -- permute (ja,a) to upper triangular form (wrt new ordering) do 8 j=jmin,jmax 7 if (r(j).eq.j) go to 8 k = r(j) r(j) = r(k) r(k) = k jak = ja(k) ja(k) = ja(j) ja(j) = jak ak = a(k) a(k) = a(j) a(j) = ak go to 7 8 continue c return end *DECK CDRV subroutine cdrv * (n, r,c,ic, ia,ja,a, b, z, nsp,isp,rsp,esp, path, flag) c*** subroutine cdrv c*** driver for subroutines for solving sparse nonsymmetric systems of c linear equations (compressed pointer storage) c c c parameters c class abbreviations are-- c n - integer variable c f - real variable c v - supplies a value to the driver c r - returns a result from the driver c i - used internally by the driver c a - array c c class - parameter c ------+---------- c - c the nonzero entries of the coefficient matrix m are stored c row-by-row in the array a. to identify the individual nonzero c entries in each row, we need to know in which column each entry c lies. the column indices which correspond to the nonzero entries c of m are stored in the array ja. i.e., if a(k) = m(i,j), then c ja(k) = j. in addition, we need to know where each row starts and c how long it is. the index positions in ja and a where the rows of c m begin are stored in the array ia. i.e., if m(i,j) is the first c nonzero entry (stored) in the i-th row and a(k) = m(i,j), then c ia(i) = k. moreover, the index in ja and a of the first location c following the last element in the last row is stored in ia(n+1). c thus, the number of entries in the i-th row is given by c ia(i+1) - ia(i), the nonzero entries of the i-th row are stored c consecutively in c a(ia(i)), a(ia(i)+1), ..., a(ia(i+1)-1), c and the corresponding column indices are stored consecutively in c ja(ia(i)), ja(ia(i)+1), ..., ja(ia(i+1)-1). c for example, the 5 by 5 matrix c ( 1. 0. 2. 0. 0.) c ( 0. 3. 0. 0. 0.) c m = ( 0. 4. 5. 6. 0.) c ( 0. 0. 0. 7. 0.) c ( 0. 0. 0. 8. 9.) c would be stored as c - 1 2 3 4 5 6 7 8 9 c ---+-------------------------- c ia - 1 3 4 7 8 10 c ja - 1 3 2 2 3 4 4 4 5 c a - 1. 2. 3. 4. 5. 6. 7. 8. 9. . c c nv - n - number of variables/equations. c fva - a - nonzero entries of the coefficient matrix m, stored c - by rows. c - size = number of nonzero entries in m. c nva - ia - pointers to delimit the rows in a. c - size = n+1. c nva - ja - column numbers corresponding to the elements of a. c - size = size of a. c fva - b - right-hand side b. b and z can the same array. c - size = n. c fra - z - solution x. b and z can be the same array. c - size = n. c c the rows and columns of the original matrix m can be c reordered (e.g., to reduce fillin or ensure numerical stability) c before calling the driver. if no reordering is done, then set c r(i) = c(i) = ic(i) = i for i=1,...,n. the solution z is returned c in the original order. c if the columns have been reordered (i.e., c(i).ne.i for some c i), then the driver will call a subroutine (nroc) which rearranges c each row of ja and a, leaving the rows in the original order, but c placing the elements of each row in increasing order with respect c to the new ordering. if path.ne.1, then nroc is assumed to have c been called already. c c nva - r - ordering of the rows of m. c - size = n. c nva - c - ordering of the columns of m. c - size = n. c nva - ic - inverse of the ordering of the columns of m. i.e., c - ic(c(i)) = i for i=1,...,n. c - size = n. c c the solution of the system of linear equations is divided into c three stages -- c nsfc -- the matrix m is processed symbolically to determine where c fillin will occur during the numeric factorization. c nnfc -- the matrix m is factored numerically into the product ldu c of a unit lower triangular matrix l, a diagonal matrix c d, and a unit upper triangular matrix u, and the system c mx = b is solved. c nnsc -- the linear system mx = b is solved using the ldu c or factorization from nnfc. c nntc -- the transposed linear system mt x = b is solved using c the ldu factorization from nnf. c for several systems whose coefficient matrices have the same c nonzero structure, nsfc need be done only once (for the first c system). then nnfc is done once for each additional system. for c several systems with the same coefficient matrix, nsfc and nnfc c need be done only once (for the first system). then nnsc or nntc c is done once for each additional right-hand side. c c nv - path - path specification. values and their meanings are -- c - 1 perform nroc, nsfc, and nnfc. c - 2 perform nnfc only (nsfc is assumed to have been c - done in a manner compatible with the storage c - allocation used in the driver). c - 3 perform nnsc only (nsfc and nnfc are assumed to c - have been done in a manner compatible with the c - storage allocation used in the driver). c - 4 perform nntc only (nsfc and nnfc are assumed to c - have been done in a manner compatible with the c - storage allocation used in the driver). c - 5 perform nroc and nsfc. c c various errors are detected by the driver and the individual c subroutines. c c nr - flag - error flag. values and their meanings are -- c - 0 no errors detected c - n+k null row in a -- row = k c - 2n+k duplicate entry in a -- row = k c - 3n+k insufficient storage in nsfc -- row = k c - 4n+1 insufficient storage in nnfc c - 5n+k null pivot -- row = k c - 6n+k insufficient storage in nsfc -- row = k c - 7n+1 insufficient storage in nnfc c - 8n+k zero pivot -- row = k c - 10n+1 insufficient storage in cdrv c - 11n+1 illegal path specification c c working storage is needed for the factored form of the matrix c m plus various temporary vectors. the arrays isp and rsp should be c equivalenced. integer storage is allocated from the beginning of c isp and real storage from the end of rsp. c c nv - nsp - declared dimension of rsp. nsp generally must c - be larger than 8n+2 + 2k (where k = (number of c - nonzero entries in m)). c nvira - isp - integer working storage divided up into various arrays c - needed by the subroutines. isp and rsp should be c - equivalenced. c - size = lratio*nsp. c fvira - rsp - real working storage divided up into various arrays c - needed by the subroutines. isp and rsp should be c - equivalenced. c - size = nsp. c nr - esp - if sufficient storage was available to perform the c - symbolic factorization (nsfc), then esp is set to c - the amount of excess storage provided (negative if c - insufficient storage was available to perform the c - numeric factorization (nnfc)). c c c conversion to double precision c c to convert these routines for double precision arrays.. c (1) use the double precision declarations in place of the real c declarations in each subprogram, as given in comment cards. c (2) change the data-loaded value of the integer lratio c in subroutine cdrv, as indicated below. c (3) change e0 to d0 in the constants in statement number 10 c in subroutine nnfc and the line following that. c integer r(*), c(*), ic(*), ia(*), ja(*), isp(*), esp, path, * flag, d, u, q, row, tmp, ar, umax c real a(*), b(*), z(*), rsp(*) double precision a(*), b(*), z(*), rsp(*) c c set lratio equal to the ratio between the length of floating point c and integer array data. e. g., lratio = 1 for (real, integer), c lratio = 2 for (double precision, integer) c data lratio/2/ c if (path.lt.1 .or. 5.lt.path) go to 111 c******initialize and divide up temporary storage ******************* il = 1 ijl = il + (n+1) iu = ijl + n iju = iu + (n+1) irl = iju + n jrl = irl + n jl = jrl + n c c ****** reorder a if necessary, call nsfc if flag is set *********** if ((path-1) * (path-5) .ne. 0) go to 5 max = (lratio*nsp + 1 - jl) - (n+1) - 5*n jlmax = max/2 q = jl + jlmax ira = q + (n+1) jra = ira + n irac = jra + n iru = irac + n jru = iru + n jutmp = jru + n jumax = lratio*nsp + 1 - jutmp esp = max/lratio if (jlmax.le.0 .or. jumax.le.0) go to 110 c do 1 i=1,n if (c(i).ne.i) go to 2 1 continue go to 3 2 ar = nsp + 1 - n call nroc * (n, ic, ia,ja,a, isp(il), rsp(ar), isp(iu), flag) if (flag.ne.0) go to 100 c 3 call nsfc * (n, r, ic, ia,ja, * jlmax, isp(il), isp(jl), isp(ijl), * jumax, isp(iu), isp(jutmp), isp(iju), * isp(q), isp(ira), isp(jra), isp(irac), * isp(irl), isp(jrl), isp(iru), isp(jru), flag) if(flag .ne. 0) go to 100 c ****** move ju next to jl ***************************************** jlmax = isp(ijl+n-1) ju = jl + jlmax jumax = isp(iju+n-1) if (jumax.le.0) go to 5 do 4 j=1,jumax 4 isp(ju+j-1) = isp(jutmp+j-1) c c ****** call remaining subroutines ********************************* 5 jlmax = isp(ijl+n-1) ju = jl + jlmax jumax = isp(iju+n-1) l = (ju + jumax - 2 + lratio) / lratio + 1 lmax = isp(il+n) - 1 d = l + lmax u = d + n row = nsp + 1 - n tmp = row - n umax = tmp - u esp = umax - (isp(iu+n) - 1) c if ((path-1) * (path-2) .ne. 0) go to 6 if (umax.lt.0) go to 110 call nnfc * (n, r, c, ic, ia, ja, a, z, b, * lmax, isp(il), isp(jl), isp(ijl), rsp(l), rsp(d), * umax, isp(iu), isp(ju), isp(iju), rsp(u), * rsp(row), rsp(tmp), isp(irl), isp(jrl), flag) if(flag .ne. 0) go to 100 c 6 if ((path-3) .ne. 0) go to 7 call nnsc * (n, r, c, isp(il), isp(jl), isp(ijl), rsp(l), * rsp(d), isp(iu), isp(ju), isp(iju), rsp(u), * z, b, rsp(tmp)) c 7 if ((path-4) .ne. 0) go to 8 call nntc * (n, r, c, isp(il), isp(jl), isp(ijl), rsp(l), * rsp(d), isp(iu), isp(ju), isp(iju), rsp(u), * z, b, rsp(tmp)) 8 return c c ** error.. error detected in nroc, nsfc, nnfc, or nnsc 100 return c ** error.. insufficient storage 110 flag = 10*n + 1 return c ** error.. illegal path specification 111 flag = 11*n + 1 return end subroutine nroc (n, ic, ia, ja, a, jar, ar, p, flag) c c ---------------------------------------------------------------- c c yale sparse matrix package - nonsymmetric codes c solving the system of equations mx = b c c i. calling sequences c the coefficient matrix can be processed by an ordering routine c (e.g., to reduce fillin or ensure numerical stability) before using c the remaining subroutines. if no reordering is done, then set c r(i) = c(i) = ic(i) = i for i=1,...,n. if an ordering subroutine c is used, then nroc should be used to reorder the coefficient matrix c the calling sequence is -- c ( (matrix ordering)) c (nroc (matrix reordering)) c nsfc (symbolic factorization to determine where fillin will c occur during numeric factorization) c nnfc (numeric factorization into product ldu of unit lower c triangular matrix l, diagonal matrix d, and unit c upper triangular matrix u, and solution of linear c system) c nnsc (solution of linear system for additional right-hand c side using ldu factorization from nnfc) c (if only one system of equations is to be solved, then the c subroutine trk should be used.) c c ii. storage of sparse matrices c the nonzero entries of the coefficient matrix m are stored c row-by-row in the array a. to identify the individual nonzero c entries in each row, we need to know in which column each entry c lies. the column indices which correspond to the nonzero entries c of m are stored in the array ja. i.e., if a(k) = m(i,j), then c ja(k) = j. in addition, we need to know where each row starts and c how long it is. the index positions in ja and a where the rows of c m begin are stored in the array ia. i.e., if m(i,j) is the first c (leftmost) entry in the i-th row and a(k) = m(i,j), then c ia(i) = k. moreover, the index in ja and a of the first location c following the last element in the last row is stored in ia(n+1). c thus, the number of entries in the i-th row is given by c ia(i+1) - ia(i), the nonzero entries of the i-th row are stored c consecutively in c a(ia(i)), a(ia(i)+1), ..., a(ia(i+1)-1), c and the corresponding column indices are stored consecutively in c ja(ia(i)), ja(ia(i)+1), ..., ja(ia(i+1)-1). c for example, the 5 by 5 matrix c ( 1. 0. 2. 0. 0.) c ( 0. 3. 0. 0. 0.) c m = ( 0. 4. 5. 6. 0.) c ( 0. 0. 0. 7. 0.) c ( 0. 0. 0. 8. 9.) c would be stored as c - 1 2 3 4 5 6 7 8 9 c ---+-------------------------- c ia - 1 3 4 7 8 10 c ja - 1 3 2 2 3 4 4 4 5 c a - 1. 2. 3. 4. 5. 6. 7. 8. 9. . c c the strict upper (lower) triangular portion of the matrix c u (l) is stored in a similar fashion using the arrays iu, ju, u c (il, jl, l) except that an additional array iju (ijl) is used to c compress storage of ju (jl) by allowing some sequences of column c (row) indices to used for more than one row (column) (n.b., l is c stored by columns). iju(k) (ijl(k)) points to the starting c location in ju (jl) of entries for the kth row (column). c compression in ju (jl) occurs in two ways. first, if a row c (column) i was merged into the current row (column) k, and the c number of elements merged in from (the tail portion of) row c (column) i is the same as the final length of row (column) k, then c the kth row (column) and the tail of row (column) i are identical c and iju(k) (ijl(k)) points to the start of the tail. second, if c some tail portion of the (k-1)st row (column) is identical to the c head of the kth row (column), then iju(k) (ijl(k)) points to the c start of that tail portion. for example, the nonzero structure of c the strict upper triangular part of the matrix c d 0 x x x c 0 d 0 x x c 0 0 d x 0 c 0 0 0 d x c 0 0 0 0 d c would be represented as c - 1 2 3 4 5 6 c ----+------------ c iu - 1 4 6 7 8 8 c ju - 3 4 5 4 c iju - 1 2 4 3 . c the diagonal entries of l and u are assumed to be equal to one and c are not stored. the array d contains the reciprocals of the c diagonal entries of the matrix d. c c iii. additional storage savings c in nsfc, r and ic can be the same array in the calling c sequence if no reordering of the coefficient matrix has been done. c in nnfc, r, c, and ic can all be the same array if no c reordering has been done. if only the rows have been reordered, c then c and ic can be the same array. if the row and column c orderings are the same, then r and c can be the same array. z and c row can be the same array. c in nnsc or nntc, r and c can be the same array if no c reordering has been done or if the row and column orderings are the c same. z and b can be the same array. however, then b will be c destroyed. c c iv. parameters c following is a list of parameters to the programs. names are c uniform among the various subroutines. class abbreviations are -- c n - integer variable c f - real variable c v - supplies a value to a subroutine c r - returns a result from a subroutine c i - used internally by a subroutine c a - array c c class - parameter c ------+---------- c fva - a - nonzero entries of the coefficient matrix m, stored c - by rows. c - size = number of nonzero entries in m. c fva - b - right-hand side b. c - size = n. c nva - c - ordering of the columns of m. c - size = n. c fvra - d - reciprocals of the diagonal entries of the matrix d. c - size = n. c nr - flag - error flag. values and their meanings are -- c - 0 no errors detected c - n+k null row in a -- row = k c - 2n+k duplicate entry in a -- row = k c - 3n+k insufficient storage for jl -- row = k c - 4n+1 insufficient storage for l c - 5n+k null pivot -- row = k c - 6n+k insufficient storage for ju -- row = k c - 7n+1 insufficient storage for u c - 8n+k zero pivot -- row = k c nva - ia - pointers to delimit the rows of a. c - size = n+1. c nvra - ijl - pointers to the first element in each column in jl, c - used to compress storage in jl. c - size = n. c nvra - iju - pointers to the first element in each row in ju, used c - to compress storage in ju. c - size = n. c nvra - il - pointers to delimit the columns of l. c - size = n+1. c nvra - iu - pointers to delimit the rows of u. c - size = n+1. c nva - ja - column numbers corresponding to the elements of a. c - size = size of a. c nvra - jl - row numbers corresponding to the elements of l. c - size = jlmax. c nv - jlmax - declared dimension of jl. jlmax must be larger than c - the number of nonzeros in the strict lower triangle c - of m plus fillin minus compression. c nvra - ju - column numbers corresponding to the elements of u. c - size = jumax. c nv - jumax - declared dimension of ju. jumax must be larger than c - the number of nonzeros in the strict upper triangle c - of m plus fillin minus compression. c fvra - l - nonzero entries in the strict lower triangular portion c - of the matrix l, stored by columns. c - size = lmax. c nv - lmax - declared dimension of l. lmax must be larger than c - the number of nonzeros in the strict lower triangle c - of m plus fillin (il(n+1)-1 after nsfc). c nv - n - number of variables/equations. c nva - r - ordering of the rows of m. c - size = n. c fvra - u - nonzero entries in the strict upper triangular portion c - of the matrix u, stored by rows. c - size = umax. c nv - umax - declared dimension of u. umax must be larger than c - the number of nonzeros in the strict upper triangle c - of m plus fillin (iu(n+1)-1 after nsfc). c fra - z - solution x. c - size = n. c c ---------------------------------------------------------------- c c*** subroutine nroc c*** reorders rows of a, leaving row order unchanged c c c input parameters.. n, ic, ia, ja, a c output parameters.. ja, a, flag c c parameters used internally.. c nia - p - at the kth step, p is a linked list of the reordered c - column indices of the kth row of a. p(n+1) points c - to the first entry in the list. c - size = n+1. c nia - jar - at the kth step,jar contains the elements of the c - reordered column indices of a. c - size = n. c fia - ar - at the kth step, ar contains the elements of the c - reordered row of a. c - size = n. c integer ic(*), ia(*), ja(*), jar(*), p(*), flag c real a(*), ar(*) double precision a(*), ar(*) c c ****** for each nonempty row ******************************* do 5 k=1,n jmin = ia(k) jmax = ia(k+1) - 1 if(jmin .gt. jmax) go to 5 p(n+1) = n + 1 c ****** insert each element in the list ********************* do 3 j=jmin,jmax newj = ic(ja(j)) i = n + 1 1 if(p(i) .ge. newj) go to 2 i = p(i) go to 1 2 if(p(i) .eq. newj) go to 102 p(newj) = p(i) p(i) = newj jar(newj) = ja(j) ar(newj) = a(j) 3 continue c ****** replace old row in ja and a ************************* i = n + 1 do 4 j=jmin,jmax i = p(i) ja(j) = jar(i) 4 a(j) = ar(i) 5 continue flag = 0 return c c ** error.. duplicate entry in a 102 flag = n + k return end subroutine nsfc * (n, r, ic, ia,ja, jlmax,il,jl,ijl, jumax,iu,ju,iju, * q, ira,jra, irac, irl,jrl, iru,jru, flag) c*** subroutine nsfc c*** symbolic ldu-factorization of nonsymmetric sparse matrix c (compressed pointer storage) c c c input variables.. n, r, ic, ia, ja, jlmax, jumax. c output variables.. il, jl, ijl, iu, ju, iju, flag. c c parameters used internally.. c nia - q - suppose m* is the result of reordering m. if c - processing of the ith row of m* (hence the ith c - row of u) is being done, q(j) is initially c - nonzero if m*(i,j) is nonzero (j.ge.i). since c - values need not be stored, each entry points to the c - next nonzero and q(n+1) points to the first. n+1 c - indicates the end of the list. for example, if n=9 c - and the 5th row of m* is c - 0 x x 0 x 0 0 x 0 c - then q will initially be c - a a a a 8 a a 10 5 (a - arbitrary). c - as the algorithm proceeds, other elements of q c - are inserted in the list because of fillin. c - q is used in an analogous manner to compute the c - ith column of l. c - size = n+1. c nia - ira, - vectors used to find the columns of m. at the kth c nia - jra, step of the factorization, irac(k) points to the c nia - irac head of a linked list in jra of row indices i c - such that i .ge. k and m(i,k) is nonzero. zero c - indicates the end of the list. ira(i) (i.ge.k) c - points to the smallest j such that j .ge. k and c - m(i,j) is nonzero. c - size of each = n. c nia - irl, - vectors used to find the rows of l. at the kth step c nia - jrl of the factorization, jrl(k) points to the head c - of a linked list in jrl of column indices j c - such j .lt. k and l(k,j) is nonzero. zero c - indicates the end of the list. irl(j) (j.lt.k) c - points to the smallest i such that i .ge. k and c - l(i,j) is nonzero. c - size of each = n. c nia - iru, - vectors used in a manner analogous to irl and jrl c nia - jru to find the columns of u. c - size of each = n. c c internal variables.. c jlptr - points to the last position used in jl. c juptr - points to the last position used in ju. c jmin,jmax - are the indices in a or u of the first and last c elements to be examined in a given row. c for example, jmin=ia(k), jmax=ia(k+1)-1. c integer cend, qm, rend, rk, vj integer ia(*), ja(*), ira(*), jra(*), il(*), jl(*), ijl(*) integer iu(*), ju(*), iju(*), irl(*), jrl(*), iru(*), jru(*) integer r(*), ic(*), q(*), irac(*), flag c c ****** initialize pointers **************************************** np1 = n + 1 jlmin = 1 jlptr = 0 il(1) = 1 jumin = 1 juptr = 0 iu(1) = 1 do 1 k=1,n irac(k) = 0 jra(k) = 0 jrl(k) = 0 1 jru(k) = 0 c ****** initialize column pointers for a *************************** do 2 k=1,n rk = r(k) iak = ia(rk) if (iak .ge. ia(rk+1)) go to 101 jaiak = ic(ja(iak)) if (jaiak .gt. k) go to 105 jra(k) = irac(jaiak) irac(jaiak) = k 2 ira(k) = iak c c ****** for each column of l and row of u ************************** do 41 k=1,n c c ****** initialize q for computing kth column of l ***************** q(np1) = np1 luk = -1 c ****** by filling in kth column of a ****************************** vj = irac(k) if (vj .eq. 0) go to 5 3 qm = np1 4 m = qm qm = q(m) if (qm .lt. vj) go to 4 if (qm .eq. vj) go to 102 luk = luk + 1 q(m) = vj q(vj) = qm vj = jra(vj) if (vj .ne. 0) go to 3 c ****** link through jru ******************************************* 5 lastid = 0 lasti = 0 ijl(k) = jlptr i = k 6 i = jru(i) if (i .eq. 0) go to 10 qm = np1 jmin = irl(i) jmax = ijl(i) + il(i+1) - il(i) - 1 long = jmax - jmin if (long .lt. 0) go to 6 jtmp = jl(jmin) if (jtmp .ne. k) long = long + 1 if (jtmp .eq. k) r(i) = -r(i) if (lastid .ge. long) go to 7 lasti = i lastid = long c ****** and merge the corresponding columns into the kth column **** 7 do 9 j=jmin,jmax vj = jl(j) 8 m = qm qm = q(m) if (qm .lt. vj) go to 8 if (qm .eq. vj) go to 9 luk = luk + 1 q(m) = vj q(vj) = qm qm = vj 9 continue go to 6 c ****** lasti is the longest column merged into the kth ************ c ****** see if it equals the entire kth column ********************* 10 qm = q(np1) if (qm .ne. k) go to 105 if (luk .eq. 0) go to 17 if (lastid .ne. luk) go to 11 c ****** if so, jl can be compressed ******************************** irll = irl(lasti) ijl(k) = irll + 1 if (jl(irll) .ne. k) ijl(k) = ijl(k) - 1 go to 17 c ****** if not, see if kth column can overlap the previous one ***** 11 if (jlmin .gt. jlptr) go to 15 qm = q(qm) do 12 j=jlmin,jlptr if (jl(j) - qm) 12, 13, 15 12 continue go to 15 13 ijl(k) = j do 14 i=j,jlptr if (jl(i) .ne. qm) go to 15 qm = q(qm) if (qm .gt. n) go to 17 14 continue jlptr = j - 1 c ****** move column indices from q to jl, update vectors *********** 15 jlmin = jlptr + 1 ijl(k) = jlmin if (luk .eq. 0) go to 17 jlptr = jlptr + luk if (jlptr .gt. jlmax) go to 103 qm = q(np1) do 16 j=jlmin,jlptr qm = q(qm) 16 jl(j) = qm 17 irl(k) = ijl(k) il(k+1) = il(k) + luk c c ****** initialize q for computing kth row of u ******************** q(np1) = np1 luk = -1 c ****** by filling in kth row of reordered a *********************** rk = r(k) jmin = ira(k) jmax = ia(rk+1) - 1 if (jmin .gt. jmax) go to 20 do 19 j=jmin,jmax vj = ic(ja(j)) qm = np1 18 m = qm qm = q(m) if (qm .lt. vj) go to 18 if (qm .eq. vj) go to 102 luk = luk + 1 q(m) = vj q(vj) = qm 19 continue c ****** link through jrl, ****************************************** 20 lastid = 0 lasti = 0 iju(k) = juptr i = k i1 = jrl(k) 21 i = i1 if (i .eq. 0) go to 26 i1 = jrl(i) qm = np1 jmin = iru(i) jmax = iju(i) + iu(i+1) - iu(i) - 1 long = jmax - jmin if (long .lt. 0) go to 21 jtmp = ju(jmin) if (jtmp .eq. k) go to 22 c ****** update irl and jrl, ***************************************** long = long + 1 cend = ijl(i) + il(i+1) - il(i) irl(i) = irl(i) + 1 if (irl(i) .ge. cend) go to 22 j = jl(irl(i)) jrl(i) = jrl(j) jrl(j) = i 22 if (lastid .ge. long) go to 23 lasti = i lastid = long c ****** and merge the corresponding rows into the kth row ********** 23 do 25 j=jmin,jmax vj = ju(j) 24 m = qm qm = q(m) if (qm .lt. vj) go to 24 if (qm .eq. vj) go to 25 luk = luk + 1 q(m) = vj q(vj) = qm qm = vj 25 continue go to 21 c ****** update jrl(k) and irl(k) *********************************** 26 if (il(k+1) .le. il(k)) go to 27 j = jl(irl(k)) jrl(k) = jrl(j) jrl(j) = k c ****** lasti is the longest row merged into the kth *************** c ****** see if it equals the entire kth row ************************ 27 qm = q(np1) if (qm .ne. k) go to 105 if (luk .eq. 0) go to 34 if (lastid .ne. luk) go to 28 c ****** if so, ju can be compressed ******************************** irul = iru(lasti) iju(k) = irul + 1 if (ju(irul) .ne. k) iju(k) = iju(k) - 1 go to 34 c ****** if not, see if kth row can overlap the previous one ******** 28 if (jumin .gt. juptr) go to 32 qm = q(qm) do 29 j=jumin,juptr if (ju(j) - qm) 29, 30, 32 29 continue go to 32 30 iju(k) = j do 31 i=j,juptr if (ju(i) .ne. qm) go to 32 qm = q(qm) if (qm .gt. n) go to 34 31 continue juptr = j - 1 c ****** move row indices from q to ju, update vectors ************** 32 jumin = juptr + 1 iju(k) = jumin if (luk .eq. 0) go to 34 juptr = juptr + luk if (juptr .gt. jumax) go to 106 qm = q(np1) do 33 j=jumin,juptr qm = q(qm) 33 ju(j) = qm 34 iru(k) = iju(k) iu(k+1) = iu(k) + luk c c ****** update iru, jru ******************************************** i = k 35 i1 = jru(i) if (r(i) .lt. 0) go to 36 rend = iju(i) + iu(i+1) - iu(i) if (iru(i) .ge. rend) go to 37 j = ju(iru(i)) jru(i) = jru(j) jru(j) = i go to 37 36 r(i) = -r(i) 37 i = i1 if (i .eq. 0) go to 38 iru(i) = iru(i) + 1 go to 35 c c ****** update ira, jra, irac ************************************** 38 i = irac(k) if (i .eq. 0) go to 41 39 i1 = jra(i) ira(i) = ira(i) + 1 if (ira(i) .ge. ia(r(i)+1)) go to 40 irai = ira(i) jairai = ic(ja(irai)) if (jairai .gt. i) go to 40 jra(i) = irac(jairai) irac(jairai) = i 40 i = i1 if (i .ne. 0) go to 39 41 continue c ijl(n) = jlptr iju(n) = juptr flag = 0 return c c ** error.. null row in a 101 flag = n + rk return c ** error.. duplicate entry in a 102 flag = 2*n + rk return c ** error.. insufficient storage for jl 103 flag = 3*n + k return c ** error.. null pivot 105 flag = 5*n + k return c ** error.. insufficient storage for ju 106 flag = 6*n + k return end subroutine nnfc * (n, r,c,ic, ia,ja,a, z, b, * lmax,il,jl,ijl,l, d, umax,iu,ju,iju,u, * row, tmp, irl,jrl, flag) c*** subroutine nnfc c*** numerical ldu-factorization of sparse nonsymmetric matrix and c solution of system of linear equations (compressed pointer c storage) c c c input variables.. n, r, c, ic, ia, ja, a, b, c il, jl, ijl, lmax, iu, ju, iju, umax c output variables.. z, l, d, u, flag c c parameters used internally.. c nia - irl, - vectors used to find the rows of l. at the kth step c nia - jrl of the factorization, jrl(k) points to the head c - of a linked list in jrl of column indices j c - such j .lt. k and l(k,j) is nonzero. zero c - indicates the end of the list. irl(j) (j.lt.k) c - points to the smallest i such that i .ge. k and c - l(i,j) is nonzero. c - size of each = n. c fia - row - holds intermediate values in calculation of u and l. c - size = n. c fia - tmp - holds new right-hand side b* for solution of the c - equation ux = b*. c - size = n. c c internal variables.. c jmin, jmax - indices of the first and last positions in a row to c be examined. c sum - used in calculating tmp. c integer rk,umax integer r(*), c(*), ic(*), ia(*), ja(*), il(*), jl(*), ijl(*) integer iu(*), ju(*), iju(*), irl(*), jrl(*), flag c real a(*), l(*), d(*), u(*), z(*), b(*), row(*) c real tmp(*), lki, sum, dk double precision a(*), l(*), d(*), u(*), z(*), b(*), row(*) double precision tmp(*), lki, sum, dk c c ****** initialize pointers and test storage *********************** if(il(n+1)-1 .gt. lmax) go to 104 if(iu(n+1)-1 .gt. umax) go to 107 do 1 k=1,n irl(k) = il(k) jrl(k) = 0 1 continue c c ****** for each row *********************************************** do 19 k=1,n c ****** reverse jrl and zero row where kth row of l will fill in *** row(k) = 0 i1 = 0 if (jrl(k) .eq. 0) go to 3 i = jrl(k) 2 i2 = jrl(i) jrl(i) = i1 i1 = i row(i) = 0 i = i2 if (i .ne. 0) go to 2 c ****** set row to zero where u will fill in *********************** 3 jmin = iju(k) jmax = jmin + iu(k+1) - iu(k) - 1 if (jmin .gt. jmax) go to 5 do 4 j=jmin,jmax 4 row(ju(j)) = 0 c ****** place kth row of a in row ********************************** 5 rk = r(k) jmin = ia(rk) jmax = ia(rk+1) - 1 do 6 j=jmin,jmax row(ic(ja(j))) = a(j) 6 continue c ****** initialize sum, and link through jrl *********************** sum = b(rk) i = i1 if (i .eq. 0) go to 10 c ****** assign the kth row of l and adjust row, sum **************** 7 lki = -row(i) c ****** if l is not required, then comment out the following line ** l(irl(i)) = -lki sum = sum + lki * tmp(i) jmin = iu(i) jmax = iu(i+1) - 1 if (jmin .gt. jmax) go to 9 mu = iju(i) - jmin do 8 j=jmin,jmax 8 row(ju(mu+j)) = row(ju(mu+j)) + lki * u(j) 9 i = jrl(i) if (i .ne. 0) go to 7 c c ****** assign kth row of u and diagonal d, set tmp(k) ************* 10 if (row(k) .eq. 0.0d0) go to 108 dk = 1.0d0 / row(k) d(k) = dk tmp(k) = sum * dk if (k .eq. n) go to 19 jmin = iu(k) jmax = iu(k+1) - 1 if (jmin .gt. jmax) go to 12 mu = iju(k) - jmin do 11 j=jmin,jmax 11 u(j) = row(ju(mu+j)) * dk 12 continue c c ****** update irl and jrl, keeping jrl in decreasing order ******** i = i1 if (i .eq. 0) go to 18 14 irl(i) = irl(i) + 1 i1 = jrl(i) if (irl(i) .ge. il(i+1)) go to 17 ijlb = irl(i) - il(i) + ijl(i) j = jl(ijlb) 15 if (i .gt. jrl(j)) go to 16 j = jrl(j) go to 15 16 jrl(i) = jrl(j) jrl(j) = i 17 i = i1 if (i .ne. 0) go to 14 18 if (irl(k) .ge. il(k+1)) go to 19 j = jl(ijl(k)) jrl(k) = jrl(j) jrl(j) = k 19 continue c c ****** solve ux = tmp by back substitution ********************** k = n do 22 i=1,n sum = tmp(k) jmin = iu(k) jmax = iu(k+1) - 1 if (jmin .gt. jmax) go to 21 mu = iju(k) - jmin do 20 j=jmin,jmax 20 sum = sum - u(j) * tmp(ju(mu+j)) 21 tmp(k) = sum z(c(k)) = sum 22 k = k-1 flag = 0 return c c ** error.. insufficient storage for l 104 flag = 4*n + 1 return c ** error.. insufficient storage for u 107 flag = 7*n + 1 return c ** error.. zero pivot 108 flag = 8*n + k return end subroutine nnsc * (n, r, c, il, jl, ijl, l, d, iu, ju, iju, u, z, b, tmp) c*** subroutine nnsc c*** numerical solution of sparse nonsymmetric system of linear c equations given ldu-factorization (compressed pointer storage) c c c input variables.. n, r, c, il, jl, ijl, l, d, iu, ju, iju, u, b c output variables.. z c c parameters used internally.. c fia - tmp - temporary vector which gets result of solving ly = b. c - size = n. c c internal variables.. c jmin, jmax - indices of the first and last positions in a row of c u or l to be used. c integer r(*), c(*), il(*), jl(*), ijl(*), iu(*), ju(*), iju(*) c real l(*), d(*), u(*), b(*), z(*), tmp(*), tmpk, sum double precision l(*), d(*), u(*), b(*), z(*), tmp(*), tmpk,sum c c ****** set tmp to reordered b ************************************* do 1 k=1,n 1 tmp(k) = b(r(k)) c ****** solve ly = b by forward substitution ********************* do 3 k=1,n jmin = il(k) jmax = il(k+1) - 1 tmpk = -d(k) * tmp(k) tmp(k) = -tmpk if (jmin .gt. jmax) go to 3 ml = ijl(k) - jmin do 2 j=jmin,jmax 2 tmp(jl(ml+j)) = tmp(jl(ml+j)) + tmpk * l(j) 3 continue c ****** solve ux = y by back substitution ************************ k = n do 6 i=1,n sum = -tmp(k) jmin = iu(k) jmax = iu(k+1) - 1 if (jmin .gt. jmax) go to 5 mu = iju(k) - jmin do 4 j=jmin,jmax 4 sum = sum + u(j) * tmp(ju(mu+j)) 5 tmp(k) = -sum z(c(k)) = -sum k = k - 1 6 continue return end subroutine nntc * (n, r, c, il, jl, ijl, l, d, iu, ju, iju, u, z, b, tmp) c*** subroutine nntc c*** numeric solution of the transpose of a sparse nonsymmetric system c of linear equations given lu-factorization (compressed pointer c storage) c c c input variables.. n, r, c, il, jl, ijl, l, d, iu, ju, iju, u, b c output variables.. z c c parameters used internally.. c fia - tmp - temporary vector which gets result of solving ut y = b c - size = n. c c internal variables.. c jmin, jmax - indices of the first and last positions in a row of c u or l to be used. c integer r(*), c(*), il(*), jl(*), ijl(*), iu(*), ju(*), iju(*) c real l(*), d(*), u(*), b(*), z(*), tmp(*), tmpk,sum double precision l(*), d(*), u(*), b(*), z(*), tmp(*), tmpk,sum c c ****** set tmp to reordered b ************************************* do 1 k=1,n 1 tmp(k) = b(c(k)) c ****** solve ut y = b by forward substitution ******************* do 3 k=1,n jmin = iu(k) jmax = iu(k+1) - 1 tmpk = -tmp(k) if (jmin .gt. jmax) go to 3 mu = iju(k) - jmin do 2 j=jmin,jmax 2 tmp(ju(mu+j)) = tmp(ju(mu+j)) + tmpk * u(j) 3 continue c ****** solve lt x = y by back substitution ********************** k = n do 6 i=1,n sum = -tmp(k) jmin = il(k) jmax = il(k+1) - 1 if (jmin .gt. jmax) go to 5 ml = ijl(k) - jmin do 4 j=jmin,jmax 4 sum = sum + l(j) * tmp(jl(ml+j)) 5 tmp(k) = -sum * d(k) z(r(k)) = tmp(k) k = k - 1 6 continue return end *DECK DSTODA SUBROUTINE DSTODA (NEQ, Y, YH, NYH, YH1, EWT, SAVF, ACOR, 1 WM, IWM, F, JAC, PJAC, SLVS) EXTERNAL F, JAC, PJAC, SLVS INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, YH1, EWT, SAVF, ACOR, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*), 1 ACOR(*), WM(*), IWM(*) INTEGER IOWND, IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IOWND2, ICOUNT, IRFLAG, JTYP, MUSED, MXORDN, MXORDS DOUBLE PRECISION CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO, 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION ROWND2, CM1, CM2, PDEST, PDLAST, RATIO, 1 PDNORM COMMON /DLS001/ CONIT, CRATE, EL(13), ELCO(13,12), 1 HOLD, RMAX, TESCO(3,12), 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 3 IOWND(6), IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 4 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 5 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 6 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSA01/ ROWND2, CM1(12), CM2(5), PDEST, PDLAST, RATIO, 1 PDNORM, 2 IOWND2(3), ICOUNT, IRFLAG, JTYP, MUSED, MXORDN, MXORDS INTEGER I, I1, IREDO, IRET, J, JB, M, NCF, NEWQ INTEGER LM1, LM1P1, LM2, LM2P1, NQM1, NQM2 DOUBLE PRECISION DCON, DDN, DEL, DELP, DSM, DUP, EXDN, EXSM, EXUP, 1 R, RH, RHDN, RHSM, RHUP, TOLD, DMNORM DOUBLE PRECISION ALPHA, DM1,DM2, EXM1,EXM2, 1 PDH, PNORM, RATE, RH1, RH1IT, RH2, RM, SM1(12) SAVE SM1 DATA SM1/0.5D0, 0.575D0, 0.55D0, 0.45D0, 0.35D0, 0.25D0, 1 0.20D0, 0.15D0, 0.10D0, 0.075D0, 0.050D0, 0.025D0/ C----------------------------------------------------------------------- C DSTODA performs one step of the integration of an initial value C problem for a system of ordinary differential equations. C Note: DSTODA is independent of the value of the iteration method C indicator MITER, when this is .ne. 0, and hence is independent C of the type of chord method used, or the Jacobian structure. C Communication with DSTODA is done with the following variables: C C Y = an array of length .ge. N used as the Y argument in C all calls to F and JAC. C NEQ = integer array containing problem size in NEQ(1), and C passed as the NEQ argument in all calls to F and JAC. C YH = an NYH by LMAX array containing the dependent variables C and their approximate scaled derivatives, where C LMAX = MAXORD + 1. YH(i,j+1) contains the approximate C j-th derivative of y(i), scaled by H**j/factorial(j) C (j = 0,1,...,NQ). On entry for the first step, the first C two columns of YH must be set from the initial values. C NYH = a constant integer .ge. N, the first dimension of YH. C YH1 = a one-dimensional array occupying the same space as YH. C EWT = an array of length N containing multiplicative weights C for local error measurements. Local errors in y(i) are C compared to 1.0/EWT(i) in various error tests. C SAVF = an array of working storage, of length N. C ACOR = a work array of length N, used for the accumulated C corrections. On a successful return, ACOR(i) contains C the estimated one-step local error in y(i). C WM,IWM = real and integer work arrays associated with matrix C operations in chord iteration (MITER .ne. 0). C PJAC = name of routine to evaluate and preprocess Jacobian matrix C and P = I - H*EL0*Jac, if a chord method is being used. C It also returns an estimate of norm(Jac) in PDNORM. C SLVS = name of routine to solve linear system in chord iteration. C CCMAX = maximum relative change in H*EL0 before PJAC is called. C H = the step size to be attempted on the next step. C H is altered by the error control algorithm during the C problem. H can be either positive or negative, but its C sign must remain constant throughout the problem. C HMIN = the minimum absolute value of the step size H to be used. C HMXI = inverse of the maximum absolute value of H to be used. C HMXI = 0.0 is allowed and corresponds to an infinite HMAX. C HMIN and HMXI may be changed at any time, but will not C take effect until the next change of H is considered. C TN = the independent variable. TN is updated on each step taken. C JSTART = an integer used for input only, with the following C values and meanings: C 0 perform the first step. C .gt.0 take a new step continuing from the last. C -1 take the next step with a new value of H, C N, METH, MITER, and/or matrix parameters. C -2 take the next step with a new value of H, C but with other inputs unchanged. C On return, JSTART is set to 1 to facilitate continuation. C KFLAG = a completion code with the following meanings: C 0 the step was succesful. C -1 the requested error could not be achieved. C -2 corrector convergence could not be achieved. C -3 fatal error in PJAC or SLVS. C A return with KFLAG = -1 or -2 means either C ABS(H) = HMIN or 10 consecutive failures occurred. C On a return with KFLAG negative, the values of TN and C the YH array are as of the beginning of the last C step, and H is the last step size attempted. C MAXORD = the maximum order of integration method to be allowed. C MAXCOR = the maximum number of corrector iterations allowed. C MSBP = maximum number of steps between PJAC calls (MITER .gt. 0). C MXNCF = maximum number of convergence failures allowed. C METH = current method. C METH = 1 means Adams method (nonstiff) C METH = 2 means BDF method (stiff) C METH may be reset by DSTODA. C MITER = corrector iteration method. C MITER = 0 means functional iteration. C MITER = JT .gt. 0 means a chord iteration corresponding C to Jacobian type JT. (The DLSODA/DLSODAR argument JT is C communicated here as JTYP, but is not used in DSTODA C except to load MITER following a method switch.) C MITER may be reset by DSTODA. C N = the number of first-order differential equations. C----------------------------------------------------------------------- KFLAG = 0 TOLD = TN NCF = 0 IERPJ = 0 IERSL = 0 JCUR = 0 ICF = 0 DELP = 0.0D0 IF (JSTART .GT. 0) GO TO 200 IF (JSTART .EQ. -1) GO TO 100 IF (JSTART .EQ. -2) GO TO 160 C----------------------------------------------------------------------- C On the first call, the order is set to 1, and other variables are C initialized. RMAX is the maximum ratio by which H can be increased C in a single step. It is initially 1.E4 to compensate for the small C initial H, but then is normally equal to 10. If a failure C occurs (in corrector convergence or error test), RMAX is set at 2 C for the next increase. C DCFODE is called to get the needed coefficients for both methods. C----------------------------------------------------------------------- LMAX = MAXORD + 1 NQ = 1 L = 2 IALTH = 2 RMAX = 10000.0D0 RC = 0.0D0 EL0 = 1.0D0 CRATE = 0.7D0 HOLD = H NSLP = 0 IPUP = MITER IRET = 3 C Initialize switching parameters. METH = 1 is assumed initially. ----- ICOUNT = 20 IRFLAG = 0 PDEST = 0.0D0 PDLAST = 0.0D0 RATIO = 5.0D0 CALL DCFODE (2, ELCO, TESCO) DO 10 I = 1,5 10 CM2(I) = TESCO(2,I)*ELCO(I+1,I) CALL DCFODE (1, ELCO, TESCO) DO 20 I = 1,12 20 CM1(I) = TESCO(2,I)*ELCO(I+1,I) GO TO 150 C----------------------------------------------------------------------- C The following block handles preliminaries needed when JSTART = -1. C IPUP is set to MITER to force a matrix update. C If an order increase is about to be considered (IALTH = 1), C IALTH is reset to 2 to postpone consideration one more step. C If the caller has changed METH, DCFODE is called to reset C the coefficients of the method. C If H is to be changed, YH must be rescaled. C If H or METH is being changed, IALTH is reset to L = NQ + 1 C to prevent further changes in H for that many steps. C----------------------------------------------------------------------- 100 IPUP = MITER LMAX = MAXORD + 1 IF (IALTH .EQ. 1) IALTH = 2 IF (METH .EQ. MUSED) GO TO 160 CALL DCFODE (METH, ELCO, TESCO) IALTH = L IRET = 1 C----------------------------------------------------------------------- C The el vector and related constants are reset C whenever the order NQ is changed, or at the start of the problem. C----------------------------------------------------------------------- 150 DO 155 I = 1,L 155 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) GO TO (160, 170, 200), IRET C----------------------------------------------------------------------- C If H is being changed, the H ratio RH is checked against C RMAX, HMIN, and HMXI, and the YH array rescaled. IALTH is set to C L = NQ + 1 to prevent a change of H for that many steps, unless C forced by a convergence or error test failure. C----------------------------------------------------------------------- 160 IF (H .EQ. HOLD) GO TO 200 RH = H/HOLD H = HOLD IREDO = 3 GO TO 175 170 RH = MAX(RH,HMIN/ABS(H)) 175 RH = MIN(RH,RMAX) RH = RH/MAX(1.0D0,ABS(H)*HMXI*RH) C----------------------------------------------------------------------- C If METH = 1, also restrict the new step size by the stability region. C If this reduces H, set IRFLAG to 1 so that if there are roundoff C problems later, we can assume that is the cause of the trouble. C----------------------------------------------------------------------- IF (METH .EQ. 2) GO TO 178 IRFLAG = 0 PDH = MAX(ABS(H)*PDLAST,0.000001D0) IF (RH*PDH*1.00001D0 .LT. SM1(NQ)) GO TO 178 RH = SM1(NQ)/PDH IRFLAG = 1 178 CONTINUE R = 1.0D0 DO 180 J = 2,L R = R*RH DO 180 I = 1,N 180 YH(I,J) = YH(I,J)*R H = H*RH RC = RC*RH IALTH = L IF (IREDO .EQ. 0) GO TO 690 C----------------------------------------------------------------------- C This section computes the predicted values by effectively C multiplying the YH array by the Pascal triangle matrix. C RC is the ratio of new to old values of the coefficient H*EL(1). C When RC differs from 1 by more than CCMAX, IPUP is set to MITER C to force PJAC to be called, if a Jacobian is involved. C In any case, PJAC is called at least every MSBP steps. C----------------------------------------------------------------------- 200 IF (ABS(RC-1.0D0) .GT. CCMAX) IPUP = MITER IF (NST .GE. NSLP+MSBP) IPUP = MITER TN = TN + H I1 = NQNYH + 1 DO 215 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 210 I = I1,NQNYH 210 YH1(I) = YH1(I) + YH1(I+NYH) 215 CONTINUE PNORM = DMNORM (N, YH1, EWT) C----------------------------------------------------------------------- C Up to MAXCOR corrector iterations are taken. A convergence test is C made on the RMS-norm of each correction, weighted by the error C weight vector EWT. The sum of the corrections is accumulated in the C vector ACOR(i). The YH array is not altered in the corrector loop. C----------------------------------------------------------------------- 220 M = 0 RATE = 0.0D0 DEL = 0.0D0 DO 230 I = 1,N 230 Y(I) = YH(I,1) CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 IF (IPUP .LE. 0) GO TO 250 C----------------------------------------------------------------------- C If indicated, the matrix P = I - H*EL(1)*J is reevaluated and C preprocessed before starting the corrector iteration. IPUP is set C to 0 as an indicator that this has been done. C----------------------------------------------------------------------- CALL PJAC (NEQ, Y, YH, NYH, EWT, ACOR, SAVF, WM, IWM, F, JAC) IPUP = 0 RC = 1.0D0 NSLP = NST CRATE = 0.7D0 IF (IERPJ .NE. 0) GO TO 430 250 DO 260 I = 1,N 260 ACOR(I) = 0.0D0 270 IF (MITER .NE. 0) GO TO 350 C----------------------------------------------------------------------- C In the case of functional iteration, update Y directly from C the result of the last function evaluation. C----------------------------------------------------------------------- DO 290 I = 1,N SAVF(I) = H*SAVF(I) - YH(I,2) 290 Y(I) = SAVF(I) - ACOR(I) DEL = DMNORM (N, Y, EWT) DO 300 I = 1,N Y(I) = YH(I,1) + EL(1)*SAVF(I) 300 ACOR(I) = SAVF(I) GO TO 400 C----------------------------------------------------------------------- C In the case of the chord method, compute the corrector error, C and solve the linear system with that as right-hand side and C P as coefficient matrix. C----------------------------------------------------------------------- 350 DO 360 I = 1,N 360 Y(I) = H*SAVF(I) - (YH(I,2) + ACOR(I)) CALL SLVS (WM, IWM, Y, SAVF) IF (IERSL .LT. 0) GO TO 430 IF (IERSL .GT. 0) GO TO 410 DEL = DMNORM (N, Y, EWT) DO 380 I = 1,N ACOR(I) = ACOR(I) + Y(I) 380 Y(I) = YH(I,1) + EL(1)*ACOR(I) C----------------------------------------------------------------------- C Test for convergence. If M .gt. 0, an estimate of the convergence C rate constant is stored in CRATE, and this is used in the test. C C We first check for a change of iterates that is the size of C roundoff error. If this occurs, the iteration has converged, and a C new rate estimate is not formed. C In all other cases, force at least two iterations to estimate a C local Lipschitz constant estimate for Adams methods. C On convergence, form PDEST = local maximum Lipschitz constant C estimate. PDLAST is the most recent nonzero estimate. C----------------------------------------------------------------------- 400 CONTINUE IF (DEL .LE. 100.0D0*PNORM*UROUND) GO TO 450 IF (M .EQ. 0 .AND. METH .EQ. 1) GO TO 405 IF (M .EQ. 0) GO TO 402 RM = 1024.0D0 IF (DEL .LE. 1024.0D0*DELP) RM = DEL/DELP RATE = MAX(RATE,RM) CRATE = MAX(0.2D0*CRATE,RM) 402 DCON = DEL*MIN(1.0D0,1.5D0*CRATE)/(TESCO(2,NQ)*CONIT) IF (DCON .GT. 1.0D0) GO TO 405 PDEST = MAX(PDEST,RATE/ABS(H*EL(1))) IF (PDEST .NE. 0.0D0) PDLAST = PDEST GO TO 450 405 CONTINUE M = M + 1 IF (M .EQ. MAXCOR) GO TO 410 IF (M .GE. 2 .AND. DEL .GT. 2.0D0*DELP) GO TO 410 DELP = DEL CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 GO TO 270 C----------------------------------------------------------------------- C The corrector iteration failed to converge. C If MITER .ne. 0 and the Jacobian is out of date, PJAC is called for C the next try. Otherwise the YH array is retracted to its values C before prediction, and H is reduced, if possible. If H cannot be C reduced or MXNCF failures have occurred, exit with KFLAG = -2. C----------------------------------------------------------------------- 410 IF (MITER .EQ. 0 .OR. JCUR .EQ. 1) GO TO 430 ICF = 1 IPUP = MITER GO TO 220 430 ICF = 2 NCF = NCF + 1 RMAX = 2.0D0 TN = TOLD I1 = NQNYH + 1 DO 445 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 440 I = I1,NQNYH 440 YH1(I) = YH1(I) - YH1(I+NYH) 445 CONTINUE IF (IERPJ .LT. 0 .OR. IERSL .LT. 0) GO TO 680 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 670 IF (NCF .EQ. MXNCF) GO TO 670 RH = 0.25D0 IPUP = MITER IREDO = 1 GO TO 170 C----------------------------------------------------------------------- C The corrector has converged. JCUR is set to 0 C to signal that the Jacobian involved may need updating later. C The local error test is made and control passes to statement 500 C if it fails. C----------------------------------------------------------------------- 450 JCUR = 0 IF (M .EQ. 0) DSM = DEL/TESCO(2,NQ) IF (M .GT. 0) DSM = DMNORM (N, ACOR, EWT)/TESCO(2,NQ) IF (DSM .GT. 1.0D0) GO TO 500 C----------------------------------------------------------------------- C After a successful step, update the YH array. C Decrease ICOUNT by 1, and if it is -1, consider switching methods. C If a method switch is made, reset various parameters, C rescale the YH array, and exit. If there is no switch, C consider changing H if IALTH = 1. Otherwise decrease IALTH by 1. C If IALTH is then 1 and NQ .lt. MAXORD, then ACOR is saved for C use in a possible order increase on the next step. C If a change in H is considered, an increase or decrease in order C by one is considered also. A change in H is made only if it is by a C factor of at least 1.1. If not, IALTH is set to 3 to prevent C testing for that many steps. C----------------------------------------------------------------------- KFLAG = 0 IREDO = 0 NST = NST + 1 HU = H NQU = NQ MUSED = METH DO 460 J = 1,L DO 460 I = 1,N 460 YH(I,J) = YH(I,J) + EL(J)*ACOR(I) ICOUNT = ICOUNT - 1 IF (ICOUNT .GE. 0) GO TO 488 IF (METH .EQ. 2) GO TO 480 C----------------------------------------------------------------------- C We are currently using an Adams method. Consider switching to BDF. C If the current order is greater than 5, assume the problem is C not stiff, and skip this section. C If the Lipschitz constant and error estimate are not polluted C by roundoff, go to 470 and perform the usual test. C Otherwise, switch to the BDF methods if the last step was C restricted to insure stability (irflag = 1), and stay with Adams C method if not. When switching to BDF with polluted error estimates, C in the absence of other information, double the step size. C C When the estimates are OK, we make the usual test by computing C the step size we could have (ideally) used on this step, C with the current (Adams) method, and also that for the BDF. C If NQ .gt. MXORDS, we consider changing to order MXORDS on switching. C Compare the two step sizes to decide whether to switch. C The step size advantage must be at least RATIO = 5 to switch. C----------------------------------------------------------------------- IF (NQ .GT. 5) GO TO 488 IF (DSM .GT. 100.0D0*PNORM*UROUND .AND. PDEST .NE. 0.0D0) 1 GO TO 470 IF (IRFLAG .EQ. 0) GO TO 488 RH2 = 2.0D0 NQM2 = MIN(NQ,MXORDS) GO TO 478 470 CONTINUE EXSM = 1.0D0/L RH1 = 1.0D0/(1.2D0*DSM**EXSM + 0.0000012D0) RH1IT = 2.0D0*RH1 PDH = PDLAST*ABS(H) IF (PDH*RH1 .GT. 0.00001D0) RH1IT = SM1(NQ)/PDH RH1 = MIN(RH1,RH1IT) IF (NQ .LE. MXORDS) GO TO 474 NQM2 = MXORDS LM2 = MXORDS + 1 EXM2 = 1.0D0/LM2 LM2P1 = LM2 + 1 DM2 = DMNORM (N, YH(1,LM2P1), EWT)/CM2(MXORDS) RH2 = 1.0D0/(1.2D0*DM2**EXM2 + 0.0000012D0) GO TO 476 474 DM2 = DSM*(CM1(NQ)/CM2(NQ)) RH2 = 1.0D0/(1.2D0*DM2**EXSM + 0.0000012D0) NQM2 = NQ 476 CONTINUE IF (RH2 .LT. RATIO*RH1) GO TO 488 C THE SWITCH TEST PASSED. RESET RELEVANT QUANTITIES FOR BDF. ---------- 478 RH = RH2 ICOUNT = 20 METH = 2 MITER = JTYP PDLAST = 0.0D0 NQ = NQM2 L = NQ + 1 GO TO 170 C----------------------------------------------------------------------- C We are currently using a BDF method. Consider switching to Adams. C Compute the step size we could have (ideally) used on this step, C with the current (BDF) method, and also that for the Adams. C If NQ .gt. MXORDN, we consider changing to order MXORDN on switching. C Compare the two step sizes to decide whether to switch. C The step size advantage must be at least 5/RATIO = 1 to switch. C If the step size for Adams would be so small as to cause C roundoff pollution, we stay with BDF. C----------------------------------------------------------------------- 480 CONTINUE EXSM = 1.0D0/L IF (MXORDN .GE. NQ) GO TO 484 NQM1 = MXORDN LM1 = MXORDN + 1 EXM1 = 1.0D0/LM1 LM1P1 = LM1 + 1 DM1 = DMNORM (N, YH(1,LM1P1), EWT)/CM1(MXORDN) RH1 = 1.0D0/(1.2D0*DM1**EXM1 + 0.0000012D0) GO TO 486 484 DM1 = DSM*(CM2(NQ)/CM1(NQ)) RH1 = 1.0D0/(1.2D0*DM1**EXSM + 0.0000012D0) NQM1 = NQ EXM1 = EXSM 486 RH1IT = 2.0D0*RH1 PDH = PDNORM*ABS(H) IF (PDH*RH1 .GT. 0.00001D0) RH1IT = SM1(NQM1)/PDH RH1 = MIN(RH1,RH1IT) RH2 = 1.0D0/(1.2D0*DSM**EXSM + 0.0000012D0) IF (RH1*RATIO .LT. 5.0D0*RH2) GO TO 488 ALPHA = MAX(0.001D0,RH1) DM1 = (ALPHA**EXM1)*DM1 IF (DM1 .LE. 1000.0D0*UROUND*PNORM) GO TO 488 C The switch test passed. Reset relevant quantities for Adams. -------- RH = RH1 ICOUNT = 20 METH = 1 MITER = 0 PDLAST = 0.0D0 NQ = NQM1 L = NQ + 1 GO TO 170 C C No method switch is being made. Do the usual step/order selection. -- 488 CONTINUE IALTH = IALTH - 1 IF (IALTH .EQ. 0) GO TO 520 IF (IALTH .GT. 1) GO TO 700 IF (L .EQ. LMAX) GO TO 700 DO 490 I = 1,N 490 YH(I,LMAX) = ACOR(I) GO TO 700 C----------------------------------------------------------------------- C The error test failed. KFLAG keeps track of multiple failures. C Restore TN and the YH array to their previous values, and prepare C to try the step again. Compute the optimum step size for this or C one lower order. After 2 or more failures, H is forced to decrease C by a factor of 0.2 or less. C----------------------------------------------------------------------- 500 KFLAG = KFLAG - 1 TN = TOLD I1 = NQNYH + 1 DO 515 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 510 I = I1,NQNYH 510 YH1(I) = YH1(I) - YH1(I+NYH) 515 CONTINUE RMAX = 2.0D0 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 660 IF (KFLAG .LE. -3) GO TO 640 IREDO = 2 RHUP = 0.0D0 GO TO 540 C----------------------------------------------------------------------- C Regardless of the success or failure of the step, factors C RHDN, RHSM, and RHUP are computed, by which H could be multiplied C at order NQ - 1, order NQ, or order NQ + 1, respectively. C In the case of failure, RHUP = 0.0 to avoid an order increase. C The largest of these is determined and the new order chosen C accordingly. If the order is to be increased, we compute one C additional scaled derivative. C----------------------------------------------------------------------- 520 RHUP = 0.0D0 IF (L .EQ. LMAX) GO TO 540 DO 530 I = 1,N 530 SAVF(I) = ACOR(I) - YH(I,LMAX) DUP = DMNORM (N, SAVF, EWT)/TESCO(3,NQ) EXUP = 1.0D0/(L+1) RHUP = 1.0D0/(1.4D0*DUP**EXUP + 0.0000014D0) 540 EXSM = 1.0D0/L RHSM = 1.0D0/(1.2D0*DSM**EXSM + 0.0000012D0) RHDN = 0.0D0 IF (NQ .EQ. 1) GO TO 550 DDN = DMNORM (N, YH(1,L), EWT)/TESCO(1,NQ) EXDN = 1.0D0/NQ RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) C If METH = 1, limit RH according to the stability region also. -------- 550 IF (METH .EQ. 2) GO TO 560 PDH = MAX(ABS(H)*PDLAST,0.000001D0) IF (L .LT. LMAX) RHUP = MIN(RHUP,SM1(L)/PDH) RHSM = MIN(RHSM,SM1(NQ)/PDH) IF (NQ .GT. 1) RHDN = MIN(RHDN,SM1(NQ-1)/PDH) PDEST = 0.0D0 560 IF (RHSM .GE. RHUP) GO TO 570 IF (RHUP .GT. RHDN) GO TO 590 GO TO 580 570 IF (RHSM .LT. RHDN) GO TO 580 NEWQ = NQ RH = RHSM GO TO 620 580 NEWQ = NQ - 1 RH = RHDN IF (KFLAG .LT. 0 .AND. RH .GT. 1.0D0) RH = 1.0D0 GO TO 620 590 NEWQ = L RH = RHUP IF (RH .LT. 1.1D0) GO TO 610 R = EL(L)/L DO 600 I = 1,N 600 YH(I,NEWQ+1) = ACOR(I)*R GO TO 630 610 IALTH = 3 GO TO 700 C If METH = 1 and H is restricted by stability, bypass 10 percent test. 620 IF (METH .EQ. 2) GO TO 622 IF (RH*PDH*1.00001D0 .GE. SM1(NEWQ)) GO TO 625 622 IF (KFLAG .EQ. 0 .AND. RH .LT. 1.1D0) GO TO 610 625 IF (KFLAG .LE. -2) RH = MIN(RH,0.2D0) C----------------------------------------------------------------------- C If there is a change of order, reset NQ, L, and the coefficients. C In any case H is reset according to RH and the YH array is rescaled. C Then exit from 690 if the step was OK, or redo the step otherwise. C----------------------------------------------------------------------- IF (NEWQ .EQ. NQ) GO TO 170 630 NQ = NEWQ L = NQ + 1 IRET = 2 GO TO 150 C----------------------------------------------------------------------- C Control reaches this section if 3 or more failures have occured. C If 10 failures have occurred, exit with KFLAG = -1. C It is assumed that the derivatives that have accumulated in the C YH array have errors of the wrong order. Hence the first C derivative is recomputed, and the order is set to 1. Then C H is reduced by a factor of 10, and the step is retried, C until it succeeds or H reaches HMIN. C----------------------------------------------------------------------- 640 IF (KFLAG .EQ. -10) GO TO 660 RH = 0.1D0 RH = MAX(HMIN/ABS(H),RH) H = H*RH DO 645 I = 1,N 645 Y(I) = YH(I,1) CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 DO 650 I = 1,N 650 YH(I,2) = H*SAVF(I) IPUP = MITER IALTH = 5 IF (NQ .EQ. 1) GO TO 200 NQ = 1 L = 2 IRET = 3 GO TO 150 C----------------------------------------------------------------------- C All returns are made through this section. H is saved in HOLD C to allow the caller to change H on the next step. C----------------------------------------------------------------------- 660 KFLAG = -1 GO TO 720 670 KFLAG = -2 GO TO 720 680 KFLAG = -3 GO TO 720 690 RMAX = 10.0D0 700 R = 1.0D0/TESCO(2,NQU) DO 710 I = 1,N 710 ACOR(I) = ACOR(I)*R 720 HOLD = H JSTART = 1 RETURN C----------------------- End of Subroutine DSTODA ---------------------- END *DECK DPRJA SUBROUTINE DPRJA (NEQ, Y, YH, NYH, EWT, FTEM, SAVF, WM, IWM, 1 F, JAC) EXTERNAL F, JAC INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, EWT, FTEM, SAVF, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), FTEM(*), SAVF(*), 1 WM(*), IWM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IOWND2, IOWNS2, JTYP, MUSED, MXORDN, MXORDS DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION ROWND2, ROWNS2, PDNORM COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSA01/ ROWND2, ROWNS2(20), PDNORM, 1 IOWND2(3), IOWNS2(2), JTYP, MUSED, MXORDN, MXORDS INTEGER I, I1, I2, IER, II, J, J1, JJ, LENP, 1 MBA, MBAND, MEB1, MEBAND, ML, ML3, MU, NP1 DOUBLE PRECISION CON, FAC, HL0, R, R0, SRUR, YI, YJ, YJJ, 1 DMNORM, DFNORM, DBNORM C----------------------------------------------------------------------- C DPRJA is called by DSTODA to compute and process the matrix C P = I - H*EL(1)*J , where J is an approximation to the Jacobian. C Here J is computed by the user-supplied routine JAC if C MITER = 1 or 4 or by finite differencing if MITER = 2 or 5. C J, scaled by -H*EL(1), is stored in WM. Then the norm of J (the C matrix norm consistent with the weighted max-norm on vectors given C by DMNORM) is computed, and J is overwritten by P. P is then C subjected to LU decomposition in preparation for later solution C of linear systems with P as coefficient matrix. This is done C by DGEFA if MITER = 1 or 2, and by DGBFA if MITER = 4 or 5. C C In addition to variables described previously, communication C with DPRJA uses the following: C Y = array containing predicted values on entry. C FTEM = work array of length N (ACOR in DSTODA). C SAVF = array containing f evaluated at predicted y. C WM = real work space for matrices. On output it contains the C LU decomposition of P. C Storage of matrix elements starts at WM(3). C WM also contains the following matrix-related data: C WM(1) = SQRT(UROUND), used in numerical Jacobian increments. C IWM = integer work space containing pivot information, starting at C IWM(21). IWM also contains the band parameters C ML = IWM(1) and MU = IWM(2) if MITER is 4 or 5. C EL0 = EL(1) (input). C PDNORM= norm of Jacobian matrix. (Output). C IERPJ = output error flag, = 0 if no trouble, .gt. 0 if C P matrix found to be singular. C JCUR = output flag = 1 to indicate that the Jacobian matrix C (or approximation) is now current. C This routine also uses the Common variables EL0, H, TN, UROUND, C MITER, N, NFE, and NJE. C----------------------------------------------------------------------- NJE = NJE + 1 IERPJ = 0 JCUR = 1 HL0 = H*EL0 GO TO (100, 200, 300, 400, 500), MITER C If MITER = 1, call JAC and multiply by scalar. ----------------------- 100 LENP = N*N DO 110 I = 1,LENP 110 WM(I+2) = 0.0D0 CALL JAC (NEQ, TN, Y, 0, 0, WM(3), N) CON = -HL0 DO 120 I = 1,LENP 120 WM(I+2) = WM(I+2)*CON GO TO 240 C If MITER = 2, make N calls to F to approximate J. -------------------- 200 FAC = DMNORM (N, SAVF, EWT) R0 = 1000.0D0*ABS(H)*UROUND*N*FAC IF (R0 .EQ. 0.0D0) R0 = 1.0D0 SRUR = WM(1) J1 = 2 DO 230 J = 1,N YJ = Y(J) R = MAX(SRUR*ABS(YJ),R0/EWT(J)) Y(J) = Y(J) + R FAC = -HL0/R CALL F (NEQ, TN, Y, FTEM) DO 220 I = 1,N 220 WM(I+J1) = (FTEM(I) - SAVF(I))*FAC Y(J) = YJ J1 = J1 + N 230 CONTINUE NFE = NFE + N 240 CONTINUE C Compute norm of Jacobian. -------------------------------------------- PDNORM = DFNORM (N, WM(3), EWT)/ABS(HL0) C Add identity