*DECK RUMACH
REAL FUNCTION RUMACH ()
C***BEGIN PROLOGUE RUMACH
C***PURPOSE Compute the unit roundoff of the machine.
C***CATEGORY R1
C***TYPE SINGLE PRECISION (RUMACH-S, DUMACH-D)
C***KEYWORDS MACHINE CONSTANTS
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C *Usage:
C REAL A, RUMACH
C A = RUMACH()
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 RUMACH
C in a machine-independent manner.
C
C***REFERENCES (NONE)
C***ROUTINES CALLED (NONE)
C***REVISION HISTORY (YYMMDD)
C 930216 DATE WRITTEN
C 930818 Added SLATEC-format prologue. (FNF)
C***END PROLOGUE RUMACH
C
C*Internal Notes:
C-----------------------------------------------------------------------
C Subroutines/functions called by RUMACH.. None
C-----------------------------------------------------------------------
C**End
C
REAL U, COMP
C***FIRST EXECUTABLE STATEMENT RUMACH
U = 1.0E0
10 U = U*0.5E0
COMP = 1.0E0 + U
IF (COMP .NE. 1.0E0) GO TO 10
RUMACH = U*2.0E0
RETURN
C----------------------- End of Function RUMACH ------------------------
END
*DECK SCFODE
SUBROUTINE SCFODE (METH, ELCO, TESCO)
C***BEGIN PROLOGUE SCFODE
C***SUBSIDIARY
C***PURPOSE Set ODE integrator coefficients.
C***TYPE SINGLE PRECISION (SCFODE-S, DCFODE-D)
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C
C SCFODE 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 SCFODE 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 SLSODE
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 SCFODE
C**End
INTEGER METH
INTEGER I, IB, NQ, NQM1, NQP1
REAL ELCO, TESCO
REAL 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 SCFODE
GO TO (100, 200), METH
C
100 ELCO(1,1) = 1.0E0
ELCO(2,1) = 1.0E0
TESCO(1,1) = 0.0E0
TESCO(2,1) = 2.0E0
TESCO(1,2) = 1.0E0
TESCO(3,12) = 0.0E0
PC(1) = 1.0E0
RQFAC = 1.0E0
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) = 0.0E0
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)/2.0E0
TSIGN = 1.0E0
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) = 1.0E0
DO 130 I = 2,NQ
130 ELCO(I+1,NQ) = RQ1FAC*PC(I)/I
AGAMQ = RQFAC*XPIN
RAGQ = 1.0E0/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) = 1.0E0
RQ1FAC = 1.0E0
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) = 0.0E0
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) = 1.0E0
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 SCFODE ----------------------
END
*DECK SINTDY
SUBROUTINE SINTDY (T, K, YH, NYH, DKY, IFLAG)
C***BEGIN PROLOGUE SINTDY
C***SUBSIDIARY
C***PURPOSE Interpolate solution derivatives.
C***TYPE SINGLE PRECISION (SINTDY-S, DINTDY-D)
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C
C SINTDY 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 SLSODE
C***ROUTINES CALLED XERRWV
C***COMMON BLOCKS SLS001
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 010412 Reduced size of Common block /SLS001/. (ACH)
C***END PROLOGUE SINTDY
C**End
INTEGER K, NYH, IFLAG
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER I, IC, J, JB, JB2, JJ, JJ1, JP1
REAL T, YH, DKY
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL C, R, S, TP
CHARACTER*80 MSG
DIMENSION YH(NYH,*), DKY(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
C***FIRST EXECUTABLE STATEMENT SINTDY
IFLAG = 0
IF (K .LT. 0 .OR. K .GT. NQ) GO TO 80
TP = TN - HU - 100.0E0*UROUND*(TN + HU)
IF ((T-TP)*(T-TN) .GT. 0.0E0) 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 = 'SINTDY- K (=I1) illegal '
CALL XERRWV (MSG, 30, 51, 0, 1, K, 0, 0, 0.0E0, 0.0E0)
IFLAG = -1
RETURN
90 MSG = 'SINTDY- T (=R1) illegal '
CALL XERRWV (MSG, 30, 52, 0, 0, 0, 0, 1, T, 0.0E0)
MSG=' T not in interval TCUR - HU (= R1) to TCUR (=R2) '
CALL XERRWV (MSG, 60, 52, 0, 0, 0, 0, 2, TP, TN)
IFLAG = -2
RETURN
C----------------------- END OF SUBROUTINE SINTDY ----------------------
END
*DECK SPREPJ
SUBROUTINE SPREPJ (NEQ, Y, YH, NYH, EWT, FTEM, SAVF, WM, IWM,
1 F, JAC)
C***BEGIN PROLOGUE SPREPJ
C***SUBSIDIARY
C***PURPOSE Compute and process Newton iteration matrix.