matrix. ------------------------------------------------- J = 3 NP1 = N + 1 DO 250 I = 1,N WM(J) = WM(J) + 1.0D0 250 J = J + NP1 C Do LU decomposition on P. -------------------------------------------- CALL DGEFA (WM(3), N, N, IWM(21), IER) IF (IER .NE. 0) IERPJ = 1 RETURN C Dummy block only, since MITER is never 3 in this routine. ------------ 300 RETURN C If MITER = 4, call JAC and multiply by scalar. ----------------------- 400 ML = IWM(1) MU = IWM(2) ML3 = ML + 3 MBAND = ML + MU + 1 MEBAND = MBAND + ML LENP = MEBAND*N DO 410 I = 1,LENP 410 WM(I+2) = 0.0D0 CALL JAC (NEQ, TN, Y, ML, MU, WM(ML3), MEBAND) CON = -HL0 DO 420 I = 1,LENP 420 WM(I+2) = WM(I+2)*CON GO TO 570 C If MITER = 5, make MBAND calls to F to approximate J. ---------------- 500 ML = IWM(1) MU = IWM(2) MBAND = ML + MU + 1 MBA = MIN(MBAND,N) MEBAND = MBAND + ML MEB1 = MEBAND - 1 SRUR = WM(1) FAC = DMNORM (N, SAVF, EWT) R0 = 1000.0D0*ABS(H)*UROUND*N*FAC IF (R0 .EQ. 0.0D0) R0 = 1.0D0 DO 560 J = 1,MBA DO 530 I = J,N,MBAND YI = Y(I) R = MAX(SRUR*ABS(YI),R0/EWT(I)) 530 Y(I) = Y(I) + R CALL F (NEQ, TN, Y, FTEM) DO 550 JJ = J,N,MBAND Y(JJ) = YH(JJ,1) YJJ = Y(JJ) R = MAX(SRUR*ABS(YJJ),R0/EWT(JJ)) FAC = -HL0/R I1 = MAX(JJ-MU,1) I2 = MIN(JJ+ML,N) II = JJ*MEB1 - ML + 2 DO 540 I = I1,I2 540 WM(II+I) = (FTEM(I) - SAVF(I))*FAC 550 CONTINUE 560 CONTINUE NFE = NFE + MBA 570 CONTINUE C Compute norm of Jacobian. -------------------------------------------- PDNORM = DBNORM (N, WM(ML+3), MEBAND, ML, MU, EWT)/ABS(HL0) C Add identity matrix. ------------------------------------------------- II = MBAND + 2 DO 580 I = 1,N WM(II) = WM(II) + 1.0D0 580 II = II + MEBAND C Do LU decomposition of P. -------------------------------------------- CALL DGBFA (WM(3), MEBAND, N, ML, MU, IWM(21), IER) IF (IER .NE. 0) IERPJ = 1 RETURN C----------------------- End of Subroutine DPRJA ----------------------- END *DECK DMNORM DOUBLE PRECISION FUNCTION DMNORM (N, V, W) C----------------------------------------------------------------------- C This function routine computes the weighted max-norm C of the vector of length N contained in the array V, with weights C contained in the array w of length N: C DMNORM = MAX(i=1,...,N) ABS(V(i))*W(i) C----------------------------------------------------------------------- INTEGER N, I DOUBLE PRECISION V, W, VM DIMENSION V(N), W(N) VM = 0.0D0 DO 10 I = 1,N 10 VM = MAX(VM,ABS(V(I))*W(I)) DMNORM = VM RETURN C----------------------- End of Function DMNORM ------------------------ END *DECK DFNORM DOUBLE PRECISION FUNCTION DFNORM (N, A, W) C----------------------------------------------------------------------- C This function computes the norm of a full N by N matrix, C stored in the array A, that is consistent with the weighted max-norm C on vectors, with weights stored in the array W: C DFNORM = MAX(i=1,...,N) ( W(i) * Sum(j=1,...,N) ABS(a(i,j))/W(j) ) C----------------------------------------------------------------------- INTEGER N, I, J DOUBLE PRECISION A, W, AN, SUM DIMENSION A(N,N), W(N) AN = 0.0D0 DO 20 I = 1,N SUM = 0.0D0 DO 10 J = 1,N 10 SUM = SUM + ABS(A(I,J))/W(J) AN = MAX(AN,SUM*W(I)) 20 CONTINUE DFNORM = AN RETURN C----------------------- End of Function DFNORM ------------------------ END *DECK DBNORM DOUBLE PRECISION FUNCTION DBNORM (N, A, NRA, ML, MU, W) C----------------------------------------------------------------------- C This function computes the norm of a banded N by N matrix, C stored in the array A, that is consistent with the weighted max-norm C on vectors, with weights stored in the array W. C ML and MU are the lower and upper half-bandwidths of the matrix. C NRA is the first dimension of the A array, NRA .ge. ML+MU+1. C In terms of the matrix elements a(i,j), the norm is given by: C DBNORM = MAX(i=1,...,N) ( W(i) * Sum(j=1,...,N) ABS(a(i,j))/W(j) ) C----------------------------------------------------------------------- INTEGER N, NRA, ML, MU INTEGER I, I1, JLO, JHI, J DOUBLE PRECISION A, W DOUBLE PRECISION AN, SUM DIMENSION A(NRA,N), W(N) AN = 0.0D0 DO 20 I = 1,N SUM = 0.0D0 I1 = I + MU + 1 JLO = MAX(I-ML,1) JHI = MIN(I+MU,N) DO 10 J = JLO,JHI 10 SUM = SUM + ABS(A(I1-J,J))/W(J) AN = MAX(AN,SUM*W(I)) 20 CONTINUE DBNORM = AN RETURN C----------------------- End of Function DBNORM ------------------------ END *DECK DSRCMA SUBROUTINE DSRCMA (RSAV, ISAV, JOB) C----------------------------------------------------------------------- C This routine saves or restores (depending on JOB) the contents of C the Common blocks DLS001, DLSA01, which are used C internally by one or more ODEPACK solvers. C C RSAV = real array of length 240 or more. C ISAV = integer array of length 46 or more. C JOB = flag indicating to save or restore the Common blocks: C JOB = 1 if Common is to be saved (written to RSAV/ISAV) C JOB = 2 if Common is to be restored (read from RSAV/ISAV) C A call with JOB = 2 presumes a prior call with JOB = 1. C----------------------------------------------------------------------- INTEGER ISAV, JOB INTEGER ILS, ILSA INTEGER I, LENRLS, LENILS, LENRLA, LENILA DOUBLE PRECISION RSAV DOUBLE PRECISION RLS, RLSA DIMENSION RSAV(*), ISAV(*) SAVE LENRLS, LENILS, LENRLA, LENILA COMMON /DLS001/ RLS(218), ILS(37) COMMON /DLSA01/ RLSA(22), ILSA(9) DATA LENRLS/218/, LENILS/37/, LENRLA/22/, LENILA/9/ C IF (JOB .EQ. 2) GO TO 100 DO 10 I = 1,LENRLS 10 RSAV(I) = RLS(I) DO 15 I = 1,LENRLA 15 RSAV(LENRLS+I) = RLSA(I) C DO 20 I = 1,LENILS 20 ISAV(I) = ILS(I) DO 25 I = 1,LENILA 25 ISAV(LENILS+I) = ILSA(I) C RETURN C 100 CONTINUE DO 110 I = 1,LENRLS 110 RLS(I) = RSAV(I) DO 115 I = 1,LENRLA 115 RLSA(I) = RSAV(LENRLS+I) C DO 120 I = 1,LENILS 120 ILS(I) = ISAV(I) DO 125 I = 1,LENILA 125 ILSA(I) = ISAV(LENILS+I) C RETURN C----------------------- End of Subroutine DSRCMA ---------------------- END *DECK DRCHEK SUBROUTINE DRCHEK (JOB, G, NEQ, Y, YH,NYH, G0, G1, GX, JROOT, IRT) EXTERNAL G INTEGER JOB, NEQ, NYH, JROOT, IRT DOUBLE PRECISION Y, YH, G0, G1, GX DIMENSION NEQ(*), Y(*), YH(NYH,*), G0(*), G1(*), GX(*), JROOT(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IOWND3, IOWNR3, IRFND, ITASKC, NGC, NGE DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION ROWNR3, T0, TLAST, TOUTC COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSR01/ ROWNR3(2), T0, TLAST, TOUTC, 1 IOWND3(3), IOWNR3(2), IRFND, ITASKC, NGC, NGE INTEGER I, IFLAG, JFLAG DOUBLE PRECISION HMING, T1, TEMP1, TEMP2, X LOGICAL ZROOT C----------------------------------------------------------------------- C This routine checks for the presence of a root in the vicinity of C the current T, in a manner depending on the input flag JOB. It calls C Subroutine DROOTS to locate the root as precisely as possible. C C In addition to variables described previously, DRCHEK C uses the following for communication: C JOB = integer flag indicating type of call: C JOB = 1 means the problem is being initialized, and DRCHEK C is to look for a root at or very near the initial T. C JOB = 2 means a continuation call to the solver was just C made, and DRCHEK is to check for a root in the C relevant part of the step last taken. C JOB = 3 means a successful step was just taken, and DRCHEK C is to look for a root in the interval of the step. C G0 = array of length NG, containing the value of g at T = T0. C G0 is input for JOB .ge. 2, and output in all cases. C G1,GX = arrays of length NG for work space. C IRT = completion flag: C IRT = 0 means no root was found. C IRT = -1 means JOB = 1 and a root was found too near to T. C IRT = 1 means a legitimate root was found (JOB = 2 or 3). C On return, T0 is the root location, and Y is the C corresponding solution vector. C T0 = value of T at one endpoint of interval of interest. Only C roots beyond T0 in the direction of integration are sought. C T0 is input if JOB .ge. 2, and output in all cases. C T0 is updated by DRCHEK, whether a root is found or not. C TLAST = last value of T returned by the solver (input only). C TOUTC = copy of TOUT (input only). C IRFND = input flag showing whether the last step taken had a root. C IRFND = 1 if it did, = 0 if not. C ITASKC = copy of ITASK (input only). C NGC = copy of NG (input only). C----------------------------------------------------------------------- IRT = 0 DO 10 I = 1,NGC 10 JROOT(I) = 0 HMING = (ABS(TN) + ABS(H))*UROUND*100.0D0 C GO TO (100, 200, 300), JOB C C Evaluate g at initial T, and check for zero values. ------------------ 100 CONTINUE T0 = TN CALL G (NEQ, T0, Y, NGC, G0) NGE = 1 ZROOT = .FALSE. DO 110 I = 1,NGC 110 IF (ABS(G0(I)) .LE. 0.0D0) ZROOT = .TRUE. IF (.NOT. ZROOT) GO TO 190 C g has a zero at T. Look at g at T + (small increment). -------------- TEMP2 = MAX(HMING/ABS(H), 0.1D0) TEMP1 = TEMP2*H T0 = T0 + TEMP1 DO 120 I = 1,N 120 Y(I) = Y(I) + TEMP2*YH(I,2) CALL G (NEQ, T0, Y, NGC, G0) NGE = NGE + 1 ZROOT = .FALSE. DO 130 I = 1,NGC 130 IF (ABS(G0(I)) .LE. 0.0D0) ZROOT = .TRUE. IF (.NOT. ZROOT) GO TO 190 C g has a zero at T and also close to T. Take error return. ----------- IRT = -1 RETURN C 190 CONTINUE RETURN C C 200 CONTINUE IF (IRFND .EQ. 0) GO TO 260 C If a root was found on the previous step, evaluate G0 = g(T0). ------- CALL DINTDY (T0, 0, YH, NYH, Y, IFLAG) CALL G (NEQ, T0, Y, NGC, G0) NGE = NGE + 1 ZROOT = .FALSE. DO 210 I = 1,NGC 210 IF (ABS(G0(I)) .LE. 0.0D0) ZROOT = .TRUE. IF (.NOT. ZROOT) GO TO 260 C g has a zero at T0. Look at g at T + (small increment). ------------- TEMP1 = SIGN(HMING,H) T0 = T0 + TEMP1 IF ((T0 - TN)*H .LT. 0.0D0) GO TO 230 TEMP2 = TEMP1/H DO 220 I = 1,N 220 Y(I) = Y(I) + TEMP2*YH(I,2) GO TO 240 230 CALL DINTDY (T0, 0, YH, NYH, Y, IFLAG) 240 CALL G (NEQ, T0, Y, NGC, G0) NGE = NGE + 1 ZROOT = .FALSE. DO 250 I = 1,NGC IF (ABS(G0(I)) .GT. 0.0D0) GO TO 250 JROOT(I) = 1 ZROOT = .TRUE. 250 CONTINUE IF (.NOT. ZROOT) GO TO 260 C g has a zero at T0 and also close to T0. Return root. --------------- IRT = 1 RETURN C G0 has no zero components. Proceed to check relevant interval. ------ 260 IF (TN .EQ. TLAST) GO TO 390 C 300 CONTINUE C Set T1 to TN or TOUTC, whichever comes first, and get g at T1. ------- IF (ITASKC.EQ.2 .OR. ITASKC.EQ.3 .OR. ITASKC.EQ.5) GO TO 310 IF ((TOUTC - TN)*H .GE. 0.0D0) GO TO 310 T1 = TOUTC IF ((T1 - T0)*H .LE. 0.0D0) GO TO 390 CALL DINTDY (T1, 0, YH, NYH, Y, IFLAG) GO TO 330 310 T1 = TN DO 320 I = 1,N 320 Y(I) = YH(I,1) 330 CALL G (NEQ, T1, Y, NGC, G1) NGE = NGE + 1 C Call DROOTS to search for root in interval from T0 to T1. ------------ JFLAG = 0 350 CONTINUE CALL DROOTS (NGC, HMING, JFLAG, T0, T1, G0, G1, GX, X, JROOT) IF (JFLAG .GT. 1) GO TO 360 CALL DINTDY (X, 0, YH, NYH, Y, IFLAG) CALL G (NEQ, X, Y, NGC, GX) NGE = NGE + 1 GO TO 350 360 T0 = X CALL DCOPY (NGC, GX, 1, G0, 1) IF (JFLAG .EQ. 4) GO TO 390 C Found a root. Interpolate to X and return. -------------------------- CALL DINTDY (X, 0, YH, NYH, Y, IFLAG) IRT = 1 RETURN C 390 CONTINUE RETURN C----------------------- End of Subroutine DRCHEK ---------------------- END *DECK DROOTS SUBROUTINE DROOTS (NG, HMIN, JFLAG, X0, X1, G0, G1, GX, X, JROOT) INTEGER NG, JFLAG, JROOT DOUBLE PRECISION HMIN, X0, X1, G0, G1, GX, X DIMENSION G0(NG), G1(NG), GX(NG), JROOT(NG) INTEGER IOWND3, IMAX, LAST, IDUM3 DOUBLE PRECISION ALPHA, X2, RDUM3 COMMON /DLSR01/ ALPHA, X2, RDUM3(3), 1 IOWND3(3), IMAX, LAST, IDUM3(4) C----------------------------------------------------------------------- C This subroutine finds the leftmost root of a set of arbitrary C functions gi(x) (i = 1,...,NG) in an interval (X0,X1). Only roots C of odd multiplicity (i.e. changes of sign of the gi) are found. C Here the sign of X1 - X0 is arbitrary, but is constant for a given C problem, and -leftmost- means nearest to X0. C The values of the vector-valued function g(x) = (gi, i=1...NG) C are communicated through the call sequence of DROOTS. C The method used is the Illinois algorithm. C C Reference: C Kathie L. Hiebert and Lawrence F. Shampine, Implicitly Defined C Output Points for Solutions of ODEs, Sandia Report SAND80-0180, C February 1980. C C Description of parameters. C C NG = number of functions gi, or the number of components of C the vector valued function g(x). Input only. C C HMIN = resolution parameter in X. Input only. When a root is C found, it is located only to within an error of HMIN in X. C Typically, HMIN should be set to something on the order of C 100 * UROUND * MAX(ABS(X0),ABS(X1)), C where UROUND is the unit roundoff of the machine. C C JFLAG = integer flag for input and output communication. C C On input, set JFLAG = 0 on the first call for the problem, C and leave it unchanged until the problem is completed. C (The problem is completed when JFLAG .ge. 2 on return.) C C On output, JFLAG has the following values and meanings: C JFLAG = 1 means DROOTS needs a value of g(x). Set GX = g(X) C and call DROOTS again. C JFLAG = 2 means a root has been found. The root is C at X, and GX contains g(X). (Actually, X is the C rightmost approximation to the root on an interval C (X0,X1) of size HMIN or less.) C JFLAG = 3 means X = X1 is a root, with one or more of the gi C being zero at X1 and no sign changes in (X0,X1). C GX contains g(X) on output. C JFLAG = 4 means no roots (of odd multiplicity) were C found in (X0,X1) (no sign changes). C C X0,X1 = endpoints of the interval where roots are sought. C X1 and X0 are input when JFLAG = 0 (first call), and C must be left unchanged between calls until the problem is C completed. X0 and X1 must be distinct, but X1 - X0 may be C of either sign. However, the notion of -left- and -right- C will be used to mean nearer to X0 or X1, respectively. C When JFLAG .ge. 2 on return, X0 and X1 are output, and C are the endpoints of the relevant interval. C C G0,G1 = arrays of length NG containing the vectors g(X0) and g(X1), C respectively. When JFLAG = 0, G0 and G1 are input and C none of the G0(i) should be zero. C When JFLAG .ge. 2 on return, G0 and G1 are output. C C GX = array of length NG containing g(X). GX is input C when JFLAG = 1, and output when JFLAG .ge. 2. C C X = independent variable value. Output only. C When JFLAG = 1 on output, X is the point at which g(x) C is to be evaluated and loaded into GX. C When JFLAG = 2 or 3, X is the root. C When JFLAG = 4, X is the right endpoint of the interval, X1. C C JROOT = integer array of length NG. Output only. C When JFLAG = 2 or 3, JROOT indicates which components C of g(x) have a root at X. JROOT(i) is 1 if the i-th C component has a root, and JROOT(i) = 0 otherwise. C----------------------------------------------------------------------- INTEGER I, IMXOLD, NXLAST DOUBLE PRECISION T2, TMAX, FRACINT, FRACSUB, ZERO,HALF,TENTH,FIVE LOGICAL ZROOT, SGNCHG, XROOT SAVE ZERO, HALF, TENTH, FIVE DATA ZERO/0.0D0/, HALF/0.5D0/, TENTH/0.1D0/, FIVE/5.0D0/ C IF (JFLAG .EQ. 1) GO TO 200 C JFLAG .ne. 1. Check for change in sign of g or zero at X1. ---------- IMAX = 0 TMAX = ZERO ZROOT = .FALSE. DO 120 I = 1,NG IF (ABS(G1(I)) .GT. ZERO) GO TO 110 ZROOT = .TRUE. GO TO 120 C At this point, G0(i) has been checked and cannot be zero. ------------ 110 IF (SIGN(1.0D0,G0(I)) .EQ. SIGN(1.0D0,G1(I))) GO TO 120 T2 = ABS(G1(I)/(G1(I)-G0(I))) IF (T2 .LE. TMAX) GO TO 120 TMAX = T2 IMAX = I 120 CONTINUE IF (IMAX .GT. 0) GO TO 130 SGNCHG = .FALSE. GO TO 140 130 SGNCHG = .TRUE. 140 IF (.NOT. SGNCHG) GO TO 400 C There is a sign change. Find the first root in the interval. -------- XROOT = .FALSE. NXLAST = 0 LAST = 1 C C Repeat until the first root in the interval is found. Loop point. --- 150 CONTINUE IF (XROOT) GO TO 300 IF (NXLAST .EQ. LAST) GO TO 160 ALPHA = 1.0D0 GO TO 180 160 IF (LAST .EQ. 0) GO TO 170 ALPHA = 0.5D0*ALPHA GO TO 180 170 ALPHA = 2.0D0*ALPHA 180 X2 = X1 - (X1 - X0)*G1(IMAX) / (G1(IMAX) - ALPHA*G0(IMAX)) C If X2 is too close to X0 or X1, adjust it inward, by a fractional ---- C distance that is between 0.1 and 0.5. -------------------------------- IF (ABS(X2 - X0) < HALF*HMIN) THEN FRACINT = ABS(X1 - X0)/HMIN FRACSUB = TENTH IF (FRACINT .LE. FIVE) FRACSUB = HALF/FRACINT X2 = X0 + FRACSUB*(X1 - X0) ENDIF IF (ABS(X1 - X2) < HALF*HMIN) THEN FRACINT = ABS(X1 - X0)/HMIN FRACSUB = TENTH IF (FRACINT .LE. FIVE) FRACSUB = HALF/FRACINT X2 = X1 - FRACSUB*(X1 - X0) ENDIF JFLAG = 1 X = X2 C Return to the calling routine to get a value of GX = g(X). ----------- RETURN C Check to see in which interval g changes sign. ----------------------- 200 IMXOLD = IMAX IMAX = 0 TMAX = ZERO ZROOT = .FALSE. DO 220 I = 1,NG IF (ABS(GX(I)) .GT. ZERO) GO TO 210 ZROOT = .TRUE. GO TO 220 C Neither G0(i) nor GX(i) can be zero at this point. ------------------- 210 IF (SIGN(1.0D0,G0(I)) .EQ. SIGN(1.0D0,GX(I))) GO TO 220 T2 = ABS(GX(I)/(GX(I) - G0(I))) IF (T2 .LE. TMAX) GO TO 220 TMAX = T2 IMAX = I 220 CONTINUE IF (IMAX .GT. 0) GO TO 230 SGNCHG = .FALSE. IMAX = IMXOLD GO TO 240 230 SGNCHG = .TRUE. 240 NXLAST = LAST IF (.NOT. SGNCHG) GO TO 250 C Sign change between X0 and X2, so replace X1 with X2. ---------------- X1 = X2 CALL DCOPY (NG, GX, 1, G1, 1) LAST = 1 XROOT = .FALSE. GO TO 270 250 IF (.NOT. ZROOT) GO TO 260 C Zero value at X2 and no sign change in (X0,X2), so X2 is a root. ----- X1 = X2 CALL DCOPY (NG, GX, 1, G1, 1) XROOT = .TRUE. GO TO 270 C No sign change between X0 and X2. Replace X0 with X2. --------------- 260 CONTINUE CALL DCOPY (NG, GX, 1, G0, 1) X0 = X2 LAST = 0 XROOT = .FALSE. 270 IF (ABS(X1-X0) .LE. HMIN) XROOT = .TRUE. GO TO 150 C C Return with X1 as the root. Set JROOT. Set X = X1 and GX = G1. ----- 300 JFLAG = 2 X = X1 CALL DCOPY (NG, G1, 1, GX, 1) DO 320 I = 1,NG JROOT(I) = 0 IF (ABS(G1(I)) .GT. ZERO) GO TO 310 JROOT(I) = 1 GO TO 320 310 IF (SIGN(1.0D0,G0(I)) .NE. SIGN(1.0D0,G1(I))) JROOT(I) = 1 320 CONTINUE RETURN C C No sign change in the interval. Check for zero at right endpoint. --- 400 IF (.NOT. ZROOT) GO TO 420 C C Zero value at X1 and no sign change in (X0,X1). Return JFLAG = 3. --- X = X1 CALL DCOPY (NG, G1, 1, GX, 1) DO 410 I = 1,NG JROOT(I) = 0 IF (ABS(G1(I)) .LE. ZERO) JROOT (I) = 1 410 CONTINUE JFLAG = 3 RETURN C C No sign changes in this interval. Set X = X1, return JFLAG = 4. ----- 420 CALL DCOPY (NG, G1, 1, GX, 1) X = X1 JFLAG = 4 RETURN C----------------------- End of Subroutine DROOTS ---------------------- END *DECK DSRCAR SUBROUTINE DSRCAR (RSAV, ISAV, JOB) C----------------------------------------------------------------------- C This routine saves or restores (depending on JOB) the contents of C the Common blocks DLS001, DLSA01, DLSR01, which are used C internally by one or more ODEPACK solvers. C C RSAV = real array of length 245 or more. C ISAV = integer array of length 55 or more. C JOB = flag indicating to save or restore the Common blocks: C JOB = 1 if Common is to be saved (written to RSAV/ISAV) C JOB = 2 if Common is to be restored (read from RSAV/ISAV) C A call with JOB = 2 presumes a prior call with JOB = 1. C----------------------------------------------------------------------- INTEGER ISAV, JOB INTEGER ILS, ILSA, ILSR INTEGER I, IOFF, LENRLS, LENILS, LENRLA, LENILA, LENRLR, LENILR DOUBLE PRECISION RSAV DOUBLE PRECISION RLS, RLSA, RLSR DIMENSION RSAV(*), ISAV(*) SAVE LENRLS, LENILS, LENRLA, LENILA, LENRLR, LENILR COMMON /DLS001/ RLS(218), ILS(37) COMMON /DLSA01/ RLSA(22), ILSA(9) COMMON /DLSR01/ RLSR(5), ILSR(9) DATA LENRLS/218/, LENILS/37/, LENRLA/22/, LENILA/9/ DATA LENRLR/5/, LENILR/9/ C IF (JOB .EQ. 2) GO TO 100 DO 10 I = 1,LENRLS 10 RSAV(I) = RLS(I) DO 15 I = 1,LENRLA 15 RSAV(LENRLS+I) = RLSA(I) IOFF = LENRLS + LENRLA DO 20 I = 1,LENRLR 20 RSAV(IOFF+I) = RLSR(I) C DO 30 I = 1,LENILS 30 ISAV(I) = ILS(I) DO 35 I = 1,LENILA 35 ISAV(LENILS+I) = ILSA(I) IOFF = LENILS + LENILA DO 40 I = 1,LENILR 40 ISAV(IOFF+I) = ILSR(I) C RETURN C 100 CONTINUE DO 110 I = 1,LENRLS 110 RLS(I) = RSAV(I) DO 115 I = 1,LENRLA 115 RLSA(I) = RSAV(LENRLS+I) IOFF = LENRLS + LENRLA DO 120 I = 1,LENRLR 120 RLSR(I) = RSAV(IOFF+I) C DO 130 I = 1,LENILS 130 ILS(I) = ISAV(I) DO 135 I = 1,LENILA 135 ILSA(I) = ISAV(LENILS+I) IOFF = LENILS + LENILA DO 140 I = 1,LENILR 140 ILSR(I) = ISAV(IOFF+I) C RETURN C----------------------- End of Subroutine DSRCAR ---------------------- END *DECK DSTODPK SUBROUTINE DSTODPK (NEQ, Y, YH, NYH, YH1, EWT, SAVF, SAVX, ACOR, 1 WM, IWM, F, JAC, PSOL) EXTERNAL F, JAC, PSOL INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, YH1, EWT, SAVF, SAVX, ACOR, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*), 1 SAVX(*), ACOR(*), WM(*), IWM(*) INTEGER IOWND, IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 1 NNI, NLI, NPS, NCFN, NCFL DOUBLE PRECISION CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO, 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION DELT, EPCON, SQRTN, RSQRTN COMMON /DLS001/ CONIT, CRATE, EL(13), ELCO(13,12), 1 HOLD, RMAX, TESCO(3,12), 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 3 IOWND(6), IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLPK01/ DELT, EPCON, SQRTN, RSQRTN, 1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 2 NNI, NLI, NPS, NCFN, NCFL C----------------------------------------------------------------------- C DSTODPK performs one step of the integration of an initial value C problem for a system of Ordinary Differential Equations. C----------------------------------------------------------------------- C The following changes were made to generate Subroutine DSTODPK C from Subroutine DSTODE: C 1. The array SAVX was added to the call sequence. C 2. PJAC and SLVS were replaced by PSOL in the call sequence. C 3. The Common block /DLPK01/ was added for communication. C 4. The test constant EPCON is loaded into Common below statement C numbers 125 and 155, and used below statement 400. C 5. The Newton iteration counter MNEWT is set below 220 and 400. C 6. The call to PJAC was replaced with a call to DPKSET (fixed name), C with a longer call sequence, called depending on JACFLG. C 7. The corrector residual is stored in SAVX (not Y) at 360, C and the solution vector is in SAVX in the 380 loop. C 8. SLVS was renamed DSOLPK and includes NEQ, SAVX, EWT, F, and JAC. C SAVX was added because DSOLPK now needs Y and SAVF undisturbed. C 9. The nonlinear convergence failure count NCFN is set at 430. C----------------------------------------------------------------------- C Note: DSTODPK is independent of the value of the iteration method C indicator MITER, when this is .ne. 0, and hence is independent C of the type of chord method used, or the Jacobian structure. C Communication with DSTODPK is done with the following variables: C C NEQ = integer array containing problem size in NEQ(1), and C passed as the NEQ argument in all calls to F and JAC. C Y = an array of length .ge. N used as the Y argument in C all calls to F and JAC. C YH = an NYH by LMAX array containing the dependent variables C and their approximate scaled derivatives, where C LMAX = MAXORD + 1. YH(i,j+1) contains the approximate C j-th derivative of y(i), scaled by H**j/factorial(j) C (j = 0,1,...,NQ). On entry for the first step, the first C two columns of YH must be set from the initial values. C NYH = a constant integer .ge. N, the first dimension of YH. C YH1 = a one-dimensional array occupying the same space as YH. C EWT = an array of length N containing multiplicative weights C for local error measurements. Local errors in y(i) are C compared to 1.0/EWT(i) in various error tests. C SAVF = an array of working storage, of length N. C Also used for input of YH(*,MAXORD+2) when JSTART = -1 C and MAXORD .lt. the current order NQ. C SAVX = an array of working storage, of length N. C ACOR = a work array of length N, used for the accumulated C corrections. On a successful return, ACOR(i) contains C the estimated one-step local error in y(i). C WM,IWM = real and integer work arrays associated with matrix C operations in chord iteration (MITER .ne. 0). C CCMAX = maximum relative change in H*EL0 before DPKSET is called. C H = the step size to be attempted on the next step. C H is altered by the error control algorithm during the C problem. H can be either positive or negative, but its C sign must remain constant throughout the problem. C HMIN = the minimum absolute value of the step size H to be used. C HMXI = inverse of the maximum absolute value of H to be used. C HMXI = 0.0 is allowed and corresponds to an infinite HMAX. C HMIN and HMXI may be changed at any time, but will not C take effect until the next change of H is considered. C TN = the independent variable. TN is updated on each step taken. C JSTART = an integer used for input only, with the following C values and meanings: C 0 perform the first step. C .gt.0 take a new step continuing from the last. C -1 take the next step with a new value of H, MAXORD, C N, METH, MITER, and/or matrix parameters. C -2 take the next step with a new value of H, C but with other inputs unchanged. C On return, JSTART is set to 1 to facilitate continuation. C KFLAG = a completion code with the following meanings: C 0 the step was succesful. C -1 the requested error could not be achieved. C -2 corrector convergence could not be achieved. C -3 fatal error in DPKSET or DSOLPK. C A return with KFLAG = -1 or -2 means either C ABS(H) = HMIN or 10 consecutive failures occurred. C On a return with KFLAG negative, the values of TN and C the YH array are as of the beginning of the last C step, and H is the last step size attempted. C MAXORD = the maximum order of integration method to be allowed. C MAXCOR = the maximum number of corrector iterations allowed. C MSBP = maximum number of steps between DPKSET calls (MITER .gt. 0). C MXNCF = maximum number of convergence failures allowed. C METH/MITER = the method flags. See description in driver. C N = the number of first-order differential equations. C----------------------------------------------------------------------- INTEGER I, I1, IREDO, IRET, J, JB, M, NCF, NEWQ DOUBLE PRECISION DCON, DDN, DEL, DELP, DSM, DUP, EXDN, EXSM, EXUP, 1 R, RH, RHDN, RHSM, RHUP, TOLD, DVNORM C KFLAG = 0 TOLD = TN NCF = 0 IERPJ = 0 IERSL = 0 JCUR = 0 ICF = 0 DELP = 0.0D0 IF (JSTART .GT. 0) GO TO 200 IF (JSTART .EQ. -1) GO TO 100 IF (JSTART .EQ. -2) GO TO 160 C----------------------------------------------------------------------- C On the first call, the order is set to 1, and other variables are C initialized. RMAX is the maximum ratio by which H can be increased C in a single step. It is initially 1.E4 to compensate for the small C initial H, but then is normally equal to 10. If a failure C occurs (in corrector convergence or error test), RMAX is set at 2 C for the next increase. C----------------------------------------------------------------------- LMAX = MAXORD + 1 NQ = 1 L = 2 IALTH = 2 RMAX = 10000.0D0 RC = 0.0D0 EL0 = 1.0D0 CRATE = 0.7D0 HOLD = H MEO = METH NSLP = 0 IPUP = MITER IRET = 3 GO TO 140 C----------------------------------------------------------------------- C The following block handles preliminaries needed when JSTART = -1. C IPUP is set to MITER to force a matrix update. C If an order increase is about to be considered (IALTH = 1), C IALTH is reset to 2 to postpone consideration one more step. C If the caller has changed METH, DCFODE is called to reset C the coefficients of the method. C If the caller has changed MAXORD to a value less than the current C order NQ, NQ is reduced to MAXORD, and a new H chosen accordingly. C If H is to be changed, YH must be rescaled. C If H or METH is being changed, IALTH is reset to L = NQ + 1 C to prevent further changes in H for that many steps. C----------------------------------------------------------------------- 100 IPUP = MITER LMAX = MAXORD + 1 IF (IALTH .EQ. 1) IALTH = 2 IF (METH .EQ. MEO) GO TO 110 CALL DCFODE (METH, ELCO, TESCO) MEO = METH IF (NQ .GT. MAXORD) GO TO 120 IALTH = L IRET = 1 GO TO 150 110 IF (NQ .LE. MAXORD) GO TO 160 120 NQ = MAXORD L = LMAX DO 125 I = 1,L 125 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) EPCON = CONIT*TESCO(2,NQ) DDN = DVNORM (N, SAVF, EWT)/TESCO(1,L) EXDN = 1.0D0/L RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) RH = MIN(RHDN,1.0D0) IREDO = 3 IF (H .EQ. HOLD) GO TO 170 RH = MIN(RH,ABS(H/HOLD)) H = HOLD GO TO 175 C----------------------------------------------------------------------- C DCFODE is called to get all the integration coefficients for the C current METH. Then the EL vector and related constants are reset C whenever the order NQ is changed, or at the start of the problem. C----------------------------------------------------------------------- 140 CALL DCFODE (METH, ELCO, TESCO) 150 DO 155 I = 1,L 155 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) EPCON = CONIT*TESCO(2,NQ) GO TO (160, 170, 200), IRET C----------------------------------------------------------------------- C If H is being changed, the H ratio RH is checked against C RMAX, HMIN, and HMXI, and the YH array rescaled. IALTH is set to C L = NQ + 1 to prevent a change of H for that many steps, unless C forced by a convergence or error test failure. C----------------------------------------------------------------------- 160 IF (H .EQ. HOLD) GO TO 200 RH = H/HOLD H = HOLD IREDO = 3 GO TO 175 170 RH = MAX(RH,HMIN/ABS(H)) 175 RH = MIN(RH,RMAX) RH = RH/MAX(1.0D0,ABS(H)*HMXI*RH) R = 1.0D0 DO 180 J = 2,L R = R*RH DO 180 I = 1,N 180 YH(I,J) = YH(I,J)*R H = H*RH RC = RC*RH IALTH = L IF (IREDO .EQ. 0) GO TO 690 C----------------------------------------------------------------------- C This section computes the predicted values by effectively C multiplying the YH array by the Pascal triangle matrix. C The flag IPUP is set according to whether matrix data is involved C (JACFLG .ne. 0) or not (JACFLG = 0), to trigger a call to DPKSET. C IPUP is set to MITER when RC differs from 1 by more than CCMAX, C and at least every MSBP steps, when JACFLG = 1. C RC is the ratio of new to old values of the coefficient H*EL(1). C----------------------------------------------------------------------- 200 IF (JACFLG .NE. 0) GO TO 202 IPUP = 0 CRATE = 0.7D0 GO TO 205 202 IF (ABS(RC-1.0D0) .GT. CCMAX) IPUP = MITER IF (NST .GE. NSLP+MSBP) IPUP = MITER 205 TN = TN + H I1 = NQNYH + 1 DO 215 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 210 I = I1,NQNYH 210 YH1(I) = YH1(I) + YH1(I+NYH) 215 CONTINUE C----------------------------------------------------------------------- C Up to MAXCOR corrector iterations are taken. A convergence test is C made on the RMS-norm of each correction, weighted by the error C weight vector EWT. The sum of the corrections is accumulated in the C vector ACOR(i). The YH array is not altered in the corrector loop. C----------------------------------------------------------------------- 220 M = 0 MNEWT = 0 DO 230 I = 1,N 230 Y(I) = YH(I,1) CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 IF (IPUP .LE. 0) GO TO 250 C----------------------------------------------------------------------- C If indicated, DPKSET is called to update any matrix data needed, C before starting the corrector iteration. C IPUP is set to 0 as an indicator that this has been done. C----------------------------------------------------------------------- CALL DPKSET (NEQ, Y, YH1, EWT, ACOR, SAVF, WM, IWM, F, JAC) IPUP = 0 RC = 1.0D0 NSLP = NST CRATE = 0.7D0 IF (IERPJ .NE. 0) GO TO 430 250 DO 260 I = 1,N 260 ACOR(I) = 0.0D0 270 IF (MITER .NE. 0) GO TO 350 C----------------------------------------------------------------------- C In the case of functional iteration, update Y directly from C the result of the last function evaluation. C----------------------------------------------------------------------- DO 290 I = 1,N SAVF(I) = H*SAVF(I) - YH(I,2) 290 Y(I) = SAVF(I) - ACOR(I) DEL = DVNORM (N, Y, EWT) DO 300 I = 1,N Y(I) = YH(I,1) + EL(1)*SAVF(I) 300 ACOR(I) = SAVF(I) GO TO 400 C----------------------------------------------------------------------- C In the case of the chord method, compute the corrector error, C and solve the linear system with that as right-hand side and C P as coefficient matrix. C----------------------------------------------------------------------- 350 DO 360 I = 1,N 360 SAVX(I) = H*SAVF(I) - (YH(I,2) + ACOR(I)) CALL DSOLPK (NEQ, Y, SAVF, SAVX, EWT, WM, IWM, F, PSOL) IF (IERSL .LT. 0) GO TO 430 IF (IERSL .GT. 0) GO TO 410 DEL = DVNORM (N, SAVX, EWT) DO 380 I = 1,N ACOR(I) = ACOR(I) + SAVX(I) 380 Y(I) = YH(I,1) + EL(1)*ACOR(I) C----------------------------------------------------------------------- C Test for convergence. If M .gt. 0, an estimate of the convergence C rate constant is stored in CRATE, and this is used in the test. C----------------------------------------------------------------------- 400 IF (M .NE. 0) CRATE = MAX(0.2D0*CRATE,DEL/DELP) DCON = DEL*MIN(1.0D0,1.5D0*CRATE)/EPCON IF (DCON .LE. 1.0D0) GO TO 450 M = M + 1 IF (M .EQ. MAXCOR) GO TO 410 IF (M .GE. 2 .AND. DEL .GT. 2.0D0*DELP) GO TO 410 MNEWT = M DELP = DEL CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 GO TO 270 C----------------------------------------------------------------------- C The corrector iteration failed to converge. C If MITER .ne. 0 and the Jacobian is out of date, DPKSET is called for C the next try. Otherwise the YH array is retracted to its values C before prediction, and H is reduced, if possible. If H cannot be C reduced or MXNCF failures have occurred, exit with KFLAG = -2. C----------------------------------------------------------------------- 410 IF (MITER.EQ.0 .OR. JCUR.EQ.1 .OR. JACFLG.EQ.0) GO TO 430 ICF = 1 IPUP = MITER GO TO 220 430 ICF = 2 NCF = NCF + 1 NCFN = NCFN + 1 RMAX = 2.0D0 TN = TOLD I1 = NQNYH + 1 DO 445 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 440 I = I1,NQNYH 440 YH1(I) = YH1(I) - YH1(I+NYH) 445 CONTINUE IF (IERPJ .LT. 0 .OR. IERSL .LT. 0) GO TO 680 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 670 IF (NCF .EQ. MXNCF) GO TO 670 RH = 0.5D0 IPUP = MITER IREDO = 1 GO TO 170 C----------------------------------------------------------------------- C The corrector has converged. JCUR is set to 0 C to signal that the Jacobian involved may need updating later. C The local error test is made and control passes to statement 500 C if it fails. C----------------------------------------------------------------------- 450 JCUR = 0 IF (M .EQ. 0) DSM = DEL/TESCO(2,NQ) IF (M .GT. 0) DSM = DVNORM (N, ACOR, EWT)/TESCO(2,NQ) IF (DSM .GT. 1.0D0) GO TO 500 C----------------------------------------------------------------------- C After a successful step, update the YH array. C Consider changing H if IALTH = 1. Otherwise decrease IALTH by 1. C If IALTH is then 1 and NQ .lt. MAXORD, then ACOR is saved for C use in a possible order increase on the next step. C If a change in H is considered, an increase or decrease in order C by one is considered also. A change in H is made only if it is by a C factor of at least 1.1. If not, IALTH is set to 3 to prevent C testing for that many steps. C----------------------------------------------------------------------- KFLAG = 0 IREDO = 0 NST = NST + 1 HU = H NQU = NQ DO 470 J = 1,L DO 470 I = 1,N 470 YH(I,J) = YH(I,J) + EL(J)*ACOR(I) IALTH = IALTH - 1 IF (IALTH .EQ. 0) GO TO 520 IF (IALTH .GT. 1) GO TO 700 IF (L .EQ. LMAX) GO TO 700 DO 490 I = 1,N 490 YH(I,LMAX) = ACOR(I) GO TO 700 C----------------------------------------------------------------------- C The error test failed. KFLAG keeps track of multiple failures. C Restore TN and the YH array to their previous values, and prepare C to try the step again. Compute the optimum step size for this or C one lower order. After 2 or more failures, H is forced to decrease C by a factor of 0.2 or less. C----------------------------------------------------------------------- 500 KFLAG = KFLAG - 1 TN = TOLD I1 = NQNYH + 1 DO 515 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 510 I = I1,NQNYH 510 YH1(I) = YH1(I) - YH1(I+NYH) 515 CONTINUE RMAX = 2.0D0 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 660 IF (KFLAG .LE. -3) GO TO 640 IREDO = 2 RHUP = 0.0D0 GO TO 540 C----------------------------------------------------------------------- C Regardless of the success or failure of the step, factors C RHDN, RHSM, and RHUP are computed, by which H could be multiplied C at order NQ - 1, order NQ, or order NQ + 1, respectively. C In the case of failure, RHUP = 0.0 to avoid an order increase. C the largest of these is determined and the new order chosen C accordingly. If the order is to be increased, we compute one C additional scaled derivative. C----------------------------------------------------------------------- 520 RHUP = 0.0D0 IF (L .EQ. LMAX) GO TO 540 DO 530 I = 1,N 530 SAVF(I) = ACOR(I) - YH(I,LMAX) DUP = DVNORM (N, SAVF, EWT)/TESCO(3,NQ) EXUP = 1.0D0/(L+1) RHUP = 1.0D0/(1.4D0*DUP**EXUP + 0.0000014D0) 540 EXSM = 1.0D0/L RHSM = 1.0D0/(1.2D0*DSM**EXSM + 0.0000012D0) RHDN = 0.0D0 IF (NQ .EQ. 1) GO TO 560 DDN = DVNORM (N, YH(1,L), EWT)/TESCO(1,NQ) EXDN = 1.0D0/NQ RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) 560 IF (RHSM .GE. RHUP) GO TO 570 IF (RHUP .GT. RHDN) GO TO 590 GO TO 580 570 IF (RHSM .LT. RHDN) GO TO 580 NEWQ = NQ RH = RHSM GO TO 620 580 NEWQ = NQ - 1 RH = RHDN IF (KFLAG .LT. 0 .AND. RH .GT. 1.0D0) RH = 1.0D0 GO TO 620 590 NEWQ = L RH = RHUP IF (RH .LT. 1.1D0) GO TO 610 R = EL(L)/L DO 600 I = 1,N 600 YH(I,NEWQ+1) = ACOR(I)*R GO TO 630 610 IALTH = 3 GO TO 700 620 IF ((KFLAG .EQ. 0) .AND. (RH .LT. 1.1D0)) GO TO 610 IF (KFLAG .LE. -2) RH = MIN(RH,0.2D0) C----------------------------------------------------------------------- C If there is a change of order, reset NQ, L, and the coefficients. C In any case H is reset according to RH and the YH array is rescaled. C Then exit from 690 if the step was OK, or redo the step otherwise. C----------------------------------------------------------------------- IF (NEWQ .EQ. NQ) GO TO 170 630 NQ = NEWQ L = NQ + 1 IRET = 2 GO TO 150 C----------------------------------------------------------------------- C Control reaches this section if 3 or more failures have occured. C If 10 failures have occurred, exit with KFLAG = -1. C It is assumed that the derivatives that have accumulated in the C YH array have errors of the wrong order. Hence the first C derivative is recomputed, and the order is set to 1. Then C H is reduced by a factor of 10, and the step is retried, C until it succeeds or H reaches HMIN. C----------------------------------------------------------------------- 640 IF (KFLAG .EQ. -10) GO TO 660 RH = 0.1D0 RH = MAX(HMIN/ABS(H),RH) H = H*RH DO 645 I = 1,N 645 Y(I) = YH(I,1) CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 DO 650 I = 1,N 650 YH(I,2) = H*SAVF(I) IPUP = MITER IALTH = 5 IF (NQ .EQ. 1) GO TO 200 NQ = 1 L = 2 IRET = 3 GO TO 150 C----------------------------------------------------------------------- C All returns are made through this section. H is saved in HOLD C to allow the caller to change H on the next step. C----------------------------------------------------------------------- 660 KFLAG = -1 GO TO 720 670 KFLAG = -2 GO TO 720 680 KFLAG = -3 GO TO 720 690 RMAX = 10.0D0 700 R = 1.0D0/TESCO(2,NQU) DO 710 I = 1,N 710 ACOR(I) = ACOR(I)*R 720 HOLD = H JSTART = 1 RETURN C----------------------- End of Subroutine DSTODPK --------------------- END *DECK DPKSET SUBROUTINE DPKSET (NEQ, Y, YSV, EWT, FTEM, SAVF, WM, IWM, F, JAC) EXTERNAL F, JAC INTEGER NEQ, IWM DOUBLE PRECISION Y, YSV, EWT, FTEM, SAVF, WM DIMENSION NEQ(*), Y(*), YSV(*), EWT(*), FTEM(*), SAVF(*), 1 WM(*), IWM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 1 NNI, NLI, NPS, NCFN, NCFL DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION DELT, EPCON, SQRTN, RSQRTN COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLPK01/ DELT, EPCON, SQRTN, RSQRTN, 1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 2 NNI, NLI, NPS, NCFN, NCFL C----------------------------------------------------------------------- C DPKSET is called by DSTODPK to interface with the user-supplied C routine JAC, to compute and process relevant parts of C the matrix P = I - H*EL(1)*J , where J is the Jacobian df/dy, C as need for preconditioning matrix operations later. C C In addition to variables described previously, communication C with DPKSET uses the following: C Y = array containing predicted values on entry. C YSV = array containing predicted y, to be saved (YH1 in DSTODPK). C FTEM = work array of length N (ACOR in DSTODPK). C SAVF = array containing f evaluated at predicted y. C WM = real work space for matrices. C Space for preconditioning data starts at WM(LOCWP). C IWM = integer work space. C Space for preconditioning data starts at IWM(LOCIWP). C IERPJ = output error flag, = 0 if no trouble, .gt. 0 if C JAC returned an error flag. C JCUR = output flag = 1 to indicate that the Jacobian matrix C (or approximation) is now current. C This routine also uses Common variables EL0, H, TN, IERPJ, JCUR, NJE. C----------------------------------------------------------------------- INTEGER IER DOUBLE PRECISION HL0 C IERPJ = 0 JCUR = 1 HL0 = EL0*H CALL JAC (F, NEQ, TN, Y, YSV, EWT, SAVF, FTEM, HL0, 1 WM(LOCWP), IWM(LOCIWP), IER) NJE = NJE + 1 IF (IER .EQ. 0) RETURN IERPJ = 1 RETURN C----------------------- End of Subroutine DPKSET ---------------------- END *DECK DSOLPK SUBROUTINE DSOLPK (NEQ, Y, SAVF, X, EWT, WM, IWM, F, PSOL) EXTERNAL F, PSOL INTEGER NEQ, IWM DOUBLE PRECISION Y, SAVF, X, EWT, WM DIMENSION NEQ(*), Y(*), SAVF(*), X(*), EWT(*), WM(*), IWM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 1 NNI, NLI, NPS, NCFN, NCFL DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION DELT, EPCON, SQRTN, RSQRTN COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLPK01/ DELT, EPCON, SQRTN, RSQRTN, 1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 2 NNI, NLI, NPS, NCFN, NCFL C----------------------------------------------------------------------- C This routine interfaces to one of DSPIOM, DSPIGMR, DPCG, DPCGS, or C DUSOL, for the solution of the linear system arising from a Newton C iteration. It is called if MITER .ne. 0. C In addition to variables described elsewhere, C communication with DSOLPK uses the following variables: C WM = real work space containing data for the algorithm C (Krylov basis vectors, Hessenberg matrix, etc.) C IWM = integer work space containing data for the algorithm C X = the right-hand side vector on input, and the solution vector C on output, of length N. C IERSL = output flag (in Common): C IERSL = 0 means no trouble occurred. C IERSL = 1 means the iterative method failed to converge. C If the preconditioner is out of date, the step C is repeated with a new preconditioner. C Otherwise, the stepsize is reduced (forcing a C new evaluation of the preconditioner) and the C step is repeated. C IERSL = -1 means there was a nonrecoverable error in the C iterative solver, and an error exit occurs. C This routine also uses the Common variables TN, EL0, H, N, MITER, C DELT, EPCON, SQRTN, RSQRTN, MAXL, KMP, MNEWT, NNI, NLI, NPS, NCFL, C LOCWP, LOCIWP. C----------------------------------------------------------------------- INTEGER IFLAG, LB, LDL, LHES, LIOM, LGMR, LPCG, LP, LQ, LR, 1 LV, LW, LWK, LZ, MAXLP1, NPSL DOUBLE PRECISION DELTA, HL0 C IERSL = 0 HL0 = H*EL0 DELTA = DELT*EPCON GO TO (100, 200, 300, 400, 900, 900, 900, 900, 900), MITER C----------------------------------------------------------------------- C Use the SPIOM algorithm to solve the linear system P*x = -f. C----------------------------------------------------------------------- 100 CONTINUE LV = 1 LB = LV + N*MAXL LHES = LB + N LWK = LHES + MAXL*MAXL CALL DCOPY (N, X, 1, WM(LB), 1) CALL DSCAL (N, RSQRTN, EWT, 1) CALL DSPIOM (NEQ, TN, Y, SAVF, WM(LB), EWT, N, MAXL, KMP, DELTA, 1 HL0, JPRE, MNEWT, F, PSOL, NPSL, X, WM(LV), WM(LHES), IWM, 2 LIOM, WM(LOCWP), IWM(LOCIWP), WM(LWK), IFLAG) NNI = NNI + 1 NLI = NLI + LIOM NPS = NPS + NPSL CALL DSCAL (N, SQRTN, EWT, 1) IF (IFLAG .NE. 0) NCFL = NCFL + 1 IF (IFLAG .GE. 2) IERSL = 1 IF (IFLAG .LT. 0) IERSL = -1 RETURN C----------------------------------------------------------------------- C Use the SPIGMR algorithm to solve the linear system P*x = -f. C----------------------------------------------------------------------- 200 CONTINUE MAXLP1 = MAXL + 1 LV = 1 LB = LV + N*MAXL LHES = LB + N + 1 LQ = LHES + MAXL*MAXLP1 LWK = LQ + 2*MAXL LDL = LWK + MIN(1,MAXL-KMP)*N CALL DCOPY (N, X, 1, WM(LB), 1) CALL DSCAL (N, RSQRTN, EWT, 1) CALL DSPIGMR (NEQ, TN, Y, SAVF, WM(LB), EWT, N, MAXL, MAXLP1, KMP, 1 DELTA, HL0, JPRE, MNEWT, F, PSOL, NPSL, X, WM(LV), WM(LHES), 2 WM(LQ), LGMR, WM(LOCWP), IWM(LOCIWP), WM(LWK), WM(LDL), IFLAG) NNI = NNI + 1 NLI = NLI + LGMR NPS = NPS + NPSL CALL DSCAL (N, SQRTN, EWT, 1) IF (IFLAG .NE. 0) NCFL = NCFL + 1 IF (IFLAG .GE. 2) IERSL = 1 IF (IFLAG .LT. 0) IERSL = -1 RETURN C----------------------------------------------------------------------- C Use DPCG to solve the linear system P*x = -f C----------------------------------------------------------------------- 300 CONTINUE LR = 1 LP = LR + N LW = LP + N LZ = LW + N LWK = LZ + N CALL DCOPY (N, X, 1, WM(LR), 1) CALL DPCG (NEQ, TN, Y, SAVF, WM(LR), EWT, N, MAXL, DELTA, HL0, 1 JPRE, MNEWT, F, PSOL, NPSL, X, WM(LP), WM(LW), WM(LZ), 2 LPCG, WM(LOCWP), IWM(LOCIWP), WM(LWK), IFLAG) NNI = NNI + 1 NLI = NLI + LPCG NPS = NPS + NPSL IF (IFLAG .NE. 0) NCFL = NCFL + 1 IF (IFLAG .GE. 2) IERSL = 1 IF (IFLAG .LT. 0) IERSL = -1 RETURN C----------------------------------------------------------------------- C Use DPCGS to solve the linear system P*x = -f C----------------------------------------------------------------------- 400 CONTINUE LR = 1 LP = LR + N LW = LP + N LZ = LW + N LWK = LZ + N CALL DCOPY (N, X, 1, WM(LR), 1) CALL DPCGS (NEQ, TN, Y, SAVF, WM(LR), EWT, N, MAXL, DELTA, HL0, 1 JPRE, MNEWT, F, PSOL, NPSL, X, WM(LP), WM(LW), WM(LZ), 2 LPCG, WM(LOCWP), IWM(LOCIWP), WM(LWK), IFLAG) NNI = NNI + 1 NLI = NLI + LPCG NPS = NPS + NPSL IF (IFLAG .NE. 0) NCFL = NCFL + 1 IF (IFLAG .GE. 2) IERSL = 1 IF (IFLAG .LT. 0) IERSL = -1 RETURN C----------------------------------------------------------------------- C Use DUSOL, which interfaces to PSOL, to solve the linear system C (no Krylov iteration). C----------------------------------------------------------------------- 900 CONTINUE LB = 1 LWK = LB + N CALL DCOPY (N, X, 1, WM(LB), 1) CALL DUSOL (NEQ, TN, Y, SAVF, WM(LB), EWT, N, DELTA, HL0, MNEWT, 1 PSOL, NPSL, X, WM(LOCWP), IWM(LOCIWP), WM(LWK), IFLAG) NNI = NNI + 1 NPS = NPS + NPSL IF (IFLAG .NE. 0) NCFL = NCFL + 1 IF (IFLAG .EQ. 3) IERSL = 1 IF (IFLAG .LT. 0) IERSL = -1 RETURN C----------------------- End of Subroutine DSOLPK ---------------------- END *DECK DSPIOM SUBROUTINE DSPIOM (NEQ, TN, Y, SAVF, B, WGHT, N, MAXL, KMP, DELTA, 1 HL0, JPRE, MNEWT, F, PSOL, NPSL, X, V, HES, IPVT, 2 LIOM, WP, IWP, WK, IFLAG) EXTERNAL F, PSOL INTEGER NEQ,N,MAXL,KMP,JPRE,MNEWT,NPSL,IPVT,LIOM,IWP,IFLAG DOUBLE PRECISION TN,Y,SAVF,B,WGHT,DELTA,HL0,X,V,HES,WP,WK DIMENSION NEQ(*), Y(*), SAVF(*), B(*), WGHT(*), X(*), V(N,*), 1 HES(MAXL,MAXL), IPVT(*), WP(*), IWP(*), WK(*) C----------------------------------------------------------------------- C This routine solves the linear system A * x = b using a scaled C preconditioned version of the Incomplete Orthogonalization Method. C An initial guess of x = 0 is assumed. C----------------------------------------------------------------------- C C On entry C C NEQ = problem size, passed to F and PSOL (NEQ(1) = N). C C TN = current value of t. C C Y = array containing current dependent variable vector. C C SAVF = array containing current value of f(t,y). C C B = the right hand side of the system A*x = b. C B is also used as work space when computing the C final approximation. C (B is the same as V(*,MAXL+1) in the call to DSPIOM.) C C WGHT = array of length N containing scale factors. C 1/WGHT(i) are the diagonal elements of the diagonal C scaling matrix D. C C N = the order of the matrix A, and the lengths C of the vectors Y, SAVF, B, WGHT, and X. C C MAXL = the maximum allowable order of the matrix HES. C C KMP = the number of previous vectors the new vector VNEW C must be made orthogonal to. KMP .le. MAXL. C C DELTA = tolerance on residuals b - A*x in weighted RMS-norm. C C HL0 = current value of (step size h) * (coefficient l0). C C JPRE = preconditioner type flag. C C MNEWT = Newton iteration counter (.ge. 0). C C WK = real work array of length N used by DATV and PSOL. C C WP = real work array used by preconditioner PSOL. C C IWP = integer work array used by preconditioner PSOL. C C On return C C X = the final computed approximation to the solution C of the system A*x = b. C C V = the N by (LIOM+1) array containing the LIOM C orthogonal vectors V(*,1) to V(*,LIOM). C C HES = the LU factorization of the LIOM by LIOM upper C Hessenberg matrix whose entries are the C scaled inner products of A*V(*,k) and V(*,i). C C IPVT = an integer array containg pivoting information. C It is loaded in DHEFA and used in DHESL. C C LIOM = the number of iterations performed, and current C order of the upper Hessenberg matrix HES. C C NPSL = the number of calls to PSOL. C C IFLAG = integer error flag: C 0 means convergence in LIOM iterations, LIOM.le.MAXL. C 1 means the convergence test did not pass in MAXL C iterations, but the residual norm is .lt. 1, C or .lt. norm(b) if MNEWT = 0, and so X is computed. C 2 means the convergence test did not pass in MAXL C iterations, residual .gt. 1, and X is undefined. C 3 means there was a recoverable error in PSOL C caused by the preconditioner being out of date. C -1 means there was a nonrecoverable error in PSOL. C C----------------------------------------------------------------------- INTEGER I, IER, INFO, J, K, LL, LM1 DOUBLE PRECISION BNRM, BNRM0, PROD, RHO, SNORMW, DNRM2, TEM C IFLAG = 0 LIOM = 0 NPSL = 0 C----------------------------------------------------------------------- C The initial residual is the vector b. Apply scaling to b, and test C for an immediate return with X = 0 or X = b. C----------------------------------------------------------------------- DO 10 I = 1,N 10 V(I,1) = B(I)*WGHT(I) BNRM0 = DNRM2 (N, V, 1) BNRM = BNRM0 IF (BNRM0 .GT. DELTA) GO TO 30 IF (MNEWT .GT. 0) GO TO 20 CALL DCOPY (N, B, 1, X, 1) RETURN 20 DO 25 I = 1,N 25 X(I) = 0.0D0 RETURN 30 CONTINUE C Apply inverse of left preconditioner to vector b. -------------------- IER = 0 IF (JPRE .EQ. 0 .OR. JPRE .EQ. 2) GO TO 55 CALL PSOL (NEQ, TN, Y, SAVF, WK, HL0, WP, IWP, B, 1, IER) NPSL = 1 IF (IER .NE. 0) GO TO 300 C Calculate norm of scaled vector V(*,1) and normalize it. ------------- DO 50 I = 1,N 50 V(I,1) = B(I)*WGHT(I) BNRM = DNRM2(N, V, 1) DELTA = DELTA*(BNRM/BNRM0) 55 TEM = 1.0D0/BNRM CALL DSCAL (N, TEM, V(1,1), 1) C Zero out the HES array. ---------------------------------------------- DO 65 J = 1,MAXL DO 60 I = 1,MAXL 60 HES(I,J) = 0.0D0 65 CONTINUE C----------------------------------------------------------------------- C Main loop on LL = l to compute the vectors V(*,2) to V(*,MAXL). C The running product PROD is needed for the convergence test. C----------------------------------------------------------------------- PROD = 1.0D0 DO 90 LL = 1,MAXL LIOM = LL C----------------------------------------------------------------------- C Call routine DATV to compute VNEW = Abar*v(l), where Abar is C the matrix A with scaling and inverse preconditioner factors applied. C Call routine DORTHOG to orthogonalize the new vector vnew = V(*,l+1). C Call routine DHEFA to update the factors of HES. C----------------------------------------------------------------------- CALL DATV (NEQ, Y, SAVF, V(1,LL), WGHT, X, F, PSOL, V(1,LL+1), 1 WK, WP, IWP, HL0, JPRE, IER, NPSL) IF (IER .NE. 0) GO TO 300 CALL DORTHOG (V(1,LL+1), V, HES, N, LL, MAXL, KMP, SNORMW) CALL DHEFA (HES, MAXL, LL, IPVT, INFO, LL) LM1 = LL - 1 IF (LL .GT. 1 .AND. IPVT(LM1) .EQ. LM1) PROD = PROD*HES(LL,LM1) IF (INFO .NE. LL) GO TO 70 C----------------------------------------------------------------------- C The last pivot in HES was found to be zero. C If vnew = 0 or l = MAXL, take an error return with IFLAG = 2. C otherwise, continue the iteration without a convergence test. C----------------------------------------------------------------------- IF (SNORMW .EQ. 0.0D0) GO TO 120 IF (LL .EQ. MAXL) GO TO 120 GO TO 80 C----------------------------------------------------------------------- C Update RHO, the estimate of the norm of the residual b - A*x(l). C test for convergence. If passed, compute approximation x(l). C If failed and l .lt. MAXL, then continue iterating. C----------------------------------------------------------------------- 70 CONTINUE RHO = BNRM*SNORMW*ABS(PROD/HES(LL,LL)) IF (RHO .LE. DELTA) GO TO 200 IF (LL .EQ. MAXL) GO TO 100 C If l .lt. MAXL, store HES(l+1,l) and normalize the vector v(*,l+1). 80 CONTINUE HES(LL+1,LL) = SNORMW TEM = 1.0D0/SNORMW CALL DSCAL (N, TEM, V(1,LL+1), 1) 90 CONTINUE C----------------------------------------------------------------------- C l has reached MAXL without passing the convergence test: C If RHO is not too large, compute a solution anyway and return with C IFLAG = 1. Otherwise return with IFLAG = 2. C----------------------------------------------------------------------- 100 CONTINUE IF (RHO .LE. 1.0D0) GO TO 150 IF (RHO .LE. BNRM .AND. MNEWT .EQ. 0) GO TO 150 120 CONTINUE IFLAG = 2 RETURN 150 IFLAG = 1 C----------------------------------------------------------------------- C Compute the approximation x(l) to the solution. C Since the vector X was used as work space, and the initial guess C of the Newton correction is zero, X must be reset to zero. C----------------------------------------------------------------------- 200 CONTINUE LL = LIOM DO 210 K = 1,LL 210 B(K) = 0.0D0 B(1) = BNRM CALL DHESL (HES, MAXL, LL, IPVT, B) DO 220 K = 1,N 220 X(K) = 0.0D0 DO 230 I = 1,LL CALL DAXPY (N, B(I), V(1,I), 1, X, 1) 230 CONTINUE DO 240 I = 1,N 240 X(I) = X(I)/WGHT(I) IF (JPRE .LE. 1) RETURN CALL PSOL (NEQ, TN, Y, SAVF, WK, HL0, WP, IWP, X, 2, IER) NPSL = NPSL + 1 IF (IER .NE. 0) GO TO 300 RETURN C----------------------------------------------------------------------- C This block handles error returns forced by routine PSOL. C----------------------------------------------------------------------- 300 CONTINUE IF (IER .LT. 0) IFLAG = -1 IF (IER .GT. 0) IFLAG = 3 RETURN C----------------------- End of Subroutine DSPIOM ---------------------- END *DECK DATV SUBROUTINE DATV (NEQ, Y, SAVF, V, WGHT, FTEM, F, PSOL, Z, VTEM, 1 WP, IWP, HL0, JPRE, IER, NPSL) EXTERNAL F, PSOL INTEGER NEQ, IWP, JPRE, IER, NPSL DOUBLE PRECISION Y, SAVF, V, WGHT, FTEM, Z, VTEM, WP, HL0 DIMENSION NEQ(*), Y(*), SAVF(*), V(*), WGHT(*), FTEM(*), Z(*), 1 VTEM(*), WP(*), IWP(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU C----------------------------------------------------------------------- C This routine computes the product C C (D-inverse)*(P1-inverse)*(I - hl0*df/dy)*(P2-inverse)*(D*v), C C where D is a diagonal scaling matrix, and P1 and P2 are the C left and right preconditioning matrices, respectively. C v is assumed to have WRMS norm equal to 1. C The product is stored in z. This is computed by a C difference quotient, a call to F, and two calls to PSOL. C----------------------------------------------------------------------- C C On entry C C NEQ = problem size, passed to F and PSOL (NEQ(1) = N). C C Y = array containing current dependent variable vector. C C SAVF = array containing current value of f(t,y). C C V = real array of length N (can be the same array as Z). C C WGHT = array of length N containing scale factors. C 1/WGHT(i) are the diagonal elements of the matrix D. C C FTEM = work array of length N. C C VTEM = work array of length N used to store the C unscaled version of V. C C WP = real work array used by preconditioner PSOL. C C IWP = integer work array used by preconditioner PSOL. C C HL0 = current value of (step size h) * (coefficient l0). C C JPRE = preconditioner type flag. C C C On return C C Z = array of length N containing desired scaled C matrix-vector product. C C IER = error flag from PSOL. C C NPSL = the number of calls to PSOL. C C In addition, this routine uses the Common variables TN, N, NFE. C----------------------------------------------------------------------- INTEGER I DOUBLE PRECISION FAC, RNORM, DNRM2, TEMPN C C Set VTEM = D * V. DO 10 I = 1,N 10 VTEM(I) = V(I)/WGHT(I) IER = 0 IF (JPRE .GE. 2) GO TO 30 C C JPRE = 0 or 1. Save Y in Z and increment Y by VTEM. CALL DCOPY (N, Y, 1, Z, 1) DO 20 I = 1,N 20 Y(I) = Z(I) + VTEM(I) FAC = HL0 GO TO 60 C C JPRE = 2 or 3. Apply inverse of right preconditioner to VTEM. 30 CONTINUE CALL PSOL (NEQ, TN, Y, SAVF, FTEM, HL0, WP, IWP, VTEM, 2, IER) NPSL = NPSL + 1 IF (IER .NE. 0) RETURN C Calculate L-2 norm of (D-inverse) * VTEM. DO 40 I = 1,N 40 Z(I) = VTEM(I)*WGHT(I) TEMPN = DNRM2 (N, Z, 1) RNORM = 1.0D0/TEMPN C Save Y in Z and increment Y by VTEM/norm. CALL DCOPY (N, Y, 1, Z, 1) DO 50 I = 1,N 50 Y(I) = Z(I) + VTEM(I)*RNORM FAC = HL0*TEMPN C C For all JPRE, call F with incremented Y argument, and restore Y. 60 CONTINUE CALL F (NEQ, TN, Y, FTEM) NFE = NFE + 1 CALL DCOPY (N, Z, 1, Y, 1) C Set Z = (identity - hl0*Jacobian) * VTEM, using difference quotient. DO 70 I = 1,N 70 Z(I) = FTEM(I) - SAVF(I) DO 80 I = 1,N 80 Z(I) = VTEM(I) - FAC*Z(I) C Apply inverse of left preconditioner to Z, if nontrivial. IF (JPRE .EQ. 0 .OR. JPRE .EQ. 2) GO TO 85 CALL PSOL (NEQ, TN, Y, SAVF, FTEM, HL0, WP, IWP, Z, 1, IER) NPSL = NPSL + 1 IF (IER .NE. 0) RETURN 85 CONTINUE C Apply D-inverse to Z and return. DO 90 I = 1,N 90 Z(I) = Z(I)*WGHT(I) RETURN C----------------------- End of Subroutine DATV ------------------------ END *DECK DORTHOG SUBROUTINE DORTHOG (VNEW, V, HES, N, LL, LDHES, KMP, SNORMW) INTEGER N, LL, LDHES, KMP DOUBLE PRECISION VNEW, V, HES, SNORMW DIMENSION VNEW(*), V(N,*), HES(LDHES,*) C----------------------------------------------------------------------- C This routine orthogonalizes the vector VNEW against the previous C KMP vectors in the V array. It uses a modified Gram-Schmidt C orthogonalization procedure with conditional reorthogonalization. C This is the version of 28 may 1986. C----------------------------------------------------------------------- C C On entry C C VNEW = the vector of length N containing a scaled product C of the Jacobian and the vector V(*,LL). C C V = the N x l array containing the previous LL C orthogonal vectors v(*,1) to v(*,LL). C C HES = an LL x LL upper Hessenberg matrix containing, C in HES(i,k), k.lt.LL, scaled inner products of C A*V(*,k) and V(*,i). C C LDHES = the leading dimension of the HES array. C C N = the order of the matrix A, and the length of VNEW. C C LL = the current order of the matrix HES. C C KMP = the number of previous vectors the new vector VNEW C must be made orthogonal to (KMP .le. MAXL). C C C On return C C VNEW = the new vector orthogonal to V(*,i0) to V(*,LL), C where i0 = MAX(1, LL-KMP+1). C C HES = upper Hessenberg matrix with column LL filled in with C scaled inner products of A*V(*,LL) and V(*,i). C C SNORMW = L-2 norm of VNEW. C C----------------------------------------------------------------------- INTEGER I, I0 DOUBLE PRECISION ARG, DDOT, DNRM2, SUMDSQ, TEM, VNRM C C Get norm of unaltered VNEW for later use. ---------------------------- VNRM = DNRM2 (N, VNEW, 1) C----------------------------------------------------------------------- C Do modified Gram-Schmidt on VNEW = A*v(LL). C Scaled inner products give new column of HES. C Projections of earlier vectors are subtracted from VNEW. C----------------------------------------------------------------------- I0 = MAX(1,LL-KMP+1) DO 10 I = I0,LL HES(I,LL) = DDOT (N, V(1,I), 1, VNEW, 1) TEM = -HES(I,LL) CALL DAXPY (N, TEM, V(1,I), 1, VNEW, 1) 10 CONTINUE C----------------------------------------------------------------------- C Compute SNORMW = norm of VNEW. C If VNEW is small compared to its input value (in norm), then C reorthogonalize VNEW to V(*,1) through V(*,LL). C Correct if relative correction exceeds 1000*(unit roundoff). C finally, correct SNORMW using the dot products involved. C----------------------------------------------------------------------- SNORMW = DNRM2 (N, VNEW, 1) IF (VNRM + 0.001D0*SNORMW .NE. VNRM) RETURN SUMDSQ = 0.0D0 DO 30 I = I0,LL TEM = -DDOT (N, V(1,I), 1, VNEW, 1) IF (HES(I,LL) + 0.001D0*TEM .EQ. HES(I,LL)) GO TO 30 HES(I,LL) = HES(I,LL) - TEM CALL DAXPY (N, TEM, V(1,I), 1, VNEW, 1) SUMDSQ = SUMDSQ + TEM**2 