C***TYPE SINGLE PRECISION (SPREPJ-S, DPREPJ-D)
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C
C SPREPJ is called by SSTODE 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 SGEFA if MITER = 1 or 2, and by SGBFA if MITER = 4 or 5.
C
C In addition to variables described in SSTODE and SLSODE prologues,
C communication with SPREPJ uses the following:
C Y = array containing predicted values on entry.
C FTEM = work array of length N (ACOR in SSTODE).
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 SLSODE
C***ROUTINES CALLED SGBFA, SGEFA, SVNORM
C***COMMON BLOCKS SLS001
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 010412 Reduced size of Common block /SLS001/. (ACH)
C***END PROLOGUE SPREPJ
C**End
EXTERNAL F, JAC
INTEGER NEQ, NYH, IWM
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
2 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
REAL Y, YH, EWT, FTEM, SAVF, WM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL CON, DI, FAC, HL0, R, R0, SRUR, YI, YJ, YJJ,
1 SVNORM
DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), FTEM(*), SAVF(*),
1 WM(*), IWM(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
C***FIRST EXECUTABLE STATEMENT SPREPJ
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) = 0.0E0
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 = SVNORM (N, SAVF, EWT)
R0 = 1000.0E0*ABS(H)*UROUND*N*FAC
IF (R0 .EQ. 0.0E0) R0 = 1.0E0
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) + 1.0E0
250 J = J + NP1
C Do LU decomposition on P. --------------------------------------------
CALL SGEFA (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*0.1E0
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 = 0.1E0*R0 - H*(WM(I+2) - SAVF(I))
WM(I+2) = 1.0E0
IF (ABS(R0) .LT. UROUND/EWT(I)) GO TO 320
IF (ABS(DI) .EQ. 0.0E0) GO TO 330
WM(I+2) = 0.1E0*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) = 0.0E0
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 = SVNORM (N, SAVF, EWT)
R0 = 1000.0E0*ABS(H)*UROUND*N*FAC
IF (R0 .EQ. 0.0E0) R0 = 1.0E0
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) + 1.0E0
580 II = II + MEBAND
C Do LU decomposition of P. --------------------------------------------
CALL SGBFA (WM(3), MEBAND, N, ML, MU, IWM(21), IER)
IF (IER .NE. 0) IERPJ = 1
RETURN
C----------------------- END OF SUBROUTINE SPREPJ ----------------------
END
*DECK SSOLSY
SUBROUTINE SSOLSY (WM, IWM, X, TEM)
C***BEGIN PROLOGUE SSOLSY
C***SUBSIDIARY
C***PURPOSE ODEPACK linear system solver.
C***TYPE SINGLE 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 SGESL 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 SGBSL.
C Communication with SSOLSY 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 SLSODE
C***ROUTINES CALLED SGBSL, SGESL
C***COMMON BLOCKS SLS001
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 010412 Reduced size of Common block /SLS001/. (ACH)
C***END PROLOGUE SSOLSY
C**End
INTEGER IWM
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER I, MEBAND, ML, MU
REAL WM, X, TEM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL DI, HL0, PHL0, R
DIMENSION WM(*), IWM(*), X(*), TEM(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
C***FIRST EXECUTABLE STATEMENT SSOLSY
IERSL = 0
GO TO (100, 100, 300, 400, 400), MITER
100 CALL SGESL (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 = 1.0E0 - R*(1.0E0 - 1.0E0/WM(I+2))
IF (ABS(DI) .EQ. 0.0E0) GO TO 390
320 WM(I+2) = 1.0E0/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 SGBSL (WM(3), MEBAND, N, ML, MU, IWM(21), X, 0)
RETURN
C----------------------- END OF SUBROUTINE SSOLSY ----------------------
END
*DECK SSRCOM
SUBROUTINE SSRCOM (RSAV, ISAV, JOB)
C***BEGIN PROLOGUE SSRCOM
C***SUBSIDIARY
C***PURPOSE Save/restore ODEPACK COMMON blocks.