30 CONTINUE IF (SUMDSQ .EQ. 0.0D0) RETURN ARG = MAX(0.0D0,SNORMW**2 - SUMDSQ) SNORMW = SQRT(ARG) C RETURN C----------------------- End of Subroutine DORTHOG --------------------- END *DECK DSPIGMR SUBROUTINE DSPIGMR (NEQ, TN, Y, SAVF, B, WGHT, N, MAXL, MAXLP1, 1 KMP, DELTA, HL0, JPRE, MNEWT, F, PSOL, NPSL, X, V, HES, Q, 2 LGMR, WP, IWP, WK, DL, IFLAG) EXTERNAL F, PSOL INTEGER NEQ,N,MAXL,MAXLP1,KMP,JPRE,MNEWT,NPSL,LGMR,IWP,IFLAG DOUBLE PRECISION TN,Y,SAVF,B,WGHT,DELTA,HL0,X,V,HES,Q,WP,WK,DL DIMENSION NEQ(*), Y(*), SAVF(*), B(*), WGHT(*), X(*), V(N,*), 1 HES(MAXLP1,*), Q(*), WP(*), IWP(*), WK(*), DL(*) C----------------------------------------------------------------------- C This routine solves the linear system A * x = b using a scaled C preconditioned version of the Generalized Minimal Residual method. C An initial guess of x = 0 is assumed. C----------------------------------------------------------------------- C C On entry C C NEQ = problem size, passed to F and PSOL (NEQ(1) = N). C C TN = current value of t. C C Y = array containing current dependent variable vector. C C SAVF = array containing current value of f(t,y). C C B = the right hand side of the system A*x = b. C B is also used as work space when computing C the final approximation. C (B is the same as V(*,MAXL+1) in the call to DSPIGMR.) C C WGHT = the vector of length N containing the nonzero C elements of the diagonal scaling matrix. C C N = the order of the matrix A, and the lengths C of the vectors WGHT, B and X. C C MAXL = the maximum allowable order of the matrix HES. C C MAXLP1 = MAXL + 1, used for dynamic dimensioning of HES. C C KMP = the number of previous vectors the new vector VNEW C must be made orthogonal to. KMP .le. MAXL. C C DELTA = tolerance on residuals b - A*x in weighted RMS-norm. C C HL0 = current value of (step size h) * (coefficient l0). C C JPRE = preconditioner type flag. C C MNEWT = Newton iteration counter (.ge. 0). C C WK = real work array used by routine DATV and PSOL. C C DL = real work array used for calculation of the residual C norm RHO when the method is incomplete (KMP .lt. MAXL). C Not needed or referenced in complete case (KMP = MAXL). C C WP = real work array used by preconditioner PSOL. C C IWP = integer work array used by preconditioner PSOL. C C On return C C X = the final computed approximation to the solution C of the system A*x = b. C C LGMR = the number of iterations performed and C the current order of the upper Hessenberg C matrix HES. C C NPSL = the number of calls to PSOL. C C V = the N by (LGMR+1) array containing the LGMR C orthogonal vectors V(*,1) to V(*,LGMR). C C HES = the upper triangular factor of the QR decomposition C of the (LGMR+1) by lgmr upper Hessenberg matrix whose C entries are the scaled inner-products of A*V(*,i) C and V(*,k). C C Q = real array of length 2*MAXL containing the components C of the Givens rotations used in the QR decomposition C of HES. It is loaded in DHEQR and used in DHELS. C C IFLAG = integer error flag: C 0 means convergence in LGMR iterations, LGMR .le. MAXL. C 1 means the convergence test did not pass in MAXL C iterations, but the residual norm is .lt. 1, C or .lt. norm(b) if MNEWT = 0, and so x is computed. C 2 means the convergence test did not pass in MAXL C iterations, residual .gt. 1, and X is undefined. C 3 means there was a recoverable error in PSOL C caused by the preconditioner being out of date. C -1 means there was a nonrecoverable error in PSOL. C C----------------------------------------------------------------------- INTEGER I, IER, INFO, IP1, I2, J, K, LL, LLP1 DOUBLE PRECISION BNRM,BNRM0,C,DLNRM,PROD,RHO,S,SNORMW,DNRM2,TEM C IFLAG = 0 LGMR = 0 NPSL = 0 C----------------------------------------------------------------------- C The initial residual is the vector b. Apply scaling to b, and test C for an immediate return with X = 0 or X = b. C----------------------------------------------------------------------- DO 10 I = 1,N 10 V(I,1) = B(I)*WGHT(I) BNRM0 = DNRM2 (N, V, 1) BNRM = BNRM0 IF (BNRM0 .GT. DELTA) GO TO 30 IF (MNEWT .GT. 0) GO TO 20 CALL DCOPY (N, B, 1, X, 1) RETURN 20 DO 25 I = 1,N 25 X(I) = 0.0D0 RETURN 30 CONTINUE C Apply inverse of left preconditioner to vector b. -------------------- IER = 0 IF (JPRE .EQ. 0 .OR. JPRE .EQ. 2) GO TO 55 CALL PSOL (NEQ, TN, Y, SAVF, WK, HL0, WP, IWP, B, 1, IER) NPSL = 1 IF (IER .NE. 0) GO TO 300 C Calculate norm of scaled vector V(*,1) and normalize it. ------------- DO 50 I = 1,N 50 V(I,1) = B(I)*WGHT(I) BNRM = DNRM2 (N, V, 1) DELTA = DELTA*(BNRM/BNRM0) 55 TEM = 1.0D0/BNRM CALL DSCAL (N, TEM, V(1,1), 1) C Zero out the HES array. ---------------------------------------------- DO 65 J = 1,MAXL DO 60 I = 1,MAXLP1 60 HES(I,J) = 0.0D0 65 CONTINUE C----------------------------------------------------------------------- C Main loop to compute the vectors V(*,2) to V(*,MAXL). C The running product PROD is needed for the convergence test. C----------------------------------------------------------------------- PROD = 1.0D0 DO 90 LL = 1,MAXL LGMR = LL C----------------------------------------------------------------------- C Call routine DATV to compute VNEW = Abar*v(ll), where Abar is C the matrix A with scaling and inverse preconditioner factors applied. C Call routine DORTHOG to orthogonalize the new vector VNEW = V(*,LL+1). C Call routine DHEQR to update the factors of HES. C----------------------------------------------------------------------- CALL DATV (NEQ, Y, SAVF, V(1,LL), WGHT, X, F, PSOL, V(1,LL+1), 1 WK, WP, IWP, HL0, JPRE, IER, NPSL) IF (IER .NE. 0) GO TO 300 CALL DORTHOG (V(1,LL+1), V, HES, N, LL, MAXLP1, KMP, SNORMW) HES(LL+1,LL) = SNORMW CALL DHEQR (HES, MAXLP1, LL, Q, INFO, LL) IF (INFO .EQ. LL) GO TO 120 C----------------------------------------------------------------------- C Update RHO, the estimate of the norm of the residual b - A*xl. C If KMP .lt. MAXL, then the vectors V(*,1),...,V(*,LL+1) are not C necessarily orthogonal for LL .gt. KMP. The vector DL must then C be computed, and its norm used in the calculation of RHO. C----------------------------------------------------------------------- PROD = PROD*Q(2*LL) RHO = ABS(PROD*BNRM) IF ((LL.GT.KMP) .AND. (KMP.LT.MAXL)) THEN IF (LL .EQ. KMP+1) THEN CALL DCOPY (N, V(1,1), 1, DL, 1) DO 75 I = 1,KMP IP1 = I + 1 I2 = I*2 S = Q(I2) C = Q(I2-1) DO 70 K = 1,N 70 DL(K) = S*DL(K) + C*V(K,IP1) 75 CONTINUE ENDIF S = Q(2*LL) C = Q(2*LL-1)/SNORMW LLP1 = LL + 1 DO 80 K = 1,N 80 DL(K) = S*DL(K) + C*V(K,LLP1) DLNRM = DNRM2 (N, DL, 1) RHO = RHO*DLNRM ENDIF C----------------------------------------------------------------------- C Test for convergence. If passed, compute approximation xl. C if failed and LL .lt. MAXL, then continue iterating. C----------------------------------------------------------------------- IF (RHO .LE. DELTA) GO TO 200 IF (LL .EQ. MAXL) GO TO 100 C----------------------------------------------------------------------- C Rescale so that the norm of V(1,LL+1) is one. C----------------------------------------------------------------------- TEM = 1.0D0/SNORMW CALL DSCAL (N, TEM, V(1,LL+1), 1) 90 CONTINUE 100 CONTINUE IF (RHO .LE. 1.0D0) GO TO 150 IF (RHO .LE. BNRM .AND. MNEWT .EQ. 0) GO TO 150 120 CONTINUE IFLAG = 2 RETURN 150 IFLAG = 1 C----------------------------------------------------------------------- C Compute the approximation xl to the solution. C Since the vector X was used as work space, and the initial guess C of the Newton correction is zero, X must be reset to zero. C----------------------------------------------------------------------- 200 CONTINUE LL = LGMR LLP1 = LL + 1 DO 210 K = 1,LLP1 210 B(K) = 0.0D0 B(1) = BNRM CALL DHELS (HES, MAXLP1, LL, Q, B) DO 220 K = 1,N 220 X(K) = 0.0D0 DO 230 I = 1,LL CALL DAXPY (N, B(I), V(1,I), 1, X, 1) 230 CONTINUE DO 240 I = 1,N 240 X(I) = X(I)/WGHT(I) IF (JPRE .LE. 1) RETURN CALL PSOL (NEQ, TN, Y, SAVF, WK, HL0, WP, IWP, X, 2, IER) NPSL = NPSL + 1 IF (IER .NE. 0) GO TO 300 RETURN C----------------------------------------------------------------------- C This block handles error returns forced by routine PSOL. C----------------------------------------------------------------------- 300 CONTINUE IF (IER .LT. 0) IFLAG = -1 IF (IER .GT. 0) IFLAG = 3 C RETURN C----------------------- End of Subroutine DSPIGMR --------------------- END *DECK DPCG SUBROUTINE DPCG (NEQ, TN, Y, SAVF, R, WGHT, N, MAXL, DELTA, HL0, 1 JPRE, MNEWT, F, PSOL, NPSL, X, P, W, Z, LPCG, WP, IWP, WK, IFLAG) EXTERNAL F, PSOL INTEGER NEQ, N, MAXL, JPRE, MNEWT, NPSL, LPCG, IWP, IFLAG DOUBLE PRECISION TN,Y,SAVF,R,WGHT,DELTA,HL0,X,P,W,Z,WP,WK DIMENSION NEQ(*), Y(*), SAVF(*), R(*), WGHT(*), X(*), P(*), W(*), 1 Z(*), WP(*), IWP(*), WK(*) C----------------------------------------------------------------------- C This routine computes the solution to the system A*x = b using a C preconditioned version of the Conjugate Gradient algorithm. C It is assumed here that the matrix A and the preconditioner C matrix M are symmetric positive definite or nearly so. C----------------------------------------------------------------------- C C On entry C C NEQ = problem size, passed to F and PSOL (NEQ(1) = N). C C TN = current value of t. C C Y = array containing current dependent variable vector. C C SAVF = array containing current value of f(t,y). C C R = the right hand side of the system A*x = b. C C WGHT = array of length N containing scale factors. C 1/WGHT(i) are the diagonal elements of the diagonal C scaling matrix D. C C N = the order of the matrix A, and the lengths C of the vectors Y, SAVF, R, WGHT, P, W, Z, WK, and X. C C MAXL = the maximum allowable number of iterates. C C DELTA = tolerance on residuals b - A*x in weighted RMS-norm. C C HL0 = current value of (step size h) * (coefficient l0). C C JPRE = preconditioner type flag. C C MNEWT = Newton iteration counter (.ge. 0). C C WK = real work array used by routine DATP. C C WP = real work array used by preconditioner PSOL. C C IWP = integer work array used by preconditioner PSOL. C C On return C C X = the final computed approximation to the solution C of the system A*x = b. C C LPCG = the number of iterations performed, and current C order of the upper Hessenberg matrix HES. C C NPSL = the number of calls to PSOL. C C IFLAG = integer error flag: C 0 means convergence in LPCG iterations, LPCG .le. MAXL. C 1 means the convergence test did not pass in MAXL C iterations, but the residual norm is .lt. 1, C or .lt. norm(b) if MNEWT = 0, and so X is computed. C 2 means the convergence test did not pass in MAXL C iterations, residual .gt. 1, and X is undefined. C 3 means there was a recoverable error in PSOL C caused by the preconditioner being out of date. C 4 means there was a zero denominator in the algorithm. C The system matrix or preconditioner matrix is not C sufficiently close to being symmetric pos. definite. C -1 means there was a nonrecoverable error in PSOL. C C----------------------------------------------------------------------- INTEGER I, IER DOUBLE PRECISION ALPHA,BETA,BNRM,PTW,RNRM,DDOT,DVNORM,ZTR,ZTR0 C IFLAG = 0 NPSL = 0 LPCG = 0 DO 10 I = 1,N 10 X(I) = 0.0D0 BNRM = DVNORM (N, R, WGHT) C Test for immediate return with X = 0 or X = b. ----------------------- IF (BNRM .GT. DELTA) GO TO 20 IF (MNEWT .GT. 0) RETURN CALL DCOPY (N, R, 1, X, 1) RETURN C 20 ZTR = 0.0D0 C Loop point for PCG iterations. --------------------------------------- 30 CONTINUE LPCG = LPCG + 1 CALL DCOPY (N, R, 1, Z, 1) IER = 0 IF (JPRE .EQ. 0) GO TO 40 CALL PSOL (NEQ, TN, Y, SAVF, WK, HL0, WP, IWP, Z, 3, IER) NPSL = NPSL + 1 IF (IER .NE. 0) GO TO 100 40 CONTINUE ZTR0 = ZTR ZTR = DDOT (N, Z, 1, R, 1) IF (LPCG .NE. 1) GO TO 50 CALL DCOPY (N, Z, 1, P, 1) GO TO 70 50 CONTINUE IF (ZTR0 .EQ. 0.0D0) GO TO 200 BETA = ZTR/ZTR0 DO 60 I = 1,N 60 P(I) = Z(I) + BETA*P(I) 70 CONTINUE C----------------------------------------------------------------------- C Call DATP to compute A*p and return the answer in W. C----------------------------------------------------------------------- CALL DATP (NEQ, Y, SAVF, P, WGHT, HL0, WK, F, W) C PTW = DDOT (N, P, 1, W, 1) IF (PTW .EQ. 0.0D0) GO TO 200 ALPHA = ZTR/PTW CALL DAXPY (N, ALPHA, P, 1, X, 1) ALPHA = -ALPHA CALL DAXPY (N, ALPHA, W, 1, R, 1) RNRM = DVNORM (N, R, WGHT) IF (RNRM .LE. DELTA) RETURN IF (LPCG .LT. MAXL) GO TO 30 IFLAG = 2 IF (RNRM .LE. 1.0D0) IFLAG = 1 IF (RNRM .LE. BNRM .AND. MNEWT .EQ. 0) IFLAG = 1 RETURN C----------------------------------------------------------------------- C This block handles error returns from PSOL. C----------------------------------------------------------------------- 100 CONTINUE IF (IER .LT. 0) IFLAG = -1 IF (IER .GT. 0) IFLAG = 3 RETURN C----------------------------------------------------------------------- C This block handles division by zero errors. C----------------------------------------------------------------------- 200 CONTINUE IFLAG = 4 RETURN C----------------------- End of Subroutine DPCG ------------------------ END *DECK DPCGS SUBROUTINE DPCGS (NEQ, TN, Y, SAVF, R, WGHT, N, MAXL, DELTA, HL0, 1 JPRE, MNEWT, F, PSOL, NPSL, X, P, W, Z, LPCG, WP, IWP, WK, IFLAG) EXTERNAL F, PSOL INTEGER NEQ, N, MAXL, JPRE, MNEWT, NPSL, LPCG, IWP, IFLAG DOUBLE PRECISION TN,Y,SAVF,R,WGHT,DELTA,HL0,X,P,W,Z,WP,WK DIMENSION NEQ(*), Y(*), SAVF(*), R(*), WGHT(*), X(*), P(*), W(*), 1 Z(*), WP(*), IWP(*), WK(*) C----------------------------------------------------------------------- C This routine computes the solution to the system A*x = b using a C scaled preconditioned version of the Conjugate Gradient algorithm. C It is assumed here that the scaled matrix D**-1 * A * D and the C scaled preconditioner D**-1 * M * D are close to being C symmetric positive definite. C----------------------------------------------------------------------- C C On entry C C NEQ = problem size, passed to F and PSOL (NEQ(1) = N). C C TN = current value of t. C C Y = array containing current dependent variable vector. C C SAVF = array containing current value of f(t,y). C C R = the right hand side of the system A*x = b. C C WGHT = array of length N containing scale factors. C 1/WGHT(i) are the diagonal elements of the diagonal C scaling matrix D. C C N = the order of the matrix A, and the lengths C of the vectors Y, SAVF, R, WGHT, P, W, Z, WK, and X. C C MAXL = the maximum allowable number of iterates. C C DELTA = tolerance on residuals b - A*x in weighted RMS-norm. C C HL0 = current value of (step size h) * (coefficient l0). C C JPRE = preconditioner type flag. C C MNEWT = Newton iteration counter (.ge. 0). C C WK = real work array used by routine DATP. C C WP = real work array used by preconditioner PSOL. C C IWP = integer work array used by preconditioner PSOL. C C On return C C X = the final computed approximation to the solution C of the system A*x = b. C C LPCG = the number of iterations performed, and current C order of the upper Hessenberg matrix HES. C C NPSL = the number of calls to PSOL. C C IFLAG = integer error flag: C 0 means convergence in LPCG iterations, LPCG .le. MAXL. C 1 means the convergence test did not pass in MAXL C iterations, but the residual norm is .lt. 1, C or .lt. norm(b) if MNEWT = 0, and so X is computed. C 2 means the convergence test did not pass in MAXL C iterations, residual .gt. 1, and X is undefined. C 3 means there was a recoverable error in PSOL C caused by the preconditioner being out of date. C 4 means there was a zero denominator in the algorithm. C the scaled matrix or scaled preconditioner is not C sufficiently close to being symmetric pos. definite. C -1 means there was a nonrecoverable error in PSOL. C C----------------------------------------------------------------------- INTEGER I, IER DOUBLE PRECISION ALPHA, BETA, BNRM, PTW, RNRM, DVNORM, ZTR, ZTR0 C IFLAG = 0 NPSL = 0 LPCG = 0 DO 10 I = 1,N 10 X(I) = 0.0D0 BNRM = DVNORM (N, R, WGHT) C Test for immediate return with X = 0 or X = b. ----------------------- IF (BNRM .GT. DELTA) GO TO 20 IF (MNEWT .GT. 0) RETURN CALL DCOPY (N, R, 1, X, 1) RETURN C 20 ZTR = 0.0D0 C Loop point for PCG iterations. --------------------------------------- 30 CONTINUE LPCG = LPCG + 1 CALL DCOPY (N, R, 1, Z, 1) IER = 0 IF (JPRE .EQ. 0) GO TO 40 CALL PSOL (NEQ, TN, Y, SAVF, WK, HL0, WP, IWP, Z, 3, IER) NPSL = NPSL + 1 IF (IER .NE. 0) GO TO 100 40 CONTINUE ZTR0 = ZTR ZTR = 0.0D0 DO 45 I = 1,N 45 ZTR = ZTR + Z(I)*R(I)*WGHT(I)**2 IF (LPCG .NE. 1) GO TO 50 CALL DCOPY (N, Z, 1, P, 1) GO TO 70 50 CONTINUE IF (ZTR0 .EQ. 0.0D0) GO TO 200 BETA = ZTR/ZTR0 DO 60 I = 1,N 60 P(I) = Z(I) + BETA*P(I) 70 CONTINUE C----------------------------------------------------------------------- C Call DATP to compute A*p and return the answer in W. C----------------------------------------------------------------------- CALL DATP (NEQ, Y, SAVF, P, WGHT, HL0, WK, F, W) C PTW = 0.0D0 DO 80 I = 1,N 80 PTW = PTW + P(I)*W(I)*WGHT(I)**2 IF (PTW .EQ. 0.0D0) GO TO 200 ALPHA = ZTR/PTW CALL DAXPY (N, ALPHA, P, 1, X, 1) ALPHA = -ALPHA CALL DAXPY (N, ALPHA, W, 1, R, 1) RNRM = DVNORM (N, R, WGHT) IF (RNRM .LE. DELTA) RETURN IF (LPCG .LT. MAXL) GO TO 30 IFLAG = 2 IF (RNRM .LE. 1.0D0) IFLAG = 1 IF (RNRM .LE. BNRM .AND. MNEWT .EQ. 0) IFLAG = 1 RETURN C----------------------------------------------------------------------- C This block handles error returns from PSOL. C----------------------------------------------------------------------- 100 CONTINUE IF (IER .LT. 0) IFLAG = -1 IF (IER .GT. 0) IFLAG = 3 RETURN C----------------------------------------------------------------------- C This block handles division by zero errors. C----------------------------------------------------------------------- 200 CONTINUE IFLAG = 4 RETURN C----------------------- End of Subroutine DPCGS ----------------------- END *DECK DATP SUBROUTINE DATP (NEQ, Y, SAVF, P, WGHT, HL0, WK, F, W) EXTERNAL F INTEGER NEQ DOUBLE PRECISION Y, SAVF, P, WGHT, HL0, WK, W DIMENSION NEQ(*), Y(*), SAVF(*), P(*), WGHT(*), WK(*), W(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU C----------------------------------------------------------------------- C This routine computes the product C C w = (I - hl0*df/dy)*p C C This is computed by a call to F and a difference quotient. C----------------------------------------------------------------------- C C On entry C C NEQ = problem size, passed to F and PSOL (NEQ(1) = N). C C Y = array containing current dependent variable vector. C C SAVF = array containing current value of f(t,y). C C P = real array of length N. C C WGHT = array of length N containing scale factors. C 1/WGHT(i) are the diagonal elements of the matrix D. C C WK = work array of length N. C C On return C C C W = array of length N containing desired C matrix-vector product. C C In addition, this routine uses the Common variables TN, N, NFE. C----------------------------------------------------------------------- INTEGER I DOUBLE PRECISION FAC, PNRM, RPNRM, DVNORM C PNRM = DVNORM (N, P, WGHT) RPNRM = 1.0D0/PNRM CALL DCOPY (N, Y, 1, W, 1) DO 20 I = 1,N 20 Y(I) = W(I) + P(I)*RPNRM CALL F (NEQ, TN, Y, WK) NFE = NFE + 1 CALL DCOPY (N, W, 1, Y, 1) FAC = HL0*PNRM DO 40 I = 1,N 40 W(I) = P(I) - FAC*(WK(I) - SAVF(I)) RETURN C----------------------- End of Subroutine DATP ------------------------ END *DECK DUSOL SUBROUTINE DUSOL (NEQ, TN, Y, SAVF, B, WGHT, N, DELTA, HL0, MNEWT, 1 PSOL, NPSL, X, WP, IWP, WK, IFLAG) EXTERNAL PSOL INTEGER NEQ, N, MNEWT, NPSL, IWP, IFLAG DOUBLE PRECISION TN, Y, SAVF, B, WGHT, DELTA, HL0, X, WP, WK DIMENSION NEQ(*), Y(*), SAVF(*), B(*), WGHT(*), X(*), 1 WP(*), IWP(*), WK(*) C----------------------------------------------------------------------- C This routine solves the linear system A * x = b using only a call C to the user-supplied routine PSOL (no Krylov iteration). C If the norm of the right-hand side vector b is smaller than DELTA, C the vector X returned is X = b (if MNEWT = 0) or X = 0 otherwise. C PSOL is called with an LR argument of 0. C----------------------------------------------------------------------- C C On entry C C NEQ = problem size, passed to F and PSOL (NEQ(1) = N). C C TN = current value of t. C C Y = array containing current dependent variable vector. C C SAVF = array containing current value of f(t,y). C C B = the right hand side of the system A*x = b. C C WGHT = the vector of length N containing the nonzero C elements of the diagonal scaling matrix. C C N = the order of the matrix A, and the lengths C of the vectors WGHT, B and X. C C DELTA = tolerance on residuals b - A*x in weighted RMS-norm. C C HL0 = current value of (step size h) * (coefficient l0). C C MNEWT = Newton iteration counter (.ge. 0). C C WK = real work array used by PSOL. C C WP = real work array used by preconditioner PSOL. C C IWP = integer work array used by preconditioner PSOL. C C On return C C X = the final computed approximation to the solution C of the system A*x = b. C C NPSL = the number of calls to PSOL. C C IFLAG = integer error flag: C 0 means no trouble occurred. C 3 means there was a recoverable error in PSOL C caused by the preconditioner being out of date. C -1 means there was a nonrecoverable error in PSOL. C C----------------------------------------------------------------------- INTEGER I, IER DOUBLE PRECISION BNRM, DVNORM C IFLAG = 0 NPSL = 0 C----------------------------------------------------------------------- C Test for an immediate return with X = 0 or X = b. C----------------------------------------------------------------------- BNRM = DVNORM (N, B, WGHT) IF (BNRM .GT. DELTA) GO TO 30 IF (MNEWT .GT. 0) GO TO 10 CALL DCOPY (N, B, 1, X, 1) RETURN 10 DO 20 I = 1,N 20 X(I) = 0.0D0 RETURN C Make call to PSOL and copy result from B to X. ----------------------- 30 IER = 0 CALL PSOL (NEQ, TN, Y, SAVF, WK, HL0, WP, IWP, B, 0, IER) NPSL = 1 IF (IER .NE. 0) GO TO 100 CALL DCOPY (N, B, 1, X, 1) RETURN C----------------------------------------------------------------------- C This block handles error returns forced by routine PSOL. C----------------------------------------------------------------------- 100 CONTINUE IF (IER .LT. 0) IFLAG = -1 IF (IER .GT. 0) IFLAG = 3 RETURN C----------------------- End of Subroutine DUSOL ----------------------- END *DECK DSRCPK SUBROUTINE DSRCPK (RSAV, ISAV, JOB) C----------------------------------------------------------------------- C This routine saves or restores (depending on JOB) the contents of C the Common blocks DLS001, DLPK01, which are used C internally by the DLSODPK solver. C C RSAV = real array of length 222 or more. C ISAV = integer array of length 50 or more. C JOB = flag indicating to save or restore the Common blocks: C JOB = 1 if Common is to be saved (written to RSAV/ISAV) C JOB = 2 if Common is to be restored (read from RSAV/ISAV) C A call with JOB = 2 presumes a prior call with JOB = 1. C----------------------------------------------------------------------- INTEGER ISAV, JOB INTEGER ILS, ILSP INTEGER I, LENILP, LENRLP, LENILS, LENRLS DOUBLE PRECISION RSAV, RLS, RLSP DIMENSION RSAV(*), ISAV(*) SAVE LENRLS, LENILS, LENRLP, LENILP COMMON /DLS001/ RLS(218), ILS(37) COMMON /DLPK01/ RLSP(4), ILSP(13) DATA LENRLS/218/, LENILS/37/, LENRLP/4/, LENILP/13/ C IF (JOB .EQ. 2) GO TO 100 CALL DCOPY (LENRLS, RLS, 1, RSAV, 1) CALL DCOPY (LENRLP, RLSP, 1, RSAV(LENRLS+1), 1) DO 20 I = 1,LENILS 20 ISAV(I) = ILS(I) DO 40 I = 1,LENILP 40 ISAV(LENILS+I) = ILSP(I) RETURN C 100 CONTINUE CALL DCOPY (LENRLS, RSAV, 1, RLS, 1) CALL DCOPY (LENRLP, RSAV(LENRLS+1), 1, RLSP, 1) DO 120 I = 1,LENILS 120 ILS(I) = ISAV(I) DO 140 I = 1,LENILP 140 ILSP(I) = ISAV(LENILS+I) RETURN C----------------------- End of Subroutine DSRCPK ---------------------- END *DECK DHEFA SUBROUTINE DHEFA (A, LDA, N, IPVT, INFO, JOB) INTEGER LDA, N, IPVT(*), INFO, JOB DOUBLE PRECISION A(LDA,*) C----------------------------------------------------------------------- C This routine is a modification of the LINPACK routine DGEFA and C performs an LU decomposition of an upper Hessenberg matrix A. C There are two options available: C C (1) performing a fresh factorization C (2) updating the LU factors by adding a row and a C column to the matrix A. C----------------------------------------------------------------------- C DHEFA factors an upper Hessenberg matrix by elimination. C C On entry C C A DOUBLE PRECISION(LDA, N) C the matrix to be factored. C C LDA INTEGER C the leading dimension of the array A . C C N INTEGER C the order of the matrix A . C C JOB INTEGER C JOB = 1 means that a fresh factorization of the C matrix A is desired. C JOB .ge. 2 means that the current factorization of A C will be updated by the addition of a row C and a column. C C On return C C A an upper triangular matrix and the multipliers C which were used to obtain it. C The factorization can be written A = L*U where C L is a product of permutation and unit lower C triangular matrices and U is upper triangular. C C IPVT INTEGER(N) C an integer vector of pivot indices. C C INFO INTEGER C = 0 normal value. C = k if U(k,k) .eq. 0.0 . This is not an error C condition for this subroutine, but it does C indicate that DHESL will divide by zero if called. C C Modification of LINPACK, by Peter Brown, LLNL. C Written 7/20/83. This version dated 6/20/01. C C BLAS called: DAXPY, IDAMAX C----------------------------------------------------------------------- INTEGER IDAMAX, J, K, KM1, KP1, L, NM1 DOUBLE PRECISION T C IF (JOB .GT. 1) GO TO 80 C C A new facorization is desired. This is essentially the LINPACK C code with the exception that we know there is only one nonzero C element below the main diagonal. C C Gaussian elimination with partial pivoting C INFO = 0 NM1 = N - 1 IF (NM1 .LT. 1) GO TO 70 DO 60 K = 1, NM1 KP1 = K + 1 C C Find L = pivot index C L = IDAMAX (2, A(K,K), 1) + K - 1 IPVT(K) = L C C Zero pivot implies this column already triangularized C IF (A(L,K) .EQ. 0.0D0) GO TO 40 C C Interchange if necessary C IF (L .EQ. K) GO TO 10 T = A(L,K) A(L,K) = A(K,K) A(K,K) = T 10 CONTINUE C C Compute multipliers C T = -1.0D0/A(K,K) A(K+1,K) = A(K+1,K)*T C C Row elimination with column indexing C DO 30 J = KP1, N T = A(L,J) IF (L .EQ. K) GO TO 20 A(L,J) = A(K,J) A(K,J) = T 20 CONTINUE CALL DAXPY (N-K, T, A(K+1,K), 1, A(K+1,J), 1) 30 CONTINUE GO TO 50 40 CONTINUE INFO = K 50 CONTINUE 60 CONTINUE 70 CONTINUE IPVT(N) = N IF (A(N,N) .EQ. 0.0D0) INFO = N RETURN C C The old factorization of A will be updated. A row and a column C has been added to the matrix A. C N-1 is now the old order of the matrix. C 80 CONTINUE NM1 = N - 1 C C Perform row interchanges on the elements of the new column, and C perform elimination operations on the elements using the multipliers. C IF (NM1 .LE. 1) GO TO 105 DO 100 K = 2,NM1 KM1 = K - 1 L = IPVT(KM1) T = A(L,N) IF (L .EQ. KM1) GO TO 90 A(L,N) = A(KM1,N) A(KM1,N) = T 90 CONTINUE A(K,N) = A(K,N) + A(K,KM1)*T 100 CONTINUE 105 CONTINUE C C Complete update of factorization by decomposing last 2x2 block. C INFO = 0 C C Find L = pivot index C L = IDAMAX (2, A(NM1,NM1), 1) + NM1 - 1 IPVT(NM1) = L C C Zero pivot implies this column already triangularized C IF (A(L,NM1) .EQ. 0.0D0) GO TO 140 C C Interchange if necessary C IF (L .EQ. NM1) GO TO 110 T = A(L,NM1) A(L,NM1) = A(NM1,NM1) A(NM1,NM1) = T 110 CONTINUE C C Compute multipliers C T = -1.0D0/A(NM1,NM1) A(N,NM1) = A(N,NM1)*T C C Row elimination with column indexing C T = A(L,N) IF (L .EQ. NM1) GO TO 120 A(L,N) = A(NM1,N) A(NM1,N) = T 120 CONTINUE A(N,N) = A(N,N) + T*A(N,NM1) GO TO 150 140 CONTINUE INFO = NM1 150 CONTINUE IPVT(N) = N IF (A(N,N) .EQ. 0.0D0) INFO = N RETURN C----------------------- End of Subroutine DHEFA ----------------------- END *DECK DHESL SUBROUTINE DHESL (A, LDA, N, IPVT, B) INTEGER LDA, N, IPVT(*) DOUBLE PRECISION A(LDA,*), B(*) C----------------------------------------------------------------------- C This is essentially the LINPACK routine DGESL except for changes C due to the fact that A is an upper Hessenberg matrix. C----------------------------------------------------------------------- C DHESL solves the real system A * x = b C using the factors computed by DHEFA. C C On entry C C A DOUBLE PRECISION(LDA, N) C the output from DHEFA. C C LDA INTEGER C the leading dimension of the array A . C C N INTEGER C the order of the matrix A . C C IPVT INTEGER(N) C the pivot vector from DHEFA. C C B DOUBLE PRECISION(N) C the right hand side vector. C C On return C C B the solution vector x . C C Modification of LINPACK, by Peter Brown, LLNL. C Written 7/20/83. This version dated 6/20/01. C C BLAS called: DAXPY C----------------------------------------------------------------------- INTEGER K, KB, L, NM1 DOUBLE PRECISION T C NM1 = N - 1 C C Solve A * x = b C First solve L*y = b C IF (NM1 .LT. 1) GO TO 30 DO 20 K = 1, NM1 L = IPVT(K) T = B(L) IF (L .EQ. K) GO TO 10 B(L) = B(K) B(K) = T 10 CONTINUE B(K+1) = B(K+1) + T*A(K+1,K) 20 CONTINUE 30 CONTINUE C C Now solve U*x = y C DO 40 KB = 1, N K = N + 1 - KB B(K) = B(K)/A(K,K) T = -B(K) CALL DAXPY (K-1, T, A(1,K), 1, B(1), 1) 40 CONTINUE RETURN C----------------------- End of Subroutine DHESL ----------------------- END *DECK DHEQR SUBROUTINE DHEQR (A, LDA, N, Q, INFO, IJOB) INTEGER LDA, N, INFO, IJOB DOUBLE PRECISION A(LDA,*), Q(*) C----------------------------------------------------------------------- C This routine performs a QR decomposition of an upper C Hessenberg matrix A. There are two options available: C C (1) performing a fresh decomposition C (2) updating the QR factors by adding a row and a C column to the matrix A. C----------------------------------------------------------------------- C DHEQR decomposes an upper Hessenberg matrix by using Givens C rotations. C C On entry C C A DOUBLE PRECISION(LDA, N) C the matrix to be decomposed. C C LDA INTEGER C the leading dimension of the array A . C C N INTEGER C A is an (N+1) by N Hessenberg matrix. C C IJOB INTEGER C = 1 means that a fresh decomposition of the C matrix A is desired. C .ge. 2 means that the current decomposition of A C will be updated by the addition of a row C and a column. C On return C C A the upper triangular matrix R. C The factorization can be written Q*A = R, where C Q is a product of Givens rotations and R is upper C triangular. C C Q DOUBLE PRECISION(2*N) C the factors c and s of each Givens rotation used C in decomposing A. C C INFO INTEGER C = 0 normal value. C = k if A(k,k) .eq. 0.0 . This is not an error C condition for this subroutine, but it does C indicate that DHELS will divide by zero C if called. C C Modification of LINPACK, by Peter Brown, LLNL. C Written 1/13/86. This version dated 6/20/01. C----------------------------------------------------------------------- INTEGER I, IQ, J, K, KM1, KP1, NM1 DOUBLE PRECISION C, S, T, T1, T2 C IF (IJOB .GT. 1) GO TO 70 C C A new facorization is desired. C C QR decomposition without pivoting C INFO = 0 DO 60 K = 1, N KM1 = K - 1 KP1 = K + 1 C C Compute kth column of R. C First, multiply the kth column of A by the previous C k-1 Givens rotations. C IF (KM1 .LT. 1) GO TO 20 DO 10 J = 1, KM1 I = 2*(J-1) + 1 T1 = A(J,K) T2 = A(J+1,K) C = Q(I) S = Q(I+1) A(J,K) = C*T1 - S*T2 A(J+1,K) = S*T1 + C*T2 10 CONTINUE C C Compute Givens components c and s C 20 CONTINUE IQ = 2*KM1 + 1 T1 = A(K,K) T2 = A(KP1,K) IF (T2 .NE. 0.0D0) GO TO 30 C = 1.0D0 S = 0.0D0 GO TO 50 30 CONTINUE IF (ABS(T2) .LT. ABS(T1)) GO TO 40 T = T1/T2 S = -1.0D0/SQRT(1.0D0+T*T) C = -S*T GO TO 50 40 CONTINUE T = T2/T1 C = 1.0D0/SQRT(1.0D0+T*T) S = -C*T 50 CONTINUE Q(IQ) = C Q(IQ+1) = S A(K,K) = C*T1 - S*T2 IF (A(K,K) .EQ. 0.0D0) INFO = K 60 CONTINUE RETURN C C The old factorization of A will be updated. A row and a column C has been added to the matrix A. C N by N-1 is now the old size of the matrix. C 70 CONTINUE NM1 = N - 1 C C Multiply the new column by the N previous Givens rotations. C DO 100 K = 1,NM1 I = 2*(K-1) + 1 T1 = A(K,N) T2 = A(K+1,N) C = Q(I) S = Q(I+1) A(K,N) = C*T1 - S*T2 A(K+1,N) = S*T1 + C*T2 100 CONTINUE C C Complete update of decomposition by forming last Givens rotation, C and multiplying it times the column vector (A(N,N), A(N+1,N)). C INFO = 0 T1 = A(N,N) T2 = A(N+1,N) IF (T2 .NE. 0.0D0) GO TO 110 C = 1.0D0 S = 0.0D0 GO TO 130 110 CONTINUE IF (ABS(T2) .LT. ABS(T1)) GO TO 120 T = T1/T2 S = -1.0D0/SQRT(1.0D0+T*T) C = -S*T GO TO 130 120 CONTINUE T = T2/T1 C = 1.0D0/SQRT(1.0D0+T*T) S = -C*T 130 CONTINUE IQ = 2*N - 1 Q(IQ) = C Q(IQ+1) = S A(N,N) = C*T1 - S*T2 IF (A(N,N) .EQ. 0.0D0) INFO = N RETURN C----------------------- End of Subroutine DHEQR ----------------------- END *DECK DHELS SUBROUTINE DHELS (A, LDA, N, Q, B) INTEGER LDA, N DOUBLE PRECISION A(LDA,*), B(*), Q(*) C----------------------------------------------------------------------- C This is part of the LINPACK routine DGESL with changes C due to the fact that A is an upper Hessenberg matrix. C----------------------------------------------------------------------- C DHELS solves the least squares problem C C min (b-A*x, b-A*x) C C using the factors computed by DHEQR. C C On entry C C A DOUBLE PRECISION(LDA, N) C the output from DHEQR which contains the upper C triangular factor R in the QR decomposition of A. C C LDA INTEGER C the leading dimension of the array A . C C N INTEGER C A is originally an (N+1) by N matrix. C C Q DOUBLE PRECISION(2*N) C The coefficients of the N givens rotations C used in the QR factorization of A. C C B DOUBLE PRECISION(N+1) C the right hand side vector. C C On return C C B the solution vector x . C C Modification of LINPACK, by Peter Brown, LLNL. C Written 1/13/86. This version dated 6/20/01. C C BLAS called: DAXPY C----------------------------------------------------------------------- INTEGER IQ, K, KB, KP1 DOUBLE PRECISION C, S, T, T1, T2 C C Minimize (b-A*x, b-A*x) C First form Q*b. C DO 20 K = 1, N KP1 = K + 1 IQ = 2*(K-1) + 1 C = Q(IQ) S = Q(IQ+1) T1 = B(K) T2 = B(KP1) B(K) = C*T1 - S*T2 B(KP1) = S*T1 + C*T2 20 CONTINUE C C Now solve R*x = Q*b. C DO 40 KB = 1, N K = N + 1 - KB B(K) = B(K)/A(K,K) T = -B(K) CALL DAXPY (K-1, T, A(1,K), 1, B(1), 1) 40 CONTINUE RETURN C----------------------- End of Subroutine DHELS ----------------------- END *DECK DLHIN SUBROUTINE DLHIN (NEQ, N, T0, Y0, YDOT, F, TOUT, UROUND, 1 EWT, ITOL, ATOL, Y, TEMP, H0, NITER, IER) EXTERNAL F DOUBLE PRECISION T0, Y0, YDOT, TOUT, UROUND, EWT, ATOL, Y, 1 TEMP, H0 INTEGER NEQ, N, ITOL, NITER, IER DIMENSION NEQ(*), Y0(*), YDOT(*), EWT(*), ATOL(*), Y(*), TEMP(*) C----------------------------------------------------------------------- C Call sequence input -- NEQ, N, T0, Y0, YDOT, F, TOUT, UROUND, C EWT, ITOL, ATOL, Y, TEMP C Call sequence output -- H0, NITER, IER C Common block variables accessed -- None C C Subroutines called by DLHIN: F, DCOPY C Function routines called by DLHIN: DVNORM C----------------------------------------------------------------------- C This routine computes the step size, H0, to be attempted on the C first step, when the user has not supplied a value for this. C C First we check that TOUT - T0 differs significantly from zero. Then C an iteration is done to approximate the initial second derivative C and this is used to define H from WRMS-norm(H**2 * yddot / 2) = 1. C A bias factor of 1/2 is applied to the resulting h. C The sign of H0 is inferred from the initial values of TOUT and T0. C C Communication with DLHIN is done with the following variables: C C NEQ = NEQ array of solver, passed to F. C N = size of ODE system, input. C T0 = initial value of independent variable, input. C Y0 = vector of initial conditions, input. C YDOT = vector of initial first derivatives, input. C F = name of subroutine for right-hand side f(t,y), input. C TOUT = first output value of independent variable C UROUND = machine unit roundoff C EWT, ITOL, ATOL = error weights and tolerance parameters C as described in the driver routine, input. C Y, TEMP = work arrays of length N. C H0 = step size to be attempted, output. C NITER = number of iterations (and of f evaluations) to compute H0, C output. C IER = the error flag, returned with the value C IER = 0 if no trouble occurred, or C IER = -1 if TOUT and t0 are considered too close to proceed. C----------------------------------------------------------------------- C C Type declarations for local variables -------------------------------- C DOUBLE PRECISION AFI, ATOLI, DELYI, HALF, HG, HLB, HNEW, HRAT, 1 HUB, HUN, PT1, T1, TDIST, TROUND, TWO, DVNORM, YDDNRM INTEGER I, ITER C----------------------------------------------------------------------- C The following Fortran-77 declaration is to cause the values of the C listed (local) variables to be saved between calls to this integrator. C----------------------------------------------------------------------- SAVE HALF, HUN, PT1, TWO DATA HALF /0.5D0/, HUN /100.0D0/, PT1 /0.1D0/, TWO /2.0D0/ C NITER = 0 TDIST = ABS(TOUT - T0) TROUND = UROUND*MAX(ABS(T0),ABS(TOUT)) IF (TDIST .LT. TWO*TROUND) GO TO 100 C C Set a lower bound on H based on the roundoff level in T0 and TOUT. --- HLB = HUN*TROUND C Set an upper bound on H based on TOUT-T0 and the initial Y and YDOT. - HUB = PT1*TDIST ATOLI = ATOL(1) DO 10 I = 1,N IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I) DELYI = PT1*ABS(Y0(I)) + ATOLI AFI = ABS(YDOT(I)) IF (AFI*HUB .GT. DELYI) HUB = DELYI/AFI 10 CONTINUE C C Set initial guess for H as geometric mean of upper and lower bounds. - ITER = 0 HG = SQRT(HLB*HUB) C If the bounds have crossed, exit with the mean value. ---------------- IF (HUB .LT. HLB) THEN H0 = HG GO TO 90 ENDIF C C Looping point for iteration. ----------------------------------------- 50 CONTINUE C Estimate the second derivative as a difference quotient in f. -------- T1 = T0 + HG DO 60 I = 1,N 60 Y(I) = Y0(I) + HG*YDOT(I) CALL F (NEQ, T1, Y, TEMP) DO 70 I = 1,N 70 TEMP(I) = (TEMP(I) - YDOT(I))/HG YDDNRM = DVNORM (N, TEMP, EWT) C Get the corresponding new value of H. -------------------------------- IF (YDDNRM*HUB*HUB .GT. TWO) THEN HNEW = SQRT(TWO/YDDNRM) ELSE HNEW = SQRT(HG*HUB) ENDIF ITER = ITER + 1 C----------------------------------------------------------------------- C Test the stopping conditions. C Stop if the new and previous H values differ by a factor of .lt. 2. C Stop if four iterations have been done. Also, stop with previous H C if hnew/hg .gt. 2 after first iteration, as this probably means that C the second derivative value is bad because of cancellation error. C----------------------------------------------------------------------- IF (ITER .GE. 4) GO TO 80 HRAT = HNEW/HG IF ( (HRAT .GT. HALF) .AND. (HRAT .LT. TWO) ) GO TO 80 IF ( (ITER .GE. 2) .AND. (HNEW .GT. TWO*HG) ) THEN HNEW = HG GO TO 80 ENDIF HG = HNEW GO TO 50 C C Iteration done. Apply bounds, bias factor, and sign. ---------------- 80 H0 = HNEW*HALF IF (H0 .LT. HLB) H0 = HLB IF (H0 .GT. HUB) H0 = HUB 90 H0 = SIGN(H0, TOUT - T0) C Restore Y array from Y0, then exit. ---------------------------------- CALL DCOPY (N, Y0, 1, Y, 1) NITER = ITER IER = 0 RETURN C Error return for TOUT - T0 too small. -------------------------------- 100 IER = -1 RETURN C----------------------- End of Subroutine DLHIN ----------------------- END *DECK DSTOKA SUBROUTINE DSTOKA (NEQ, Y, YH, NYH, YH1, EWT, SAVF, SAVX, ACOR, 1 WM, IWM, F, JAC, PSOL) EXTERNAL F, JAC, PSOL INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, YH1, EWT, SAVF, SAVX, ACOR, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*), 1 SAVX(*), ACOR(*), WM(*), IWM(*) INTEGER IOWND, IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER NEWT, NSFI, NSLJ, NJEV INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 1 NNI, NLI, NPS, NCFN, NCFL DOUBLE PRECISION CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO, 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION STIFR DOUBLE PRECISION DELT, EPCON, SQRTN, RSQRTN COMMON /DLS001/ CONIT, CRATE, EL(13), ELCO(13,12), 1 HOLD, RMAX, TESCO(3,12), 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 3 IOWND(6), IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLS002/ STIFR, NEWT, NSFI, NSLJ, NJEV COMMON /DLPK01/ DELT, EPCON, SQRTN, RSQRTN, 1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 2 NNI, NLI, NPS, NCFN, NCFL C----------------------------------------------------------------------- C DSTOKA performs one step of the integration of an initial value C problem for a system of Ordinary Differential Equations. C C This routine was derived from Subroutine DSTODPK in the DLSODPK C package by the addition of automatic functional/Newton iteration C switching and logic for re-use of Jacobian data. C----------------------------------------------------------------------- C Note: DSTOKA is independent of the value of the iteration method C indicator MITER, when this is .ne. 0, and hence is independent C of the type of chord method used, or the Jacobian structure. C Communication with DSTOKA is done with the following variables: C C NEQ = integer array containing problem size in NEQ(1), and C passed as the NEQ argument in all calls to F and JAC. C Y = an array of length .ge. N used as the Y argument in C all calls to F and JAC. C YH = an NYH by LMAX array containing the dependent variables C and their approximate scaled derivatives, where C LMAX = MAXORD + 1. YH(i,j+1) contains the approximate C j-th derivative of y(i), scaled by H**j/factorial(j) C (j = 0,1,...,NQ). On entry for the first step, the first C two columns of YH must be set from the initial values. C NYH = a constant integer .ge. N, the first dimension of YH. C YH1 = a one-dimensional array occupying the same space as YH. C EWT = an array of length N containing multiplicative weights C for local error measurements. Local errors in y(i) are C compared to 1.0/EWT(i) in various error tests. C SAVF = an array of working storage, of length N. C Also used for input of YH(*,MAXORD+2) when JSTART = -1 C and MAXORD .lt. the current order NQ. C SAVX = an array of working storage, of length N. C ACOR = a work array of length N, used for the accumulated C corrections. On a successful return, ACOR(i) contains C the estimated one-step local error in y(i). C WM,IWM = real and integer work arrays associated with matrix C operations in chord iteration (MITER .ne. 0). C CCMAX = maximum relative change in H*EL0 before DSETPK is called. C H = the step size to be attempted on the next step. C H is altered by the error control algorithm during the C problem. H can be either positive or negative, but its C sign must remain constant throughout the problem. C HMIN = the minimum absolute value of the step size H to be used. C HMXI = inverse of the maximum absolute value of H to be used. C HMXI = 0.0 is allowed and corresponds to an infinite HMAX. C HMIN and HMXI may be changed at any time, but will not C take effect until the next change of H is considered. C TN = the independent variable. TN is updated on each step taken. C JSTART = an integer used for input only, with the following C values and meanings: C 0 perform the first step. C .gt.0 take a new step continuing from the last. C -1 take the next step with a new value of H, MAXORD, C N, METH, MITER, and/or matrix parameters. C -2 take the next step with a new value of H, C but with other inputs unchanged. C On return, JSTART is set to 1 to facilitate continuation. C KFLAG = a completion code with the following meanings: C 0 the step was succesful. C -1 the requested error could not be achieved. C -2 corrector convergence could not be achieved. C -3 fatal error in DSETPK or DSOLPK. C A return with KFLAG = -1 or -2 means either C ABS(H) = HMIN or 10 consecutive failures occurred. C On a return with KFLAG negative, the values of TN and C the YH array are as of the beginning of the last C step, and H is the last step size attempted. C MAXORD = the maximum order of integration method to be allowed. C MAXCOR = the maximum number of corrector iterations allowed. C MSBP = maximum number of steps between DSETPK calls (MITER .gt. 0). C MXNCF = maximum number of convergence failures allowed. C METH/MITER = the method flags. See description in driver. C N = the number of first-order differential equations. C----------------------------------------------------------------------- INTEGER I, I1, IREDO, IRET, J, JB, JOK, M, NCF, NEWQ, NSLOW DOUBLE PRECISION DCON, DDN, DEL, DELP, DRC, DSM, DUP, EXDN, EXSM, 1 EXUP, DFNORM, R, RH, RHDN, RHSM, RHUP, ROC, STIFF, TOLD, DVNORM C KFLAG = 0 TOLD = TN NCF = 0 IERPJ = 0 IERSL = 0 JCUR = 0 ICF = 0 DELP = 0.0D0 IF (JSTART .GT. 0) GO TO 200 IF (JSTART .EQ. -1) GO TO 100 IF (JSTART .EQ. -2) GO TO 160 C----------------------------------------------------------------------- C On the first call, the order is set to 1, and other variables are C initialized. RMAX is the maximum ratio by which H can be increased C in a single step. It is initially 1.E4 to compensate for the small C initial H, but then is normally equal to 10. If a failure C occurs (in corrector convergence or error test), RMAX is set at 2 C for the next increase. C----------------------------------------------------------------------- LMAX = MAXORD + 1 NQ = 1 L = 2 IALTH = 2 RMAX = 10000.0D0 RC = 0.0D0 EL0 = 1.0D0 CRATE = 0.7D0 HOLD = H MEO = METH NSLP = 0 NSLJ = 0 IPUP = 0 IRET = 3 NEWT = 0 STIFR = 0.0D0 GO TO 140 C----------------------------------------------------------------------- C The following block handles preliminaries needed when JSTART = -1. C IPUP is set to MITER to force a matrix update. C If an order increase is about to be considered (IALTH = 1), C IALTH is reset to 2 to postpone consideration one more step. C If the caller has changed METH, DCFODE is called to reset C the coefficients of the method. C If the caller has changed MAXORD to a value less than the current C order NQ, NQ is reduced to MAXORD, and a new H chosen accordingly. C If H is to be changed, YH must be rescaled. C If H or METH is being changed, IALTH is reset to L = NQ + 1 C to prevent further changes in H for that many steps. C----------------------------------------------------------------------- 100 IPUP = MITER LMAX = MAXORD + 1 IF (IALTH .EQ. 1) IALTH = 2 IF (METH .EQ. MEO) GO TO 110 CALL DCFODE (METH, ELCO, TESCO) MEO = METH IF (NQ .GT. MAXORD) GO TO 120 IALTH = L IRET = 1 GO TO 150 110 IF (NQ .LE. MAXORD) GO TO 160 120 NQ = MAXORD L = LMAX DO 125 I = 1,L 125 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) EPCON = CONIT*TESCO(2,NQ) DDN = DVNORM (N, SAVF, EWT)/TESCO(1,L) EXDN = 1.0D0/L RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) RH = MIN(RHDN,1.0D0) IREDO = 3 IF (H .EQ. HOLD) GO TO 170 RH = MIN(RH,ABS(H/HOLD)) H = HOLD GO TO 175 C----------------------------------------------------------------------- C DCFODE is called to get all the integration coefficients for the C current METH. Then the EL vector and related constants are reset C whenever the order NQ is changed, or at the start of the problem. C----------------------------------------------------------------------- 140 CALL DCFODE (METH, ELCO, TESCO) 150 DO 155 I = 1,L 155 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) EPCON = CONIT*TESCO(2,NQ) GO TO (160, 170, 200), IRET C----------------------------------------------------------------------- C If H is being changed, the H ratio RH is checked against C RMAX, HMIN, and HMXI, and the YH array rescaled. IALTH is set to C L = NQ + 1 to prevent a change of H for that many steps, unless C forced by a convergence or error test failure. C----------------------------------------------------------------------- 160 IF (H .EQ. HOLD) GO TO 200 RH = H/HOLD H = HOLD IREDO = 3 GO TO 175 170 RH = MAX(RH,HMIN/ABS(H)) 175 RH = MIN(RH,RMAX) RH = RH/MAX(1.0D0,ABS(H)*HMXI*RH) R = 1.0D0 DO 180 J = 2,L R = R*RH DO 180 I = 1,N 180 YH(I,J) = YH(I,J)*R H = H*RH RC = RC*RH IALTH = L IF (IREDO .EQ. 0) GO TO 690 C----------------------------------------------------------------------- C This section computes the predicted values by effectively C multiplying the YH array by the Pascal triangle matrix. C The flag IPUP is set according to whether matrix data is involved C (NEWT .gt. 0 .and. JACFLG .ne. 0) or not, to trigger a call to DSETPK. C IPUP is set to MITER when RC differs from 1 by more than CCMAX, C and at least every MSBP steps, when JACFLG = 1. C RC is the ratio of new to old values of the coefficient H*EL(1). C----------------------------------------------------------------------- 200 IF (NEWT .EQ. 0 .OR. JACFLG .EQ. 0) THEN DRC = 0.0D0 IPUP = 0 CRATE = 0.7D0 ELSE DRC = ABS(RC - 1.0D0) IF (DRC .GT. CCMAX) IPUP = MITER IF (NST .GE. NSLP+MSBP) IPUP = MITER ENDIF TN = TN + H I1 = NQNYH + 1 DO 215 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 210 I = I1,NQNYH 210 YH1(I) = YH1(I) + YH1(I+NYH) 215 CONTINUE C----------------------------------------------------------------------- C Up to MAXCOR corrector iterations are taken. A convergence test is C made on the RMS-norm of each correction, weighted by the error C weight vector EWT. The sum of the corrections is accumulated in the C vector ACOR(i). The YH array is not altered in the corrector loop. C Within the corrector loop, an estimated rate of convergence (ROC) C and a stiffness ratio estimate (STIFF) are kept. Corresponding C global estimates are kept as CRATE and stifr. C----------------------------------------------------------------------- 220 M = 0 MNEWT = 0 STIFF = 0.0D0 ROC = 0.05D0 NSLOW = 0 DO 230 I = 1,N 230 Y(I) = YH(I,1) CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 IF (NEWT .EQ. 0 .OR. IPUP .LE. 0) GO TO 250 C----------------------------------------------------------------------- C If indicated, DSETPK is called to update any matrix data needed, C before starting the corrector iteration. C JOK is set to indicate if the matrix data need not be recomputed. C IPUP is set to 0 as an indicator that the matrix data is up to date. C----------------------------------------------------------------------- JOK = 1 IF (NST .EQ. 0 .OR. NST .GT. NSLJ+50) JOK = -1 IF (ICF .EQ. 1 .AND. DRC .LT. 0.2D0) JOK = -1 IF (ICF .EQ. 2) JOK = -1 IF (JOK .EQ. -1) THEN NSLJ = NST NJEV = NJEV + 1 ENDIF CALL DSETPK (NEQ, Y, YH1, EWT, ACOR, SAVF, JOK, WM, IWM, F, JAC) IPUP = 0 RC = 1.0D0 DRC = 0.0D0 NSLP = NST CRATE = 0.7D0 IF (IERPJ .NE. 0) GO TO 430 250 DO 260 I = 1,N 260 ACOR(I) = 0.0D0 270 IF (NEWT .NE. 0) GO TO 350 C----------------------------------------------------------------------- C In the case of functional iteration, update Y directly from C the result of the last function evaluation, and STIFF is set to 1.0. C----------------------------------------------------------------------- DO 290 I = 1,N SAVF(I) = H*SAVF(I) - YH(I,2) 290 Y(I) = SAVF(I) - ACOR(I) DEL = DVNORM (N, Y, EWT) DO 300 I = 1,N Y(I) = YH(I,1) + EL(1)*SAVF(I) 300 ACOR(I) = SAVF(I) STIFF = 1.0D0 GO TO 400 C----------------------------------------------------------------------- C In the case of the chord method, compute the corrector error, C and solve the linear system with that as right-hand side and C P as coefficient matrix. STIFF is set to the ratio of the norms C of the residual and the correction vector. C----------------------------------------------------------------------- 350 DO 360 I = 1,N 360 SAVX(I) = H*SAVF(I) - (YH(I,2) + ACOR(I)) DFNORM = DVNORM (N, SAVX, EWT) CALL DSOLPK (NEQ, Y, SAVF, SAVX, EWT, WM, IWM, F, PSOL) IF (IERSL .LT. 0) GO TO 430 IF (IERSL .GT. 0) GO TO 410 DEL = DVNORM (N, SAVX, EWT) IF (DEL .GT. 1.0D-8) STIFF = MAX(STIFF, DFNORM/DEL) DO 380 I = 1,N ACOR(I) = ACOR(I) + SAVX(I) 380 Y(I) = YH(I,1) + EL(1)*ACOR(I) C----------------------------------------------------------------------- C Test for convergence. If M .gt. 0, an estimate of the convergence C rate constant is made for the iteration switch, and is also used C in the convergence test. If the iteration seems to be diverging or C converging at a slow rate (.gt. 0.8 more than once), it is stopped. C----------------------------------------------------------------------- 400 IF (M .NE. 0) THEN ROC = MAX(0.05D0, DEL/DELP) CRATE = MAX(0.2D0*CRATE,ROC) ENDIF DCON = DEL*MIN(1.0D0,1.5D0*CRATE)/EPCON IF (DCON .LE. 1.0D0) GO TO 450 M = M + 1 IF (M .EQ. MAXCOR) GO TO 410 IF (M .GE. 2 .AND. DEL .GT. 2.0D0*DELP) GO TO 410 IF (ROC .GT. 10.0D0) GO TO 410 IF (ROC .GT. 0.8D0) NSLOW = NSLOW + 1 IF (NSLOW .GE. 2) GO TO 410 MNEWT = M DELP = DEL CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 GO TO 270 C----------------------------------------------------------------------- C The corrector iteration failed to converge. C If functional iteration is being done (NEWT = 0) and MITER .gt. 0 C (and this is not the first step), then switch to Newton C (NEWT = MITER), and retry the step. (Setting STIFR = 1023 insures C that a switch back will not occur for 10 step attempts.) C If Newton iteration is being done, but using a preconditioner that C is out of date (JACFLG .ne. 0 .and. JCUR = 0), then signal for a C re-evalutation of the preconditioner, and retry the step. C In all other cases, the YH array is retracted to its values C before prediction, and H is reduced, if possible. If H cannot be C reduced or MXNCF failures have occurred, exit with KFLAG = -2. C----------------------------------------------------------------------- 410 ICF = 1 IF (NEWT .EQ. 0) THEN IF (NST .EQ. 0) GO TO 430 IF (MITER .EQ. 0) GO TO 430 NEWT = MITER STIFR = 1023.0D0 IPUP = MITER GO TO 220 ENDIF IF (JCUR.EQ.1 .OR. JACFLG.EQ.0) GO TO 430 IPUP = MITER GO TO 220 430 ICF = 2 NCF = NCF + 1 NCFN = NCFN + 1 RMAX = 2.0D0 TN = TOLD I1 = NQNYH + 1 DO 445 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 440 I = I1,NQNYH 440 YH1(I) = YH1(I) - YH1(I+NYH) 445 CONTINUE IF (IERPJ .LT. 0 .OR. IERSL .LT. 0) GO TO 680 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 670 IF (NCF .EQ. MXNCF) GO TO 670 RH = 0.5D0 IPUP = MITER IREDO = 1 GO TO 170 C----------------------------------------------------------------------- C The corrector has converged. JCUR is set to 0 to signal that the C preconditioner involved may need updating later. C The stiffness ratio STIFR is updated using the latest STIFF value. C The local error test is made and control passes to statement 500 C if it fails. C----------------------------------------------------------------------- 450 JCUR = 0 IF (NEWT .GT. 0) STIFR = 0.5D0*(STIFR + STIFF) IF (M .EQ. 0) DSM = DEL/TESCO(2,NQ) IF (M .GT. 0) DSM = DVNORM (N, ACOR, EWT)/TESCO(2,NQ) IF (DSM .GT. 1.0D0) GO TO 500 C----------------------------------------------------------------------- C After a successful step, update the YH array. C If Newton iteration is being done and STIFR is less than 1.5, C then switch to functional iteration. C Consider changing H if IALTH = 1. Otherwise decrease IALTH by 1. C If IALTH is then 1 and NQ .lt. MAXORD, then ACOR is saved for C use in a possible order increase on the next step. C If a change in H is considered, an increase or decrease in order C by one is considered also. A change in H is made only if it is by a C factor of at least 1.1. If not, IALTH is set to 3 to prevent C testing for that many steps. C----------------------------------------------------------------------- KFLAG = 0 IREDO = 0 NST = NST + 1 IF (NEWT .EQ. 0) NSFI = NSFI + 1 IF (NEWT .GT. 0 .AND. STIFR .LT. 1.5D0) NEWT = 0 HU = H NQU = NQ DO 470 J = 1,L DO 470 I = 1,N 470 YH(I,J) = YH(I,J) + EL(J)*ACOR(I) IALTH = IALTH - 1 IF (IALTH .EQ. 0) GO TO 520 IF (IALTH .GT. 1) GO TO 700 IF (L .EQ. LMAX) GO TO 700 DO 490 I = 1,N 490 YH(I,LMAX) = ACOR(I) GO TO 700 C----------------------------------------------------------------------- C The error test failed. KFLAG keeps track of multiple failures. C Restore TN and the YH array to their previous values, and prepare C to try the step again. Compute the optimum step size for this or C one lower order. After 2 or more failures, H is forced to decrease C by a factor of 0.2 or less. C----------------------------------------------------------------------- 500 KFLAG = KFLAG - 1 TN = TOLD I1 = NQNYH + 1 DO 515 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 510 I = I1,NQNYH 510 YH1(I) = YH1(I) - YH1(I+NYH) 515 CONTINUE RMAX = 2.0D0 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 660 IF (KFLAG .LE. -3) GO TO 640 IREDO = 2 RHUP = 0.0D0 GO TO 540 C----------------------------------------------------------------------- C Regardless of the success or failure of the step, factors C RHDN, RHSM, and RHUP are computed, by which H could be multiplied C at order NQ - 1, order NQ, or order NQ + 1, respectively. C in the case of failure, RHUP = 0.0 to avoid an order increase. C the largest of these is determined and the new order chosen C accordingly. If the order is to be increased, we compute one C additional scaled derivative. C----------------------------------------------------------------------- 520 RHUP = 0.0D0 IF (L .EQ. LMAX) GO TO 540 DO 530 I = 1,N 530 SAVF(I) = ACOR(I) - YH(I,LMAX) DUP = DVNORM (N, SAVF, EWT)/TESCO(3,NQ) EXUP = 1.0D0/(L+1) RHUP = 1.0D0/(1.4D0*DUP**EXUP + 0.0000014D0) 540 EXSM = 1.0D0/L RHSM = 1.0D0/(1.2D0*DSM**EXSM + 0.0000012D0) RHDN = 0.0D0 IF (NQ .EQ. 1) GO TO 560 DDN = DVNORM (N, YH(1,L), EWT)/TESCO(1,NQ) EXDN = 1.0D0/NQ RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) 560 IF (RHSM .GE. RHUP) GO TO 570 IF (RHUP .GT. RHDN) GO TO 590 GO TO 580 570 IF (RHSM .LT. RHDN) GO TO 580 NEWQ = NQ RH = RHSM GO TO 620 580 NEWQ = NQ - 1 RH = RHDN IF (KFLAG .LT. 0 .AND. RH .GT. 1.0D0) RH = 1.0D0 GO TO 620 590 NEWQ = L RH = RHUP IF (RH .LT. 1.1D0) GO TO 610 R = EL(L)/L DO 600 I = 1,N 600 YH(I,NEWQ+1) = ACOR(I)*R GO TO 630 610 IALTH = 3 GO TO 700 620 IF ((KFLAG .EQ. 0) .AND. (RH .LT. 1.1D0)) GO TO 610 IF (KFLAG .LE. -2) RH = MIN(RH,0.2D0) C----------------------------------------------------------------------- C If there is a change of order, reset NQ, L, and the coefficients. C In any case H is reset according to RH and the YH array is rescaled. C Then exit from 690 if the step was OK, or redo the step otherwise. C----------------------------------------------------------------------- IF (NEWQ .EQ. NQ) GO TO 170 630 NQ = NEWQ L = NQ + 1 IRET = 2 GO TO 150 C----------------------------------------------------------------------- C Control reaches this section if 3 or more failures have occured. C If 10 failures have occurred, exit with KFLAG = -1. C It is assumed that the derivatives that have accumulated in the C YH array have errors of the wrong order. Hence the first C derivative is recomputed, and the order is set to 1. Then C H is reduced by a factor of 10, and the step is retried, C until it succeeds or H reaches HMIN. C----------------------------------------------------------------------- 640 IF (KFLAG .EQ. -10) GO TO 660 RH = 0.1D0 RH = MAX(HMIN/ABS(H),RH) H = H*RH DO 645 I = 1,N 645 Y(I) = YH(I,1) CALL F (NEQ, TN, Y, SAVF) NFE = NFE + 1 DO 650 I = 1,N 650 YH(I,2) = H*SAVF(I) IPUP = MITER IALTH = 5 IF (NQ .EQ. 1) GO TO 200 NQ = 1 L = 2 IRET = 3 GO TO 150 C----------------------------------------------------------------------- C All returns are made through this section. H is saved in HOLD C to allow the caller to change H on the next step. C----------------------------------------------------------------------- 660 KFLAG = -1 GO TO 720 670 KFLAG = -2 GO TO 720 680 KFLAG = -3 GO TO 720 690 RMAX = 10.0D0 700 R = 1.0D0/TESCO(2,NQU) DO 710 I = 1,N 710 ACOR(I) = ACOR(I)*R 720 HOLD = H JSTART = 1 RETURN C----------------------- End of Subroutine DSTOKA ---------------------- END *DECK DSETPK SUBROUTINE DSETPK (NEQ, Y, YSV, EWT, FTEM, SAVF, JOK, WM, IWM, 1 F, JAC) EXTERNAL F, JAC INTEGER NEQ, JOK, IWM DOUBLE PRECISION Y, YSV, EWT, FTEM, SAVF, WM DIMENSION NEQ(*), Y(*), YSV(*), EWT(*), FTEM(*), SAVF(*), 1 WM(*), IWM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 1 NNI, NLI, NPS, NCFN, NCFL DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION DELT, EPCON, SQRTN, RSQRTN COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLPK01/ DELT, EPCON, SQRTN, RSQRTN, 1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT, 2 NNI, NLI, NPS, NCFN, NCFL C----------------------------------------------------------------------- C DSETPK is called by DSTOKA to interface with the user-supplied C routine JAC, to compute and process relevant parts of C the matrix P = I - H*EL(1)*J , where J is the Jacobian df/dy, C as need for preconditioning matrix operations later. C C In addition