C***TYPE SINGLE 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 SLS001, which is used internally
C by one or more ODEPACK solvers.
C
C RSAV = real array of length 9 or more.
C ISAV = integer array of length 25 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 SLSODE
C***ROUTINES CALLED (NONE)
C***COMMON BLOCKS SLS001
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 010412 Reduced Common block length by 209+12. (ACH)
C***END PROLOGUE SSRCOM
C**End
INTEGER ISAV, JOB
INTEGER ILS
INTEGER I, LENILS, LENRLS
REAL RSAV, RLS
DIMENSION RSAV(*), ISAV(*)
COMMON /SLS001/ RLS(9), ILS(25)
DATA LENRLS/9/, LENILS/25/
C
C***FIRST EXECUTABLE STATEMENT SSRCOM
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 SSRCOM ----------------------
END
*DECK SSTODE
SUBROUTINE SSTODE (NEQ, Y, YH, NYH, YH1, EWT, SAVF, ACOR,
1 WM, IWM, F, JAC, PJAC, SLVS)
C***BEGIN PROLOGUE SSTODE
C***SUBSIDIARY
C***PURPOSE Performs one step of an ODEPACK integration.
C***TYPE SINGLE PRECISION (SSTODE-S, DSTODE-D)
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C
C SSTODE performs one step of the integration of an initial value
C problem for a system of ordinary differential equations.
C Note: SSTODE 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 SSTODE 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 SLSODE
C***ROUTINES CALLED SCFODE, SVNORM
C***COMMON BLOCKS SLS001
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 010413 Reduced size of Common block /SLS001/. (ACH)
C***END PROLOGUE SSTODE
C**End
EXTERNAL F, JAC, PJAC, SLVS
INTEGER NEQ, NYH, IWM
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER I, I1, IREDO, IRET, J, JB, M, NCF, NEWQ
INTEGER IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
REAL Y, YH, YH1, EWT, SAVF, ACOR, WM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL DCON, DDN, DEL, DELP, DSM, DUP, EXDN, EXSM, EXUP,
1 R, RH, RHDN, RHSM, RHUP, TOLD, SVNORM,
2 CONIT, CRATE, EL(13), ELCO(13,12), HOLD, RMAX, TESCO(3,12)
DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*),
1 ACOR(*), WM(*), IWM(*)
SAVE CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO,
1 IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
C
C***FIRST EXECUTABLE STATEMENT SSTODE
KFLAG = 0
TOLD = TN
NCF = 0
IERPJ = 0
IERSL = 0
JCUR = 0
ICF = 0
DELP = 0.0E0
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 = 10000.0E0
RC = 0.0E0
EL0 = 1.0E0
CRATE = 0.7E0
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, SCFODE 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 SCFODE (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 = 0.5E0/(NQ+2)
DDN = SVNORM (N, SAVF, EWT)/TESCO(1,L)
EXDN = 1.0E0/L
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
RH = MIN(RHDN,1.0E0)
IREDO = 3
IF (H .EQ. HOLD) GO TO 170
RH = MIN(RH,ABS(H/HOLD))
H = HOLD
GO TO 175
C-----------------------------------------------------------------------
C SCFODE 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 SCFODE (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 = 0.5E0/(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(1.0E0,ABS(H)*HMXI*RH)
R = 1.0E0
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-1.0E0) .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 = 1.0E0
NSLP = NST
CRATE = 0.7E0
IF (IERPJ .NE. 0) GO TO 430
250 DO 260 I = 1,N
260 ACOR(I) = 0.0E0
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 = SVNORM (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 = SVNORM (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(0.2E0*CRATE,DEL/DELP)