to variables described previously, communication C with DSETPK uses the following: C Y = array containing predicted values on entry. C YSV = array containing predicted y, to be saved (YH1 in DSTOKA). C FTEM = work array of length N (ACOR in DSTOKA). C SAVF = array containing f evaluated at predicted y. C JOK = input flag showing whether it was judged that Jacobian matrix C data need not be recomputed (JOK = 1) or needs to be C (JOK = -1). C WM = real work space for matrices. C Space for preconditioning data starts at WM(LOCWP). C IWM = integer work space. C Space for preconditioning data starts at IWM(LOCIWP). C IERPJ = output error flag, = 0 if no trouble, .gt. 0 if C JAC returned an error flag. C JCUR = output flag to indicate whether the matrix data involved C is now current (JCUR = 1) or not (JCUR = 0). C This routine also uses Common variables EL0, H, TN, IERPJ, JCUR, NJE. C----------------------------------------------------------------------- INTEGER IER DOUBLE PRECISION HL0 C IERPJ = 0 JCUR = 0 IF (JOK .EQ. -1) JCUR = 1 HL0 = EL0*H CALL JAC (F, NEQ, TN, Y, YSV, EWT, SAVF, FTEM, HL0, JOK, 1 WM(LOCWP), IWM(LOCIWP), IER) NJE = NJE + 1 IF (IER .EQ. 0) RETURN IERPJ = 1 RETURN C----------------------- End of Subroutine DSETPK ---------------------- END *DECK DSRCKR SUBROUTINE DSRCKR (RSAV, ISAV, JOB) C----------------------------------------------------------------------- C This routine saves or restores (depending on JOB) the contents of C the Common blocks DLS001, DLS002, DLSR01, DLPK01, which C are used internally by the DLSODKR solver. C C RSAV = real array of length 228 or more. C ISAV = integer array of length 63 or more. C JOB = flag indicating to save or restore the Common blocks: C JOB = 1 if Common is to be saved (written to RSAV/ISAV) C JOB = 2 if Common is to be restored (read from RSAV/ISAV) C A call with JOB = 2 presumes a prior call with JOB = 1. C----------------------------------------------------------------------- INTEGER ISAV, JOB INTEGER ILS, ILS2, ILSR, ILSP INTEGER I, IOFF, LENILP, LENRLP, LENILS, LENRLS, LENILR, LENRLR DOUBLE PRECISION RSAV, RLS, RLS2, RLSR, RLSP DIMENSION RSAV(*), ISAV(*) SAVE LENRLS, LENILS, LENRLP, LENILP, LENRLR, LENILR COMMON /DLS001/ RLS(218), ILS(37) COMMON /DLS002/ RLS2, ILS2(4) COMMON /DLSR01/ RLSR(5), ILSR(9) COMMON /DLPK01/ RLSP(4), ILSP(13) DATA LENRLS/218/, LENILS/37/, LENRLP/4/, LENILP/13/ DATA LENRLR/5/, LENILR/9/ C IF (JOB .EQ. 2) GO TO 100 CALL DCOPY (LENRLS, RLS, 1, RSAV, 1) RSAV(LENRLS+1) = RLS2 CALL DCOPY (LENRLR, RLSR, 1, RSAV(LENRLS+2), 1) CALL DCOPY (LENRLP, RLSP, 1, RSAV(LENRLS+LENRLR+2), 1) DO 20 I = 1,LENILS 20 ISAV(I) = ILS(I) ISAV(LENILS+1) = ILS2(1) ISAV(LENILS+2) = ILS2(2) ISAV(LENILS+3) = ILS2(3) ISAV(LENILS+4) = ILS2(4) IOFF = LENILS + 2 DO 30 I = 1,LENILR 30 ISAV(IOFF+I) = ILSR(I) IOFF = IOFF + LENILR DO 40 I = 1,LENILP 40 ISAV(IOFF+I) = ILSP(I) RETURN C 100 CONTINUE CALL DCOPY (LENRLS, RSAV, 1, RLS, 1) RLS2 = RSAV(LENRLS+1) CALL DCOPY (LENRLR, RSAV(LENRLS+2), 1, RLSR, 1) CALL DCOPY (LENRLP, RSAV(LENRLS+LENRLR+2), 1, RLSP, 1) DO 120 I = 1,LENILS 120 ILS(I) = ISAV(I) ILS2(1) = ISAV(LENILS+1) ILS2(2) = ISAV(LENILS+2) ILS2(3) = ISAV(LENILS+3) ILS2(4) = ISAV(LENILS+4) IOFF = LENILS + 2 DO 130 I = 1,LENILR 130 ILSR(I) = ISAV(IOFF+I) IOFF = IOFF + LENILR DO 140 I = 1,LENILP 140 ILSP(I) = ISAV(IOFF+I) RETURN C----------------------- End of Subroutine DSRCKR ---------------------- END *DECK DAINVG SUBROUTINE DAINVG (RES, ADDA, NEQ, T, Y, YDOT, MITER, 1 ML, MU, PW, IPVT, IER ) EXTERNAL RES, ADDA INTEGER NEQ, MITER, ML, MU, IPVT, IER INTEGER I, LENPW, MLP1, NROWPW DOUBLE PRECISION T, Y, YDOT, PW DIMENSION Y(*), YDOT(*), PW(*), IPVT(*) C----------------------------------------------------------------------- C This subroutine computes the initial value C of the vector YDOT satisfying C A * YDOT = g(t,y) C when A is nonsingular. It is called by DLSODI for C initialization only, when ISTATE = 0 . C DAINVG returns an error flag IER: C IER = 0 means DAINVG was successful. C IER .ge. 2 means RES returned an error flag IRES = IER. C IER .lt. 0 means the a-matrix was found to be singular. C----------------------------------------------------------------------- C IF (MITER .GE. 4) GO TO 100 C C Full matrix case ----------------------------------------------------- C LENPW = NEQ*NEQ DO 10 I = 1, LENPW 10 PW(I) = 0.0D0 C IER = 1 CALL RES ( NEQ, T, Y, PW, YDOT, IER ) IF (IER .GT. 1) RETURN C CALL ADDA ( NEQ, T, Y, 0, 0, PW, NEQ ) CALL DGEFA ( PW, NEQ, NEQ, IPVT, IER ) IF (IER .EQ. 0) GO TO 20 IER = -IER RETURN 20 CALL DGESL ( PW, NEQ, NEQ, IPVT, YDOT, 0 ) RETURN C C Band matrix case ----------------------------------------------------- C 100 CONTINUE NROWPW = 2*ML + MU + 1 LENPW = NEQ * NROWPW DO 110 I = 1, LENPW 110 PW(I) = 0.0D0 C IER = 1 CALL RES ( NEQ, T, Y, PW, YDOT, IER ) IF (IER .GT. 1) RETURN C MLP1 = ML + 1 CALL ADDA ( NEQ, T, Y, ML, MU, PW(MLP1), NROWPW ) CALL DGBFA ( PW, NROWPW, NEQ, ML, MU, IPVT, IER ) IF (IER .EQ. 0) GO TO 120 IER = -IER RETURN 120 CALL DGBSL ( PW, NROWPW, NEQ, ML, MU, IPVT, YDOT, 0 ) RETURN C----------------------- End of Subroutine DAINVG ---------------------- END *DECK DSTODI SUBROUTINE DSTODI (NEQ, Y, YH, NYH, YH1, EWT, SAVF, SAVR, 1 ACOR, WM, IWM, RES, ADDA, JAC, PJAC, SLVS ) EXTERNAL RES, ADDA, JAC, PJAC, SLVS INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, YH1, EWT, SAVF, SAVR, ACOR, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*), 1 SAVR(*), ACOR(*), WM(*), IWM(*) INTEGER IOWND, IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU DOUBLE PRECISION CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO, 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND COMMON /DLS001/ CONIT, CRATE, EL(13), ELCO(13,12), 1 HOLD, RMAX, TESCO(3,12), 2 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 3 IOWND(6), IALTH, IPUP, LMAX, MEO, NQNYH, NSLP, 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER I, I1, IREDO, IRES, IRET, J, JB, KGO, M, NCF, NEWQ DOUBLE PRECISION DCON, DDN, DEL, DELP, DSM, DUP, 1 ELJH, EL1H, EXDN, EXSM, EXUP, 2 R, RH, RHDN, RHSM, RHUP, TOLD, DVNORM C----------------------------------------------------------------------- C DSTODI performs one step of the integration of an initial value C problem for a system of Ordinary Differential Equations. C Note: DSTODI is independent of the value of the iteration method C indicator MITER, and hence is independent C of the type of chord method used, or the Jacobian structure. C Communication with DSTODI is done with the following variables: C C NEQ = integer array containing problem size in NEQ(1), and C passed as the NEQ argument in all calls to RES, ADDA, C and JAC. C Y = an array of length .ge. N used as the Y argument in C all calls to RES, JAC, and ADDA. C NEQ = integer array containing problem size in NEQ(1), and C passed as the NEQ argument in all calls tO RES, G, ADDA, C and JAC. C YH = an NYH by LMAX array containing the dependent variables C and their approximate scaled derivatives, where C LMAX = MAXORD + 1. YH(i,j+1) contains the approximate C j-th derivative of y(i), scaled by H**j/factorial(j) C (j = 0,1,...,NQ). On entry for the first step, the first C two columns of YH must be set from the initial values. C NYH = a constant integer .ge. N, the first dimension of YH. C YH1 = a one-dimensional array occupying the same space as YH. C EWT = an array of length N containing multiplicative weights C for local error measurements. Local errors in y(i) are C compared to 1.0/EWT(i) in various error tests. C SAVF = an array of working storage, of length N. also used for C input of YH(*,MAXORD+2) when JSTART = -1 and MAXORD is less C than the current order NQ. C Same as YDOTI in the driver. C SAVR = an array of working storage, of length N. C ACOR = a work array of length N used for the accumulated C corrections. On a succesful return, ACOR(i) contains C the estimated one-step local error in y(i). C WM,IWM = real and integer work arrays associated with matrix C operations in chord iteration. C PJAC = name of routine to evaluate and preprocess Jacobian matrix. C SLVS = name of routine to solve linear system in chord iteration. C CCMAX = maximum relative change in H*EL0 before PJAC is called. C H = the step size to be attempted on the next step. C H is altered by the error control algorithm during the C problem. H can be either positive or negative, but its C sign must remain constant throughout the problem. C HMIN = the minimum absolute value of the step size H to be used. C HMXI = inverse of the maximum absolute value of H to be used. C HMXI = 0.0 is allowed and corresponds to an infinite HMAX. C HMIN and HMXI may be changed at any time, but will not C take effect until the next change of H is considered. C TN = the independent variable. TN is updated on each step taken. C JSTART = an integer used for input only, with the following C values and meanings: C 0 perform the first step. C .gt.0 take a new step continuing from the last. C -1 take the next step with a new value of H, MAXORD, C N, METH, MITER, and/or matrix parameters. C -2 take the next step with a new value of H, C but with other inputs unchanged. C On return, JSTART is set to 1 to facilitate continuation. C KFLAG = a completion code with the following meanings: C 0 the step was succesful. C -1 the requested error could not be achieved. C -2 corrector convergence could not be achieved. C -3 RES ordered immediate return. C -4 error condition from RES could not be avoided. C -5 fatal error in PJAC or SLVS. C A return with KFLAG = -1, -2, or -4 means either C ABS(H) = HMIN or 10 consecutive failures occurred. C On a return with KFLAG negative, the values of TN and C the YH array are as of the beginning of the last C step, and H is the last step size attempted. C MAXORD = the maximum order of integration method to be allowed. C MAXCOR = the maximum number of corrector iterations allowed. C MSBP = maximum number of steps between PJAC calls. C MXNCF = maximum number of convergence failures allowed. C METH/MITER = the method flags. See description in driver. C N = the number of first-order differential equations. C----------------------------------------------------------------------- KFLAG = 0 TOLD = TN NCF = 0 IERPJ = 0 IERSL = 0 JCUR = 0 ICF = 0 DELP = 0.0D0 IF (JSTART .GT. 0) GO TO 200 IF (JSTART .EQ. -1) GO TO 100 IF (JSTART .EQ. -2) GO TO 160 C----------------------------------------------------------------------- C On the first call, the order is set to 1, and other variables are C initialized. RMAX is the maximum ratio by which H can be increased C in a single step. It is initially 1.E4 to compensate for the small C initial H, but then is normally equal to 10. If a failure C occurs (in corrector convergence or error test), RMAX is set at 2 C for the next increase. C----------------------------------------------------------------------- LMAX = MAXORD + 1 NQ = 1 L = 2 IALTH = 2 RMAX = 10000.0D0 RC = 0.0D0 EL0 = 1.0D0 CRATE = 0.7D0 HOLD = H MEO = METH NSLP = 0 IPUP = MITER IRET = 3 GO TO 140 C----------------------------------------------------------------------- C The following block handles preliminaries needed when JSTART = -1. C IPUP is set to MITER to force a matrix update. C If an order increase is about to be considered (IALTH = 1), C IALTH is reset to 2 to postpone consideration one more step. C If the caller has changed METH, DCFODE is called to reset C the coefficients of the method. C If the caller has changed MAXORD to a value less than the current C order NQ, NQ is reduced to MAXORD, and a new H chosen accordingly. C If H is to be changed, YH must be rescaled. C If H or METH is being changed, IALTH is reset to L = NQ + 1 C to prevent further changes in H for that many steps. C----------------------------------------------------------------------- 100 IPUP = MITER LMAX = MAXORD + 1 IF (IALTH .EQ. 1) IALTH = 2 IF (METH .EQ. MEO) GO TO 110 CALL DCFODE (METH, ELCO, TESCO) MEO = METH IF (NQ .GT. MAXORD) GO TO 120 IALTH = L IRET = 1 GO TO 150 110 IF (NQ .LE. MAXORD) GO TO 160 120 NQ = MAXORD L = LMAX DO 125 I = 1,L 125 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) DDN = DVNORM (N, SAVF, EWT)/TESCO(1,L) EXDN = 1.0D0/L RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) RH = MIN(RHDN,1.0D0) IREDO = 3 IF (H .EQ. HOLD) GO TO 170 RH = MIN(RH,ABS(H/HOLD)) H = HOLD GO TO 175 C----------------------------------------------------------------------- C DCFODE is called to get all the integration coefficients for the C current METH. Then the EL vector and related constants are reset C whenever the order NQ is changed, or at the start of the problem. C----------------------------------------------------------------------- 140 CALL DCFODE (METH, ELCO, TESCO) 150 DO 155 I = 1,L 155 EL(I) = ELCO(I,NQ) NQNYH = NQ*NYH RC = RC*EL(1)/EL0 EL0 = EL(1) CONIT = 0.5D0/(NQ+2) GO TO (160, 170, 200), IRET C----------------------------------------------------------------------- C If H is being changed, the H ratio RH is checked against C RMAX, HMIN, and HMXI, and the YH array rescaled. IALTH is set to C L = NQ + 1 to prevent a change of H for that many steps, unless C forced by a convergence or error test failure. C----------------------------------------------------------------------- 160 IF (H .EQ. HOLD) GO TO 200 RH = H/HOLD H = HOLD IREDO = 3 GO TO 175 170 RH = MAX(RH,HMIN/ABS(H)) 175 RH = MIN(RH,RMAX) RH = RH/MAX(1.0D0,ABS(H)*HMXI*RH) R = 1.0D0 DO 180 J = 2,L R = R*RH DO 180 I = 1,N 180 YH(I,J) = YH(I,J)*R H = H*RH RC = RC*RH IALTH = L IF (IREDO .EQ. 0) GO TO 690 C----------------------------------------------------------------------- C This section computes the predicted values by effectively C multiplying the YH array by the Pascal triangle matrix. C RC is the ratio of new to old values of the coefficient H*EL(1). C When RC differs from 1 by more than CCMAX, IPUP is set to MITER C to force PJAC to be called. C In any case, PJAC is called at least every MSBP steps. C----------------------------------------------------------------------- 200 IF (ABS(RC-1.0D0) .GT. CCMAX) IPUP = MITER IF (NST .GE. NSLP+MSBP) IPUP = MITER TN = TN + H I1 = NQNYH + 1 DO 215 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 210 I = I1,NQNYH 210 YH1(I) = YH1(I) + YH1(I+NYH) 215 CONTINUE C----------------------------------------------------------------------- C Up to MAXCOR corrector iterations are taken. A convergence test is C made on the RMS-norm of each correction, weighted by H and the C error weight vector EWT. The sum of the corrections is accumulated C in ACOR(i). The YH array is not altered in the corrector loop. C----------------------------------------------------------------------- 220 M = 0 DO 230 I = 1,N SAVF(I) = YH(I,2) / H 230 Y(I) = YH(I,1) IF (IPUP .LE. 0) GO TO 240 C----------------------------------------------------------------------- C If indicated, the matrix P = A - H*EL(1)*dr/dy is reevaluated and C preprocessed before starting the corrector iteration. IPUP is set C to 0 as an indicator that this has been done. C----------------------------------------------------------------------- CALL PJAC (NEQ, Y, YH, NYH, EWT, ACOR, SAVR, SAVF, WM, IWM, 1 RES, JAC, ADDA ) IPUP = 0 RC = 1.0D0 NSLP = NST CRATE = 0.7D0 IF (IERPJ .EQ. 0) GO TO 250 IF (IERPJ .LT. 0) GO TO 435 IRES = IERPJ GO TO (430, 435, 430), IRES C Get residual at predicted values, if not already done in PJAC. ------- 240 IRES = 1 CALL RES ( NEQ, TN, Y, SAVF, SAVR, IRES ) NFE = NFE + 1 KGO = ABS(IRES) GO TO ( 250, 435, 430 ) , KGO 250 DO 260 I = 1,N 260 ACOR(I) = 0.0D0 C----------------------------------------------------------------------- C Solve the linear system with the current residual as C right-hand side and P as coefficient matrix. C----------------------------------------------------------------------- 270 CONTINUE CALL SLVS (WM, IWM, SAVR, SAVF) IF (IERSL .LT. 0) GO TO 430 IF (IERSL .GT. 0) GO TO 410 EL1H = EL(1) * H DEL = DVNORM (N, SAVR, EWT) * ABS(H) DO 380 I = 1,N ACOR(I) = ACOR(I) + SAVR(I) SAVF(I) = ACOR(I) + YH(I,2)/H 380 Y(I) = YH(I,1) + EL1H*ACOR(I) C----------------------------------------------------------------------- C Test for convergence. If M .gt. 0, an estimate of the convergence C rate constant is stored in CRATE, and this is used in the test. C----------------------------------------------------------------------- IF (M .NE. 0) CRATE = MAX(0.2D0*CRATE,DEL/DELP) DCON = DEL*MIN(1.0D0,1.5D0*CRATE)/(TESCO(2,NQ)*CONIT) IF (DCON .LE. 1.0D0) GO TO 460 M = M + 1 IF (M .EQ. MAXCOR) GO TO 410 IF (M .GE. 2 .AND. DEL .GT. 2.0D0*DELP) GO TO 410 DELP = DEL IRES = 1 CALL RES ( NEQ, TN, Y, SAVF, SAVR, IRES ) NFE = NFE + 1 KGO = ABS(IRES) GO TO ( 270, 435, 410 ) , KGO C----------------------------------------------------------------------- C The correctors failed to converge, or RES has returned abnormally. C on a convergence failure, if the Jacobian is out of date, PJAC is C called for the next try. Otherwise the YH array is retracted to its C values before prediction, and H is reduced, if possible. C take an error exit if IRES = 2, or H cannot be reduced, or MXNCF C failures have occurred, or a fatal error occurred in PJAC or SLVS. C----------------------------------------------------------------------- 410 ICF = 1 IF (JCUR .EQ. 1) GO TO 430 IPUP = MITER GO TO 220 430 ICF = 2 NCF = NCF + 1 RMAX = 2.0D0 435 TN = TOLD I1 = NQNYH + 1 DO 445 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 440 I = I1,NQNYH 440 YH1(I) = YH1(I) - YH1(I+NYH) 445 CONTINUE IF (IRES .EQ. 2) GO TO 680 IF (IERPJ .LT. 0 .OR. IERSL .LT. 0) GO TO 685 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 450 IF (NCF .EQ. MXNCF) GO TO 450 RH = 0.25D0 IPUP = MITER IREDO = 1 GO TO 170 450 IF (IRES .EQ. 3) GO TO 680 GO TO 670 C----------------------------------------------------------------------- C The corrector has converged. JCUR is set to 0 C to signal that the Jacobian involved may need updating later. C The local error test is made and control passes to statement 500 C if it fails. C----------------------------------------------------------------------- 460 JCUR = 0 IF (M .EQ. 0) DSM = DEL/TESCO(2,NQ) IF (M .GT. 0) DSM = ABS(H) * DVNORM (N, ACOR, EWT)/TESCO(2,NQ) IF (DSM .GT. 1.0D0) GO TO 500 C----------------------------------------------------------------------- C After a successful step, update the YH array. C Consider changing H if IALTH = 1. Otherwise decrease IALTH by 1. C If IALTH is then 1 and NQ .lt. MAXORD, then ACOR is saved for C use in a possible order increase on the next step. C If a change in H is considered, an increase or decrease in order C by one is considered also. A change in H is made only if it is by a C factor of at least 1.1. If not, IALTH is set to 3 to prevent C testing for that many steps. C----------------------------------------------------------------------- KFLAG = 0 IREDO = 0 NST = NST + 1 HU = H NQU = NQ DO 470 J = 1,L ELJH = EL(J)*H DO 470 I = 1,N 470 YH(I,J) = YH(I,J) + ELJH*ACOR(I) IALTH = IALTH - 1 IF (IALTH .EQ. 0) GO TO 520 IF (IALTH .GT. 1) GO TO 700 IF (L .EQ. LMAX) GO TO 700 DO 490 I = 1,N 490 YH(I,LMAX) = ACOR(I) GO TO 700 C----------------------------------------------------------------------- C The error test failed. KFLAG keeps track of multiple failures. C restore TN and the YH array to their previous values, and prepare C to try the step again. Compute the optimum step size for this or C one lower order. After 2 or more failures, H is forced to decrease C by a factor of 0.1 or less. C----------------------------------------------------------------------- 500 KFLAG = KFLAG - 1 TN = TOLD I1 = NQNYH + 1 DO 515 JB = 1,NQ I1 = I1 - NYH CDIR\$ IVDEP DO 510 I = I1,NQNYH 510 YH1(I) = YH1(I) - YH1(I+NYH) 515 CONTINUE RMAX = 2.0D0 IF (ABS(H) .LE. HMIN*1.00001D0) GO TO 660 IF (KFLAG .LE. -7) GO TO 660 IREDO = 2 RHUP = 0.0D0 GO TO 540 C----------------------------------------------------------------------- C Regardless of the success or failure of the step, factors C RHDN, RHSM, and RHUP are computed, by which H could be multiplied C at order NQ - 1, order NQ, or order NQ + 1, respectively. C In the case of failure, RHUP = 0.0 to avoid an order increase. C The largest of these is determined and the new order chosen C accordingly. If the order is to be increased, we compute one C additional scaled derivative. C----------------------------------------------------------------------- 520 RHUP = 0.0D0 IF (L .EQ. LMAX) GO TO 540 DO 530 I = 1,N 530 SAVF(I) = ACOR(I) - YH(I,LMAX) DUP = ABS(H) * DVNORM (N, SAVF, EWT)/TESCO(3,NQ) EXUP = 1.0D0/(L+1) RHUP = 1.0D0/(1.4D0*DUP**EXUP + 0.0000014D0) 540 EXSM = 1.0D0/L RHSM = 1.0D0/(1.2D0*DSM**EXSM + 0.0000012D0) RHDN = 0.0D0 IF (NQ .EQ. 1) GO TO 560 DDN = DVNORM (N, YH(1,L), EWT)/TESCO(1,NQ) EXDN = 1.0D0/NQ RHDN = 1.0D0/(1.3D0*DDN**EXDN + 0.0000013D0) 560 IF (RHSM .GE. RHUP) GO TO 570 IF (RHUP .GT. RHDN) GO TO 590 GO TO 580 570 IF (RHSM .LT. RHDN) GO TO 580 NEWQ = NQ RH = RHSM GO TO 620 580 NEWQ = NQ - 1 RH = RHDN IF (KFLAG .LT. 0 .AND. RH .GT. 1.0D0) RH = 1.0D0 GO TO 620 590 NEWQ = L RH = RHUP IF (RH .LT. 1.1D0) GO TO 610 R = H*EL(L)/L DO 600 I = 1,N 600 YH(I,NEWQ+1) = ACOR(I)*R GO TO 630 610 IALTH = 3 GO TO 700 620 IF ((KFLAG .EQ. 0) .AND. (RH .LT. 1.1D0)) GO TO 610 IF (KFLAG .LE. -2) RH = MIN(RH,0.1D0) C----------------------------------------------------------------------- C If there is a change of order, reset NQ, L, and the coefficients. C In any case H is reset according to RH and the YH array is rescaled. C Then exit from 690 if the step was OK, or redo the step otherwise. C----------------------------------------------------------------------- IF (NEWQ .EQ. NQ) GO TO 170 630 NQ = NEWQ L = NQ + 1 IRET = 2 GO TO 150 C----------------------------------------------------------------------- C All returns are made through this section. H is saved in HOLD C to allow the caller to change H on the next step. C----------------------------------------------------------------------- 660 KFLAG = -1 GO TO 720 670 KFLAG = -2 GO TO 720 680 KFLAG = -1 - IRES GO TO 720 685 KFLAG = -5 GO TO 720 690 RMAX = 10.0D0 700 R = H/TESCO(2,NQU) DO 710 I = 1,N 710 ACOR(I) = ACOR(I)*R 720 HOLD = H JSTART = 1 RETURN C----------------------- End of Subroutine DSTODI ---------------------- END *DECK DPREPJI SUBROUTINE DPREPJI (NEQ, Y, YH, NYH, EWT, RTEM, SAVR, S, WM, IWM, 1 RES, JAC, ADDA) EXTERNAL RES, JAC, ADDA INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, EWT, RTEM, SAVR, S, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), RTEM(*), 1 S(*), SAVR(*), WM(*), IWM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER I, I1, I2, IER, II, IRES, J, J1, JJ, LENP, 1 MBA, MBAND, MEB1, MEBAND, ML, ML3, MU DOUBLE PRECISION CON, FAC, HL0, R, SRUR, YI, YJ, YJJ C----------------------------------------------------------------------- C DPREPJI is called by DSTODI to compute and process the matrix C P = A - H*EL(1)*J , where J is an approximation to the Jacobian dr/dy, C where r = g(t,y) - A(t,y)*s. Here J is computed by the user-supplied C routine JAC if MITER = 1 or 4, or by finite differencing if MITER = C 2 or 5. J is stored in WM, rescaled, and ADDA is called to generate C P. P is then subjected to LU decomposition in preparation C for later solution of linear systems with P as coefficient C matrix. This is done by DGEFA if MITER = 1 or 2, and by C DGBFA if MITER = 4 or 5. C C In addition to variables described previously, communication C with DPREPJI uses the following: C Y = array containing predicted values on entry. C RTEM = work array of length N (ACOR in DSTODI). C SAVR = array used for output only. On output it contains the C residual evaluated at current values of t and y. C S = array containing predicted values of dy/dt (SAVF in DSTODI). C WM = real work space for matrices. On output it contains the C LU decomposition of P. C Storage of matrix elements starts at WM(3). C WM also contains the following matrix-related data: C WM(1) = SQRT(UROUND), used in numerical Jacobian increments. C IWM = integer work space containing pivot information, starting at C IWM(21). IWM also contains the band parameters C ML = IWM(1) and MU = IWM(2) if MITER is 4 or 5. C EL0 = el(1) (input). C IERPJ = output error flag. C = 0 if no trouble occurred, C = 1 if the P matrix was found to be singular, C = IRES (= 2 or 3) if RES returned IRES = 2 or 3. C JCUR = output flag = 1 to indicate that the Jacobian matrix C (or approximation) is now current. C This routine also uses the Common variables EL0, H, TN, UROUND, C MITER, N, NFE, and NJE. C----------------------------------------------------------------------- NJE = NJE + 1 HL0 = H*EL0 IERPJ = 0 JCUR = 1 GO TO (100, 200, 300, 400, 500), MITER C If MITER = 1, call RES, then JAC, and multiply by scalar. ------------ 100 IRES = 1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 LENP = N*N DO 110 I = 1,LENP 110 WM(I+2) = 0.0D0 CALL JAC ( NEQ, TN, Y, S, 0, 0, WM(3), N ) CON = -HL0 DO 120 I = 1,LENP 120 WM(I+2) = WM(I+2)*CON GO TO 240 C If MITER = 2, make N + 1 calls to RES to approximate J. -------------- 200 CONTINUE IRES = -1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 SRUR = WM(1) J1 = 2 DO 230 J = 1,N YJ = Y(J) R = MAX(SRUR*ABS(YJ),0.01D0/EWT(J)) Y(J) = Y(J) + R FAC = -HL0/R CALL RES ( NEQ, TN, Y, S, RTEM, IRES ) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 DO 220 I = 1,N 220 WM(I+J1) = (RTEM(I) - SAVR(I))*FAC Y(J) = YJ J1 = J1 + N 230 CONTINUE IRES = 1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 C Add matrix A. -------------------------------------------------------- 240 CONTINUE CALL ADDA(NEQ, TN, Y, 0, 0, WM(3), N) C Do LU decomposition on P. -------------------------------------------- CALL DGEFA (WM(3), N, N, IWM(21), IER) IF (IER .NE. 0) IERPJ = 1 RETURN C Dummy section for MITER = 3 300 RETURN C If MITER = 4, call RES, then JAC, and multiply by scalar. ------------ 400 IRES = 1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 ML = IWM(1) MU = IWM(2) ML3 = ML + 3 MBAND = ML + MU + 1 MEBAND = MBAND + ML LENP = MEBAND*N DO 410 I = 1,LENP 410 WM(I+2) = 0.0D0 CALL JAC ( NEQ, TN, Y, S, ML, MU, WM(ML3), MEBAND) CON = -HL0 DO 420 I = 1,LENP 420 WM(I+2) = WM(I+2)*CON GO TO 570 C If MITER = 5, make ML + MU + 2 calls to RES to approximate J. -------- 500 CONTINUE IRES = -1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 ML = IWM(1) MU = IWM(2) ML3 = ML + 3 MBAND = ML + MU + 1 MBA = MIN(MBAND,N) MEBAND = MBAND + ML MEB1 = MEBAND - 1 SRUR = WM(1) DO 560 J = 1,MBA DO 530 I = J,N,MBAND YI = Y(I) R = MAX(SRUR*ABS(YI),0.01D0/EWT(I)) 530 Y(I) = Y(I) + R CALL RES ( NEQ, TN, Y, S, RTEM, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 DO 550 JJ = J,N,MBAND Y(JJ) = YH(JJ,1) YJJ = Y(JJ) R = MAX(SRUR*ABS(YJJ),0.01D0/EWT(JJ)) FAC = -HL0/R I1 = MAX(JJ-MU,1) I2 = MIN(JJ+ML,N) II = JJ*MEB1 - ML + 2 DO 540 I = I1,I2 540 WM(II+I) = (RTEM(I) - SAVR(I))*FAC 550 CONTINUE 560 CONTINUE IRES = 1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 C Add matrix A. -------------------------------------------------------- 570 CONTINUE CALL ADDA(NEQ, TN, Y, ML, MU, WM(ML3), MEBAND) C Do LU decomposition of P. -------------------------------------------- CALL DGBFA (WM(3), MEBAND, N, ML, MU, IWM(21), IER) IF (IER .NE. 0) IERPJ = 1 RETURN C Error return for IRES = 2 or IRES = 3 return from RES. --------------- 600 IERPJ = IRES RETURN C----------------------- End of Subroutine DPREPJI --------------------- END *DECK DAIGBT SUBROUTINE DAIGBT (RES, ADDA, NEQ, T, Y, YDOT, 1 MB, NB, PW, IPVT, IER ) EXTERNAL RES, ADDA INTEGER NEQ, MB, NB, IPVT, IER INTEGER I, LENPW, LBLOX, LPB, LPC DOUBLE PRECISION T, Y, YDOT, PW DIMENSION Y(*), YDOT(*), PW(*), IPVT(*), NEQ(*) C----------------------------------------------------------------------- C This subroutine computes the initial value C of the vector YDOT satisfying C A * YDOT = g(t,y) C when A is nonsingular. It is called by DLSOIBT for C initialization only, when ISTATE = 0 . C DAIGBT returns an error flag IER: C IER = 0 means DAIGBT was successful. C IER .ge. 2 means RES returned an error flag IRES = IER. C IER .lt. 0 means the A matrix was found to have a singular C diagonal block (hence YDOT could not be solved for). C----------------------------------------------------------------------- LBLOX = MB*MB*NB LPB = 1 + LBLOX LPC = LPB + LBLOX LENPW = 3*LBLOX DO 10 I = 1,LENPW 10 PW(I) = 0.0D0 IER = 1 CALL RES (NEQ, T, Y, PW, YDOT, IER) IF (IER .GT. 1) RETURN CALL ADDA (NEQ, T, Y, MB, NB, PW(1), PW(LPB), PW(LPC) ) CALL DDECBT (MB, NB, PW, PW(LPB), PW(LPC), IPVT, IER) IF (IER .EQ. 0) GO TO 20 IER = -IER RETURN 20 CALL DSOLBT (MB, NB, PW, PW(LPB), PW(LPC), YDOT, IPVT) RETURN C----------------------- End of Subroutine DAIGBT ---------------------- END *DECK DPJIBT SUBROUTINE DPJIBT (NEQ, Y, YH, NYH, EWT, RTEM, SAVR, S, WM, IWM, 1 RES, JAC, ADDA) EXTERNAL RES, JAC, ADDA INTEGER NEQ, NYH, IWM DOUBLE PRECISION Y, YH, EWT, RTEM, SAVR, S, WM DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), RTEM(*), 1 S(*), SAVR(*), WM(*), IWM(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER I, IER, IIA, IIB, IIC, IPA, IPB, IPC, IRES, J, J1, J2, 1 K, K1, LENP, LBLOX, LPB, LPC, MB, MBSQ, MWID, NB DOUBLE PRECISION CON, FAC, HL0, R, SRUR C----------------------------------------------------------------------- C DPJIBT is called by DSTODI to compute and process the matrix C P = A - H*EL(1)*J , where J is an approximation to the Jacobian dr/dy, C and r = g(t,y) - A(t,y)*s. Here J is computed by the user-supplied C routine JAC if MITER = 1, or by finite differencing if MITER = 2. C J is stored in WM, rescaled, and ADDA is called to generate P. C P is then subjected to LU decomposition by DDECBT in preparation C for later solution of linear systems with P as coefficient matrix. C C In addition to variables described previously, communication C with DPJIBT uses the following: C Y = array containing predicted values on entry. C RTEM = work array of length N (ACOR in DSTODI). C SAVR = array used for output only. On output it contains the C residual evaluated at current values of t and y. C S = array containing predicted values of dy/dt (SAVF in