DCON = DEL*MIN(1.0E0,1.5E0*CRATE)/(TESCO(2,NQ)*CONIT)
IF (DCON .LE. 1.0E0) GO TO 450
M = M + 1
IF (M .EQ. MAXCOR) GO TO 410
IF (M .GE. 2 .AND. DEL .GT. 2.0E0*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 = 2.0E0
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*1.00001E0) GO TO 670
IF (NCF .EQ. MXNCF) GO TO 670
RH = 0.25E0
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 = SVNORM (N, ACOR, EWT)/TESCO(2,NQ)
IF (DSM .GT. 1.0E0) 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 = 2.0E0
IF (ABS(H) .LE. HMIN*1.00001E0) GO TO 660
IF (KFLAG .LE. -3) GO TO 640
IREDO = 2
RHUP = 0.0E0
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 = 0.0E0
IF (L .EQ. LMAX) GO TO 540
DO 530 I = 1,N
530 SAVF(I) = ACOR(I) - YH(I,LMAX)
DUP = SVNORM (N, SAVF, EWT)/TESCO(3,NQ)
EXUP = 1.0E0/(L+1)
RHUP = 1.0E0/(1.4E0*DUP**EXUP + 0.0000014E0)
540 EXSM = 1.0E0/L
RHSM = 1.0E0/(1.2E0*DSM**EXSM + 0.0000012E0)
RHDN = 0.0E0
IF (NQ .EQ. 1) GO TO 560
DDN = SVNORM (N, YH(1,L), EWT)/TESCO(1,NQ)
EXDN = 1.0E0/NQ
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
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. 1.0E0) RH = 1.0E0
GO TO 620
590 NEWQ = L
RH = RHUP
IF (RH .LT. 1.1E0) 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. 1.1E0)) GO TO 610
IF (KFLAG .LE. -2) RH = MIN(RH,0.2E0)
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 = 0.1E0
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 = 10.0E0
700 R = 1.0E0/TESCO(2,NQU)
DO 710 I = 1,N
710 ACOR(I) = ACOR(I)*R
720 HOLD = H
JSTART = 1
RETURN
C----------------------- END OF SUBROUTINE SSTODE ----------------------
END
*DECK SEWSET
SUBROUTINE SEWSET (N, ITOL, RTOL, ATOL, YCUR, EWT)
C***BEGIN PROLOGUE SEWSET
C***SUBSIDIARY
C***PURPOSE Set error weight vector.
C***TYPE SINGLE 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 SLSODE
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 SEWSET
C**End
INTEGER N, ITOL
INTEGER I
REAL RTOL, ATOL, YCUR, EWT
DIMENSION RTOL(*), ATOL(*), YCUR(N), EWT(N)
C
C***FIRST EXECUTABLE STATEMENT SEWSET
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 SEWSET ----------------------
END
*DECK SVNORM
REAL FUNCTION SVNORM (N, V, W)
C***BEGIN PROLOGUE SVNORM
C***SUBSIDIARY
C***PURPOSE Weighted root-mean-square vector norm.
C***TYPE SINGLE 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 SVNORM = SQRT( (1/N) * SUM( V(i)*W(i) )**2 )
C
C***SEE ALSO SLSODE
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 SVNORM
C**End
INTEGER N, I
REAL V, W, SUM
DIMENSION V(N), W(N)
C
C***FIRST EXECUTABLE STATEMENT SVNORM
SUM = 0.0E0
DO 10 I = 1,N
10 SUM = SUM + (V(I)*W(I))**2
SVNORM = SQRT(SUM/N)
RETURN
C----------------------- END OF FUNCTION SVNORM ------------------------
END
*DECK SIPREP
SUBROUTINE SIPREP (NEQ, Y, RWORK, IA, JA, IPFLAG, F, JAC)
EXTERNAL F, JAC
INTEGER NEQ, IA, JA, IPFLAG
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 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
INTEGER I, IMAX, LEWTN, LYHD, LYHN
REAL Y, RWORK
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL RLSS
DIMENSION NEQ(*), Y(*), RWORK(*), IA(*), JA(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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
COMMON /SLSS01/ 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 routine serves as an interface between the driver and
C Subroutine SPREP. It is called only if MITER is 1 or 2.
C Tasks performed here are:
C * call SPREP,
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 SPREP error flag IPPER
C if there was trouble in Subroutine SPREP.