DSTODI). C WM = real work space for matrices. On output it contains the C LU decomposition of P. C Storage of matrix elements starts at WM(3). C WM also contains the following matrix-related data: C WM(1) = SQRT(UROUND), used in numerical Jacobian increments. C IWM = integer work space containing pivot information, starting at C IWM(21). IWM also contains block structure parameters C MB = IWM(1) and NB = IWM(2). C EL0 = EL(1) (input). C IERPJ = output error flag. C = 0 if no trouble occurred, C = 1 if the P matrix was found to be unfactorable, C = IRES (= 2 or 3) if RES returned IRES = 2 or 3. C JCUR = output flag = 1 to indicate that the Jacobian matrix C (or approximation) is now current. C This routine also uses the Common variables EL0, H, TN, UROUND, C MITER, N, NFE, and NJE. C----------------------------------------------------------------------- NJE = NJE + 1 HL0 = H*EL0 IERPJ = 0 JCUR = 1 MB = IWM(1) NB = IWM(2) MBSQ = MB*MB LBLOX = MBSQ*NB LPB = 3 + LBLOX LPC = LPB + LBLOX LENP = 3*LBLOX GO TO (100, 200), MITER C If MITER = 1, call RES, then JAC, and multiply by scalar. ------------ 100 IRES = 1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 DO 110 I = 1,LENP 110 WM(I+2) = 0.0D0 CALL JAC (NEQ, TN, Y, S, MB, NB, WM(3), WM(LPB), WM(LPC)) CON = -HL0 DO 120 I = 1,LENP 120 WM(I+2) = WM(I+2)*CON GO TO 260 C C If MITER = 2, make 3*MB + 1 calls to RES to approximate J. ----------- 200 CONTINUE IRES = -1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 MWID = 3*MB SRUR = WM(1) DO 205 I = 1,LENP 205 WM(2+I) = 0.0D0 DO 250 K = 1,3 DO 240 J = 1,MB C Increment Y(I) for group of column indices, and call RES. ---- J1 = J+(K-1)*MB DO 210 I = J1,N,MWID R = MAX(SRUR*ABS(Y(I)),0.01D0/EWT(I)) Y(I) = Y(I) + R 210 CONTINUE CALL RES (NEQ, TN, Y, S, RTEM, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 DO 215 I = 1,N 215 RTEM(I) = RTEM(I) - SAVR(I) K1 = K DO 230 I = J1,N,MWID C Get Jacobian elements in column I (block-column K1). ------- Y(I) = YH(I,1) R = MAX(SRUR*ABS(Y(I)),0.01D0/EWT(I)) FAC = -HL0/R C Compute and load elements PA(*,J,K1). ---------------------- IIA = I - J IPA = 2 + (J-1)*MB + (K1-1)*MBSQ DO 221 J2 = 1,MB 221 WM(IPA+J2) = RTEM(IIA+J2)*FAC IF (K1 .LE. 1) GO TO 223 C Compute and load elements PB(*,J,K1-1). -------------------- IIB = IIA - MB IPB = IPA + LBLOX - MBSQ DO 222 J2 = 1,MB 222 WM(IPB+J2) = RTEM(IIB+J2)*FAC 223 CONTINUE IF (K1 .GE. NB) GO TO 225 C Compute and load elements PC(*,J,K1+1). -------------------- IIC = IIA + MB IPC = IPA + 2*LBLOX + MBSQ DO 224 J2 = 1,MB 224 WM(IPC+J2) = RTEM(IIC+J2)*FAC 225 CONTINUE IF (K1 .NE. 3) GO TO 227 C Compute and load elements PC(*,J,1). ----------------------- IPC = IPA - 2*MBSQ + 2*LBLOX DO 226 J2 = 1,MB 226 WM(IPC+J2) = RTEM(J2)*FAC 227 CONTINUE IF (K1 .NE. NB-2) GO TO 229 C Compute and load elements PB(*,J,NB). ---------------------- IIB = N - MB IPB = IPA + 2*MBSQ + LBLOX DO 228 J2 = 1,MB 228 WM(IPB+J2) = RTEM(IIB+J2)*FAC 229 K1 = K1 + 3 230 CONTINUE 240 CONTINUE 250 CONTINUE C RES call for first corrector iteration. ------------------------------ IRES = 1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 C Add matrix A. -------------------------------------------------------- 260 CONTINUE CALL ADDA (NEQ, TN, Y, MB, NB, WM(3), WM(LPB), WM(LPC)) C Do LU decomposition on P. -------------------------------------------- CALL DDECBT (MB, NB, WM(3), WM(LPB), WM(LPC), IWM(21), IER) IF (IER .NE. 0) IERPJ = 1 RETURN C Error return for IRES = 2 or IRES = 3 return from RES. --------------- 600 IERPJ = IRES RETURN C----------------------- End of Subroutine DPJIBT ---------------------- END *DECK DSLSBT SUBROUTINE DSLSBT (WM, IWM, X, TEM) INTEGER IWM INTEGER LBLOX, LPB, LPC, MB, NB DOUBLE PRECISION WM, X, TEM DIMENSION WM(*), IWM(*), X(*), TEM(*) C----------------------------------------------------------------------- C This routine acts as an interface between the core integrator C routine and the DSOLBT routine for the solution of the linear system C arising from chord iteration. C Communication with DSLSBT uses the following variables: C WM = real work space containing the LU decomposition, C starting at WM(3). C IWM = integer work space containing pivot information, starting at C IWM(21). IWM also contains block structure parameters C MB = IWM(1) and NB = IWM(2). C X = the right-hand side vector on input, and the solution vector C on output, of length N. C TEM = vector of work space of length N, not used in this version. C----------------------------------------------------------------------- MB = IWM(1) NB = IWM(2) LBLOX = MB*MB*NB LPB = 3 + LBLOX LPC = LPB + LBLOX CALL DSOLBT (MB, NB, WM(3), WM(LPB), WM(LPC), X, IWM(21)) RETURN C----------------------- End of Subroutine DSLSBT ---------------------- END *DECK DDECBT SUBROUTINE DDECBT (M, N, A, B, C, IP, IER) INTEGER M, N, IP(M,N), IER DOUBLE PRECISION A(M,M,N), B(M,M,N), C(M,M,N) C----------------------------------------------------------------------- C Block-tridiagonal matrix decomposition routine. C Written by A. C. Hindmarsh. C Latest revision: November 10, 1983 (ACH) C Reference: UCID-30150 C Solution of Block-Tridiagonal Systems of Linear C Algebraic Equations C A.C. Hindmarsh C February 1977 C The input matrix contains three blocks of elements in each block-row, C including blocks in the (1,3) and (N,N-2) block positions. C DDECBT uses block Gauss elimination and Subroutines DGEFA and DGESL C for solution of blocks. Partial pivoting is done within C block-rows only. C C Note: this version uses LINPACK routines DGEFA/DGESL instead of C of dec/sol for solution of blocks, and it uses the BLAS routine DDOT C for dot product calculations. C C Input: C M = order of each block. C N = number of blocks in each direction of the matrix. C N must be 4 or more. The complete matrix has order M*N. C A = M by M by N array containing diagonal blocks. C A(i,j,k) contains the (i,j) element of the k-th block. C B = M by M by N array containing the super-diagonal blocks C (in B(*,*,k) for k = 1,...,N-1) and the block in the (N,N-2) C block position (in B(*,*,N)). C C = M by M by N array containing the subdiagonal blocks C (in C(*,*,k) for k = 2,3,...,N) and the block in the C (1,3) block position (in C(*,*,1)). C IP = integer array of length M*N for working storage. C Output: C A,B,C = M by M by N arrays containing the block-LU decomposition C of the input matrix. C IP = M by N array of pivot information. IP(*,k) contains C information for the k-th digonal block. C IER = 0 if no trouble occurred, or C = -1 if the input value of M or N was illegal, or C = k if a singular matrix was found in the k-th diagonal block. C Use DSOLBT to solve the associated linear system. C C External routines required: DGEFA and DGESL (from LINPACK) and C DDOT (from the BLAS, or Basic Linear Algebra package). C----------------------------------------------------------------------- INTEGER NM1, NM2, KM1, I, J, K DOUBLE PRECISION DP, DDOT IF (M .LT. 1 .OR. N .LT. 4) GO TO 210 NM1 = N - 1 NM2 = N - 2 C Process the first block-row. ----------------------------------------- CALL DGEFA (A, M, M, IP, IER) K = 1 IF (IER .NE. 0) GO TO 200 DO 10 J = 1,M CALL DGESL (A, M, M, IP, B(1,J,1), 0) CALL DGESL (A, M, M, IP, C(1,J,1), 0) 10 CONTINUE C Adjust B(*,*,2). ----------------------------------------------------- DO 40 J = 1,M DO 30 I = 1,M DP = DDOT (M, C(I,1,2), M, C(1,J,1), 1) B(I,J,2) = B(I,J,2) - DP 30 CONTINUE 40 CONTINUE C Main loop. Process block-rows 2 to N-1. ----------------------------- DO 100 K = 2,NM1 KM1 = K - 1 DO 70 J = 1,M DO 60 I = 1,M DP = DDOT (M, C(I,1,K), M, B(1,J,KM1), 1) A(I,J,K) = A(I,J,K) - DP 60 CONTINUE 70 CONTINUE CALL DGEFA (A(1,1,K), M, M, IP(1,K), IER) IF (IER .NE. 0) GO TO 200 DO 80 J = 1,M 80 CALL DGESL (A(1,1,K), M, M, IP(1,K), B(1,J,K), 0) 100 CONTINUE C Process last block-row and return. ----------------------------------- DO 130 J = 1,M DO 120 I = 1,M DP = DDOT (M, B(I,1,N), M, B(1,J,NM2), 1) C(I,J,N) = C(I,J,N) - DP 120 CONTINUE 130 CONTINUE DO 160 J = 1,M DO 150 I = 1,M DP = DDOT (M, C(I,1,N), M, B(1,J,NM1), 1) A(I,J,N) = A(I,J,N) - DP 150 CONTINUE 160 CONTINUE CALL DGEFA (A(1,1,N), M, M, IP(1,N), IER) K = N IF (IER .NE. 0) GO TO 200 RETURN C Error returns. ------------------------------------------------------- 200 IER = K RETURN 210 IER = -1 RETURN C----------------------- End of Subroutine DDECBT ---------------------- END *DECK DSOLBT SUBROUTINE DSOLBT (M, N, A, B, C, Y, IP) INTEGER M, N, IP(M,N) DOUBLE PRECISION A(M,M,N), B(M,M,N), C(M,M,N), Y(M,N) C----------------------------------------------------------------------- C Solution of block-tridiagonal linear system. C Coefficient matrix must have been previously processed by DDECBT. C M, N, A,B,C, and IP must not have been changed since call to DDECBT. C Written by A. C. Hindmarsh. C Input: C M = order of each block. C N = number of blocks in each direction of matrix. C A,B,C = M by M by N arrays containing block LU decomposition C of coefficient matrix from DDECBT. C IP = M by N integer array of pivot information from DDECBT. C Y = array of length M*N containg the right-hand side vector C (treated as an M by N array here). C Output: C Y = solution vector, of length M*N. C C External routines required: DGESL (LINPACK) and DDOT (BLAS). C----------------------------------------------------------------------- C INTEGER NM1, NM2, I, K, KB, KM1, KP1 DOUBLE PRECISION DP, DDOT NM1 = N - 1 NM2 = N - 2 C Forward solution sweep. ---------------------------------------------- CALL DGESL (A, M, M, IP, Y, 0) DO 30 K = 2,NM1 KM1 = K - 1 DO 20 I = 1,M DP = DDOT (M, C(I,1,K), M, Y(1,KM1), 1) Y(I,K) = Y(I,K) - DP 20 CONTINUE CALL DGESL (A(1,1,K), M, M, IP(1,K), Y(1,K), 0) 30 CONTINUE DO 50 I = 1,M DP = DDOT (M, C(I,1,N), M, Y(1,NM1), 1) 1 + DDOT (M, B(I,1,N), M, Y(1,NM2), 1) Y(I,N) = Y(I,N) - DP 50 CONTINUE CALL DGESL (A(1,1,N), M, M, IP(1,N), Y(1,N), 0) C Backward solution sweep. --------------------------------------------- DO 80 KB = 1,NM1 K = N - KB KP1 = K + 1 DO 70 I = 1,M DP = DDOT (M, B(I,1,K), M, Y(1,KP1), 1) Y(I,K) = Y(I,K) - DP 70 CONTINUE 80 CONTINUE DO 100 I = 1,M DP = DDOT (M, C(I,1,1), M, Y(1,3), 1) Y(I,1) = Y(I,1) - DP 100 CONTINUE RETURN C----------------------- End of Subroutine DSOLBT ---------------------- END *DECK DIPREPI SUBROUTINE DIPREPI (NEQ, Y, S, RWORK, IA, JA, IC, JC, IPFLAG, 1 RES, JAC, ADDA) EXTERNAL RES, JAC, ADDA INTEGER NEQ, IA, JA, IC, JC, IPFLAG DOUBLE PRECISION Y, S, RWORK DIMENSION NEQ(*), Y(*), S(*), RWORK(*), IA(*), JA(*), IC(*), JC(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION RLSS COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSS01/ RLSS(6), 1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU INTEGER I, IMAX, LEWTN, LYHD, LYHN C----------------------------------------------------------------------- C This routine serves as an interface between the driver and C Subroutine DPREPI. Tasks performed here are: C * call DPREPI, C * reset the required WM segment length LENWK, C * move YH back to its final location (following WM in RWORK), C * reset pointers for YH, SAVR, EWT, and ACOR, and C * move EWT to its new position if ISTATE = 0 or 1. C IPFLAG is an output error indication flag. IPFLAG = 0 if there was C no trouble, and IPFLAG is the value of the DPREPI error flag IPPER C if there was trouble in Subroutine DPREPI. C----------------------------------------------------------------------- IPFLAG = 0 C Call DPREPI to do matrix preprocessing operations. ------------------- CALL DPREPI (NEQ, Y, S, RWORK(LYH), RWORK(LSAVF), RWORK(LEWT), 1 RWORK(LACOR), IA, JA, IC, JC, RWORK(LWM), RWORK(LWM), IPFLAG, 2 RES, JAC, ADDA) LENWK = MAX(LREQ,LWMIN) IF (IPFLAG .LT. 0) RETURN C If DPREPI was successful, move YH to end of required space for WM. --- LYHN = LWM + LENWK IF (LYHN .GT. LYH) RETURN LYHD = LYH - LYHN IF (LYHD .EQ. 0) GO TO 20 IMAX = LYHN - 1 + LENYHM DO 10 I=LYHN,IMAX 10 RWORK(I) = RWORK(I+LYHD) LYH = LYHN C Reset pointers for SAVR, EWT, and ACOR. ------------------------------ 20 LSAVF = LYH + LENYH LEWTN = LSAVF + N LACOR = LEWTN + N IF (ISTATC .EQ. 3) GO TO 40 C If ISTATE = 1, move EWT (left) to its new position. ------------------ IF (LEWTN .GT. LEWT) RETURN DO 30 I=1,N 30 RWORK(I+LEWTN-1) = RWORK(I+LEWT-1) 40 LEWT = LEWTN RETURN C----------------------- End of Subroutine DIPREPI --------------------- END *DECK DPREPI SUBROUTINE DPREPI (NEQ, Y, S, YH, SAVR, EWT, RTEM, IA, JA, IC, JC, 1 WK, IWK, IPPER, RES, JAC, ADDA) EXTERNAL RES, JAC, ADDA INTEGER NEQ, IA, JA, IC, JC, IWK, IPPER DOUBLE PRECISION Y, S, YH, SAVR, EWT, RTEM, WK DIMENSION NEQ(*), Y(*), S(*), YH(*), SAVR(*), EWT(*), RTEM(*), 1 IA(*), JA(*), IC(*), JC(*), WK(*), IWK(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION RLSS COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSS01/ RLSS(6), 1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU INTEGER I, IBR, IER, IPIL, IPIU, IPTT1, IPTT2, J, K, KNEW, KAMAX, 1 KAMIN, KCMAX, KCMIN, LDIF, LENIGP, LENWK1, LIWK, LJFO, MAXG, 2 NP1, NZSUT DOUBLE PRECISION ERWT, FAC, YJ C----------------------------------------------------------------------- C This routine performs preprocessing related to the sparse linear C systems that must be solved. C The operations that are performed here are: C * compute sparseness structure of the iteration matrix C P = A - con*J according to MOSS, C * compute grouping of column indices (MITER = 2), C * compute a new ordering of rows and columns of the matrix, C * reorder JA corresponding to the new ordering, C * perform a symbolic LU factorization of the matrix, and C * set pointers for segments of the IWK/WK array. C In addition to variables described previously, DPREPI uses the C following for communication: C YH = the history array. Only the first column, containing the C current Y vector, is used. Used only if MOSS .ne. 0. C S = array of length NEQ, identical to YDOTI in the driver, used C only if MOSS .ne. 0. C SAVR = a work array of length NEQ, used only if MOSS .ne. 0. C EWT = array of length NEQ containing (inverted) error weights. C Used only if MOSS = 2 or 4 or if ISTATE = MOSS = 1. C RTEM = a work array of length NEQ, identical to ACOR in the driver, C used only if MOSS = 2 or 4. C WK = a real work array of length LENWK, identical to WM in C the driver. C IWK = integer work array, assumed to occupy the same space as WK. C LENWK = the length of the work arrays WK and IWK. C ISTATC = a copy of the driver input argument ISTATE (= 1 on the C first call, = 3 on a continuation call). C IYS = flag value from ODRV or CDRV. C IPPER = output error flag , with the following values and meanings: C = 0 no error. C = -1 insufficient storage for internal structure pointers. C = -2 insufficient storage for JGROUP. C = -3 insufficient storage for ODRV. C = -4 other error flag from ODRV (should never occur). C = -5 insufficient storage for CDRV. C = -6 other error flag from CDRV. C = -7 if the RES routine returned error flag IRES = IER = 2. C = -8 if the RES routine returned error flag IRES = IER = 3. C----------------------------------------------------------------------- IBIAN = LRAT*2 IPIAN = IBIAN + 1 NP1 = N + 1 IPJAN = IPIAN + NP1 IBJAN = IPJAN - 1 LENWK1 = LENWK - N LIWK = LENWK*LRAT IF (MOSS .EQ. 0) LIWK = LIWK - N IF (MOSS .EQ. 1 .OR. MOSS .EQ. 2) LIWK = LENWK1*LRAT IF (IPJAN+N-1 .GT. LIWK) GO TO 310 IF (MOSS .EQ. 0) GO TO 30 C IF (ISTATC .EQ. 3) GO TO 20 C ISTATE = 1 and MOSS .ne. 0. Perturb Y for structure determination. C Initialize S with random nonzero elements for structure determination. DO 10 I=1,N ERWT = 1.0D0/EWT(I) FAC = 1.0D0 + 1.0D0/(I + 1.0D0) Y(I) = Y(I) + FAC*SIGN(ERWT,Y(I)) S(I) = 1.0D0 + FAC*ERWT 10 CONTINUE GO TO (70, 100, 150, 200), MOSS C 20 CONTINUE C ISTATE = 3 and MOSS .ne. 0. Load Y from YH(*,1) and S from YH(*,2). -- DO 25 I = 1,N Y(I) = YH(I) 25 S(I) = YH(N+I) GO TO (70, 100, 150, 200), MOSS C C MOSS = 0. Process user's IA,JA and IC,JC. ---------------------------- 30 KNEW = IPJAN KAMIN = IA(1) KCMIN = IC(1) IWK(IPIAN) = 1 DO 60 J = 1,N DO 35 I = 1,N 35 IWK(LIWK+I) = 0 KAMAX = IA(J+1) - 1 IF (KAMIN .GT. KAMAX) GO TO 45 DO 40 K = KAMIN,KAMAX I = JA(K) IWK(LIWK+I) = 1 IF (KNEW .GT. LIWK) GO TO 310 IWK(KNEW) = I KNEW = KNEW + 1 40 CONTINUE 45 KAMIN = KAMAX + 1 KCMAX = IC(J+1) - 1 IF (KCMIN .GT. KCMAX) GO TO 55 DO 50 K = KCMIN,KCMAX I = JC(K) IF (IWK(LIWK+I) .NE. 0) GO TO 50 IF (KNEW .GT. LIWK) GO TO 310 IWK(KNEW) = I KNEW = KNEW + 1 50 CONTINUE 55 IWK(IPIAN+J) = KNEW + 1 - IPJAN KCMIN = KCMAX + 1 60 CONTINUE GO TO 240 C C MOSS = 1. Compute structure from user-supplied Jacobian routine JAC. - 70 CONTINUE C A dummy call to RES allows user to create temporaries for use in JAC. IER = 1 CALL RES (NEQ, TN, Y, S, SAVR, IER) IF (IER .GT. 1) GO TO 370 DO 75 I = 1,N SAVR(I) = 0.0D0 75 WK(LENWK1+I) = 0.0D0 K = IPJAN IWK(IPIAN) = 1 DO 95 J = 1,N CALL ADDA (NEQ, TN, Y, J, IWK(IPIAN), IWK(IPJAN), WK(LENWK1+1)) CALL JAC (NEQ, TN, Y, S, J, IWK(IPIAN), IWK(IPJAN), SAVR) DO 90 I = 1,N LJFO = LENWK1 + I IF (WK(LJFO) .EQ. 0.0D0) GO TO 80 WK(LJFO) = 0.0D0 SAVR(I) = 0.0D0 GO TO 85 80 IF (SAVR(I) .EQ. 0.0D0) GO TO 90 SAVR(I) = 0.0D0 85 IF (K .GT. LIWK) GO TO 310 IWK(K) = I K = K+1 90 CONTINUE IWK(IPIAN+J) = K + 1 - IPJAN 95 CONTINUE GO TO 240 C C MOSS = 2. Compute structure from results of N + 1 calls to RES. ------ 100 DO 105 I = 1,N 105 WK(LENWK1+I) = 0.0D0 K = IPJAN IWK(IPIAN) = 1 IER = -1 IF (MITER .EQ. 1) IER = 1 CALL RES (NEQ, TN, Y, S, SAVR, IER) IF (IER .GT. 1) GO TO 370 DO 130 J = 1,N CALL ADDA (NEQ, TN, Y, J, IWK(IPIAN), IWK(IPJAN), WK(LENWK1+1)) YJ = Y(J) ERWT = 1.0D0/EWT(J) Y(J) = YJ + SIGN(ERWT,YJ) CALL RES (NEQ, TN, Y, S, RTEM, IER) IF (IER .GT. 1) RETURN Y(J) = YJ DO 120 I = 1,N LJFO = LENWK1 + I IF (WK(LJFO) .EQ. 0.0D0) GO TO 110 WK(LJFO) = 0.0D0 GO TO 115 110 IF (RTEM(I) .EQ. SAVR(I)) GO TO 120 115 IF (K .GT. LIWK) GO TO 310 IWK(K) = I K = K + 1 120 CONTINUE IWK(IPIAN+J) = K + 1 - IPJAN 130 CONTINUE GO TO 240 C C MOSS = 3. Compute structure from the user's IA/JA and JAC routine. --- 150 CONTINUE C A dummy call to RES allows user to create temporaries for use in JAC. IER = 1 CALL RES (NEQ, TN, Y, S, SAVR, IER) IF (IER .GT. 1) GO TO 370 DO 155 I = 1,N 155 SAVR(I) = 0.0D0 KNEW = IPJAN KAMIN = IA(1) IWK(IPIAN) = 1 DO 190 J = 1,N CALL JAC (NEQ, TN, Y, S, J, IWK(IPIAN), IWK(IPJAN), SAVR) KAMAX = IA(J+1) - 1 IF (KAMIN .GT. KAMAX) GO TO 170 DO 160 K = KAMIN,KAMAX I = JA(K) SAVR(I) = 0.0D0 IF (KNEW .GT. LIWK) GO TO 310 IWK(KNEW) = I KNEW = KNEW + 1 160 CONTINUE 170 KAMIN = KAMAX + 1 DO 180 I = 1,N IF (SAVR(I) .EQ. 0.0D0) GO TO 180 SAVR(I) = 0.0D0 IF (KNEW .GT. LIWK) GO TO 310 IWK(KNEW) = I KNEW = KNEW + 1 180 CONTINUE IWK(IPIAN+J) = KNEW + 1 - IPJAN 190 CONTINUE GO TO 240 C C MOSS = 4. Compute structure from user's IA/JA and N + 1 RES calls. --- 200 KNEW = IPJAN KAMIN = IA(1) IWK(IPIAN) = 1 IER = -1 IF (MITER .EQ. 1) IER = 1 CALL RES (NEQ, TN, Y, S, SAVR, IER) IF (IER .GT. 1) GO TO 370 DO 235 J = 1,N YJ = Y(J) ERWT = 1.0D0/EWT(J) Y(J) = YJ + SIGN(ERWT,YJ) CALL RES (NEQ, TN, Y, S, RTEM, IER) IF (IER .GT. 1) RETURN Y(J) = YJ KAMAX = IA(J+1) - 1 IF (KAMIN .GT. KAMAX) GO TO 225 DO 220 K = KAMIN,KAMAX I = JA(K) RTEM(I) = SAVR(I) IF (KNEW .GT. LIWK) GO TO 310 IWK(KNEW) = I KNEW = KNEW + 1 220 CONTINUE 225 KAMIN = KAMAX + 1 DO 230 I = 1,N IF (RTEM(I) .EQ. SAVR(I)) GO TO 230 IF (KNEW .GT. LIWK) GO TO 310 IWK(KNEW) = I KNEW = KNEW + 1 230 CONTINUE IWK(IPIAN+J) = KNEW + 1 - IPJAN 235 CONTINUE C 240 CONTINUE IF (MOSS .EQ. 0 .OR. ISTATC .EQ. 3) GO TO 250 C If ISTATE = 0 or 1 and MOSS .ne. 0, restore Y from YH. --------------- DO 245 I = 1,N 245 Y(I) = YH(I) 250 NNZ = IWK(IPIAN+N) - 1 IPPER = 0 NGP = 0 LENIGP = 0 IPIGP = IPJAN + NNZ IF (MITER .NE. 2) GO TO 260 C C Compute grouping of column indices (MITER = 2). ---------------------- C MAXG = NP1 IPJGP = IPJAN + NNZ IBJGP = IPJGP - 1 IPIGP = IPJGP + N IPTT1 = IPIGP + NP1 IPTT2 = IPTT1 + N LREQ = IPTT2 + N - 1 IF (LREQ .GT. LIWK) GO TO 320 CALL JGROUP (N, IWK(IPIAN), IWK(IPJAN), MAXG, NGP, IWK(IPIGP), 1 IWK(IPJGP), IWK(IPTT1), IWK(IPTT2), IER) IF (IER .NE. 0) GO TO 320 LENIGP = NGP + 1 C C Compute new ordering of rows/columns of Jacobian. -------------------- 260 IPR = IPIGP + LENIGP IPC = IPR IPIC = IPC + N IPISP = IPIC + N IPRSP = (IPISP-2)/LRAT + 2 IESP = LENWK + 1 - IPRSP IF (IESP .LT. 0) GO TO 330 IBR = IPR - 1 DO 270 I = 1,N 270 IWK(IBR+I) = I NSP = LIWK + 1 - IPISP CALL ODRV(N, IWK(IPIAN), IWK(IPJAN), WK, IWK(IPR), IWK(IPIC), NSP, 1 IWK(IPISP), 1, IYS) IF (IYS .EQ. 11*N+1) GO TO 340 IF (IYS .NE. 0) GO TO 330 C C Reorder JAN and do symbolic LU factorization of matrix. -------------- IPA = LENWK + 1 - NNZ NSP = IPA - IPRSP LREQ = MAX(12*N/LRAT, 6*N/LRAT+2*N+NNZ) + 3 LREQ = LREQ + IPRSP - 1 + NNZ IF (LREQ .GT. LENWK) GO TO 350 IBA = IPA - 1 DO 280 I = 1,NNZ 280 WK(IBA+I) = 0.0D0 IPISP = LRAT*(IPRSP - 1) + 1 CALL CDRV(N,IWK(IPR),IWK(IPC),IWK(IPIC),IWK(IPIAN),IWK(IPJAN), 1 WK(IPA),WK(IPA),WK(IPA),NSP,IWK(IPISP),WK(IPRSP),IESP,5,IYS) LREQ = LENWK - IESP IF (IYS .EQ. 10*N+1) GO TO 350 IF (IYS .NE. 0) GO TO 360 IPIL = IPISP IPIU = IPIL + 2*N + 1 NZU = IWK(IPIL+N) - IWK(IPIL) NZL = IWK(IPIU+N) - IWK(IPIU) IF (LRAT .GT. 1) GO TO 290 CALL ADJLR (N, IWK(IPISP), LDIF) LREQ = LREQ + LDIF 290 CONTINUE IF (LRAT .EQ. 2 .AND. NNZ .EQ. N) LREQ = LREQ + 1 NSP = NSP + LREQ - LENWK IPA = LREQ + 1 - NNZ IBA = IPA - 1 IPPER = 0 RETURN C 310 IPPER = -1 LREQ = 2 + (2*N + 1)/LRAT LREQ = MAX(LENWK+1,LREQ) RETURN C 320 IPPER = -2 LREQ = (LREQ - 1)/LRAT + 1 RETURN C 330 IPPER = -3 CALL CNTNZU (N, IWK(IPIAN), IWK(IPJAN), NZSUT) LREQ = LENWK - IESP + (3*N + 4*NZSUT - 1)/LRAT + 1 RETURN C 340 IPPER = -4 RETURN C 350 IPPER = -5 RETURN C 360 IPPER = -6 LREQ = LENWK RETURN C 370 IPPER = -IER - 5 LREQ = 2 + (2*N + 1)/LRAT RETURN C----------------------- End of Subroutine DPREPI ---------------------- END *DECK DAINVGS SUBROUTINE DAINVGS (NEQ, T, Y, WK, IWK, TEM, YDOT, IER, RES, ADDA) EXTERNAL RES, ADDA INTEGER NEQ, IWK, IER INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU INTEGER I, IMUL, J, K, KMIN, KMAX DOUBLE PRECISION T, Y, WK, TEM, YDOT DOUBLE PRECISION RLSS DIMENSION Y(*), WK(*), IWK(*), TEM(*), YDOT(*) COMMON /DLSS01/ RLSS(6), 1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU C----------------------------------------------------------------------- C This subroutine computes the initial value of the vector YDOT C satisfying C A * YDOT = g(t,y) C when A is nonsingular. It is called by DLSODIS for initialization C only, when ISTATE = 0. The matrix A is subjected to LU C decomposition in CDRV. Then the system A*YDOT = g(t,y) is solved C in CDRV. C In addition to variables described previously, communication C with DAINVGS uses the following: C Y = array of initial values. C WK = real work space for matrices. On output it contains A and C its LU decomposition. The LU decomposition is not entirely C sparse unless the structure of the matrix A is identical to C the structure of the Jacobian matrix dr/dy. C Storage of matrix elements starts at WK(3). C WK(1) = SQRT(UROUND), not used here. C IWK = integer work space for matrix-related data, assumed to C be equivalenced to WK. In addition, WK(IPRSP) and WK(IPISP) C are assumed to have identical locations. C TEM = vector of work space of length N (ACOR in DSTODI). C YDOT = output vector containing the initial dy/dt. YDOT(i) contains C dy(i)/dt when the matrix A is non-singular. C IER = output error flag with the following values and meanings: C = 0 if DAINVGS was successful. C = 1 if the A-matrix was found to be singular. C = 2 if RES returned an error flag IRES = IER = 2. C = 3 if RES returned an error flag IRES = IER = 3. C = 4 if insufficient storage for CDRV (should not occur here). C = 5 if other error found in CDRV (should not occur here). C----------------------------------------------------------------------- C DO 10 I = 1,NNZ 10 WK(IBA+I) = 0.0D0 C IER = 1 CALL RES (NEQ, T, Y, WK(IPA), YDOT, IER) IF (IER .GT. 1) RETURN C KMIN = IWK(IPIAN) DO 30 J = 1,NEQ KMAX = IWK(IPIAN+J) - 1 DO 15 K = KMIN,KMAX I = IWK(IBJAN+K) 15 TEM(I) = 0.0D0 CALL ADDA (NEQ, T, Y, J, IWK(IPIAN), IWK(IPJAN), TEM) DO 20 K = KMIN,KMAX I = IWK(IBJAN+K) 20 WK(IBA+K) = TEM(I) KMIN = KMAX + 1 30 CONTINUE NLU = NLU + 1 IER = 0 DO 40 I = 1,NEQ 40 TEM(I) = 0.0D0 C C Numerical factorization of matrix A. --------------------------------- CALL CDRV (NEQ,IWK(IPR),IWK(IPC),IWK(IPIC),IWK(IPIAN),IWK(IPJAN), 1 WK(IPA),TEM,TEM,NSP,IWK(IPISP),WK(IPRSP),IESP,2,IYS) IF (IYS .EQ. 0) GO TO 50 IMUL = (IYS - 1)/NEQ IER = 5 IF (IMUL .EQ. 8) IER = 1 IF (IMUL .EQ. 10) IER = 4 RETURN C C Solution of the linear system. --------------------------------------- 50 CALL CDRV (NEQ,IWK(IPR),IWK(IPC),IWK(IPIC),IWK(IPIAN),IWK(IPJAN), 1 WK(IPA),YDOT,YDOT,NSP,IWK(IPISP),WK(IPRSP),IESP,4,IYS) IF (IYS .NE. 0) IER = 5 RETURN C----------------------- End of Subroutine DAINVGS --------------------- END *DECK DPRJIS SUBROUTINE DPRJIS (NEQ, Y, YH, NYH, EWT, RTEM, SAVR, S, WK, IWK, 1 RES, JAC, ADDA) EXTERNAL RES, JAC, ADDA INTEGER NEQ, NYH, IWK DOUBLE PRECISION Y, YH, EWT, RTEM, SAVR, S, WK DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), RTEM(*), 1 S(*), SAVR(*), WK(*), IWK(*) INTEGER IOWND, IOWNS, 1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 2 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU INTEGER IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 1 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 2 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 3 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU DOUBLE PRECISION ROWNS, 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND DOUBLE PRECISION RLSS COMMON /DLS001/ ROWNS(209), 1 CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND, 2 IOWND(6), IOWNS(6), 3 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L, 4 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER, 5 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU COMMON /DLSS01/ RLSS(6), 1 IPLOST, IESP, ISTATC, IYS, IBA, IBIAN, IBJAN, IBJGP, 2 IPIAN, IPJAN, IPJGP, IPIGP, IPR, IPC, IPIC, IPISP, IPRSP, IPA, 3 LENYH, LENYHM, LENWK, LREQ, LRAT, LREST, LWMIN, MOSS, MSBJ, 4 NSLJ, NGP, NLU, NNZ, NSP, NZL, NZU INTEGER I, IMUL, IRES, J, JJ, JMAX, JMIN, K, KMAX, KMIN, NG DOUBLE PRECISION CON, FAC, HL0, R, SRUR C----------------------------------------------------------------------- C DPRJIS is called to compute and process the matrix C P = A - H*EL(1)*J, where J is an approximation to the Jacobian dr/dy, C where r = g(t,y) - A(t,y)*s. J is computed by columns, either by C the user-supplied routine JAC if MITER = 1, or by finite differencing C if MITER = 2. J is stored in WK, rescaled, and ADDA is called to C generate P. The matrix P is subjected to LU decomposition in CDRV. C P and its LU decomposition are stored separately in WK. C C In addition to variables described previously, communication C with DPRJIS uses the following: C Y = array containing predicted values on entry. C RTEM = work array of length N (ACOR in DSTODI). C SAVR = array containing r evaluated at predicted y. On output it C contains the residual evaluated at current values of t and y. C S = array containing predicted values of dy/dt (SAVF in DSTODI). C WK = real work space for matrices. On output it contains P and C its sparse LU decomposition. Storage of matrix elements C starts at WK(3). C WK also contains the following matrix-related data. C WK(1) = SQRT(UROUND), used in numerical Jacobian increments. C IWK = integer work space for matrix-related data, assumed to be C equivalenced to WK. In addition, WK(IPRSP) and IWK(IPISP) C are assumed to have identical locations. C EL0 = EL(1) (input). C IERPJ = output error flag (in COMMON). C = 0 if no error. C = 1 if zero pivot found in CDRV. C = IRES (= 2 or 3) if RES returned IRES = 2 or 3. C = -1 if insufficient storage for CDRV (should not occur). C = -2 if other error found in CDRV (should not occur here). C JCUR = output flag = 1 to indicate that the Jacobian matrix C (or approximation) is now current. C This routine also uses other variables in Common. C----------------------------------------------------------------------- HL0 = H*EL0 CON = -HL0 JCUR = 1 NJE = NJE + 1 GO TO (100, 200), MITER C C If MITER = 1, call RES, then call JAC and ADDA for each column. ------ 100 IRES = 1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 KMIN = IWK(IPIAN) DO 130 J = 1,N KMAX = IWK(IPIAN+J)-1 DO 110 I = 1,N 110 RTEM(I) = 0.0D0 CALL JAC (NEQ, TN, Y, S, J, IWK(IPIAN), IWK(IPJAN), RTEM) DO 120 I = 1,N 120 RTEM(I) = RTEM(I)*CON CALL ADDA (NEQ, TN, Y, J, IWK(IPIAN), IWK(IPJAN), RTEM) DO 125 K = KMIN,KMAX I = IWK(IBJAN+K) WK(IBA+K) = RTEM(I) 125 CONTINUE KMIN = KMAX + 1 130 CONTINUE GO TO 290 C C If MITER = 2, make NGP + 1 calls to RES to approximate J and P. ------ 200 CONTINUE IRES = -1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 SRUR = WK(1) JMIN = IWK(IPIGP) DO 240 NG = 1,NGP JMAX = IWK(IPIGP+NG) - 1 DO 210 J = JMIN,JMAX JJ = IWK(IBJGP+J) R = MAX(SRUR*ABS(Y(JJ)),0.01D0/EWT(JJ)) 210 Y(JJ) = Y(JJ) + R CALL RES (NEQ,TN,Y,S,RTEM,IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 DO 230 J = JMIN,JMAX JJ = IWK(IBJGP+J) Y(JJ) = YH(JJ,1) R = MAX(SRUR*ABS(Y(JJ)),0.01D0/EWT(JJ)) FAC = -HL0/R KMIN = IWK(IBIAN+JJ) KMAX = IWK(IBIAN+JJ+1) - 1 DO 220 K = KMIN,KMAX I = IWK(IBJAN+K) RTEM(I) = (RTEM(I) - SAVR(I))*FAC 220 CONTINUE CALL ADDA (NEQ, TN, Y, JJ, IWK(IPIAN), IWK(IPJAN), RTEM) DO 225 K = KMIN,KMAX I = IWK(IBJAN+K) WK(IBA+K) = RTEM(I) 225 CONTINUE 230 CONTINUE JMIN = JMAX + 1 240 CONTINUE IRES = 1 CALL RES (NEQ, TN, Y, S, SAVR, IRES) NFE = NFE + 1 IF (IRES .GT. 1) GO TO 600 C C Do numerical factorization of P matrix. ------------------------------ 290 NLU = NLU + 1 IERPJ = 0 DO 295 I = 1,N 295 RTEM(I) = 0.0D0 CALL CDRV (N,IWK(IPR),IWK(IPC),IWK(IPIC),IWK(IPIAN),IWK(IPJAN), 1 WK(IPA),RTEM,RTEM,NSP,IWK(IPISP),WK(IPRSP),IESP,2,IYS) IF (IYS .EQ. 0) RETURN IMUL = (IYS - 1)/N IERPJ = -2 IF (IMUL .EQ. 8) IERPJ = 1 IF (IMUL .EQ. 10) IERPJ = -1 RETURN C Error return for IRES = 2 or IRES = 3 return from RES. --------------- 600 IERPJ = IRES RETURN C----------------------- End of Subroutine DPRJIS ---------------------- END