C-----------------------------------------------------------------------
IPFLAG = 0
C Call SPREP to do matrix preprocessing operations. --------------------
CALL SPREP (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 SPREP 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 SIPREP ----------------------
END
*DECK SPREP
SUBROUTINE SPREP (NEQ, Y, YH, SAVF, EWT, FTEM, IA, JA,
1 WK, IWK, IPPER, F, JAC)
EXTERNAL F,JAC
INTEGER NEQ, IA, JA, IWK, IPPER
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 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
INTEGER I, IBR, IER, IPIL, IPIU, IPTT1, IPTT2, J, JFOUND, K,
1 KNEW, KMAX, KMIN, LDIF, LENIGP, LIWK, MAXG, NP1, NZSUT
REAL Y, YH, SAVF, EWT, FTEM, WK
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH
REAL DQ, DYJ, ERWT, FAC, YJ
DIMENSION NEQ(*), Y(*), YH(*), SAVF(*), EWT(*), FTEM(*),
1 IA(*), JA(*), WK(*), IWK(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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
COMMON /SLSS01/ 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
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, SPREP 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 SODRV or SCDRV.
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 SODRV.
C -4 other error flag from SODRV (should never occur).
C -5 insufficient storage for SCDRV.
C -6 other error flag from SCDRV.
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 = 1.0E0/EWT(I)
FAC = 1.0E0 + 1.0E0/(I + 1.0E0)
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) = 0.0E0
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 = 1.0E0/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 SODRV (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) = 0.0E0
IPISP = LRAT*(IPRSP - 1) + 1
CALL SCDRV (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 SPREP -----------------------
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 SODRV.
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 SPRJS
SUBROUTINE SPRJS (NEQ,Y,YH,NYH,EWT,FTEM,SAVF,WK,IWK,F,JAC)
EXTERNAL F,JAC
INTEGER NEQ, NYH, IWK
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 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
INTEGER I, IMUL, J, JJ, JOK, JMAX, JMIN, K, KMAX, KMIN, NG
REAL Y, YH, EWT, FTEM, SAVF, WK
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL CON0, CONMIN, CCMXJ, PSMALL, RBIG, SETH
REAL CON, DI, FAC, HL0, PIJ, R, R0, RCON, RCONT,
1 SRUR, SVNORM
DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), FTEM(*), SAVF(*),
1 WK(*), IWK(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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
COMMON /SLSS01/ 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
C-----------------------------------------------------------------------
C SPRJS 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 SCDRV.
C P and its LU decomposition are stored (separately) in WK.
C
C In addition to variables described previously, communication
C with SPRJS uses the following:
C Y = array containing predicted values on entry.
C FTEM = work array of length N (ACOR in SSTODE).
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 SCDRV.
C = 2 if a singular matrix arose with MITER = 3.
C = -1 if insufficient storage for SCDRV (should not occur here).
C = -2 if other error found in SCDRV (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 - 1.0E0) .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) = 0.0E0
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) + 1.0E0
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 = SVNORM(N, SAVF, EWT)
R0 = 1000.0E0 * ABS(H) * UROUND * N * FAC
IF (R0 .EQ. 0.0E0) R0 = 1.0E0
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) + 1.0E0
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 - 1.0E0
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 + 1.0E0
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) = 0.0E0
CALL SCDRV (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*0.1E0
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 = 0.1E0*R0 - H*(WK(I+2) - SAVF(I))
WK(I+2) = 1.0E0
IF (ABS(R0) .LT. UROUND/EWT(I)) GO TO 320
IF (ABS(DI) .EQ. 0.0E0) GO TO 330
WK(I+2) = 0.1E0*R0/DI
320 CONTINUE
RETURN
330 IERPJ = 2
RETURN
C----------------------- End of Subroutine SPRJS -----------------------
END
*DECK SSOLSS
SUBROUTINE SSOLSS (WK, IWK, X, TEM)
INTEGER IWK
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 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
INTEGER I
REAL WK, X, TEM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL RLSS
REAL DI, HL0, PHL0, R
DIMENSION WK(*), IWK(*), X(*), TEM(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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
COMMON /SLSS01/ 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 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 SCDRV 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 SSOLSS 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 SCDRV 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 SCDRV (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 = 1.0E0 - R*(1.0E0 - 1.0E0/WK(I+2))
IF (ABS(DI) .EQ. 0.0E0) GO TO 390
320 WK(I+2) = 1.0E0/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 SSOLSS ----------------------
END
*DECK SSRCMS
SUBROUTINE SSRCMS (RSAV, ISAV, JOB)
C-----------------------------------------------------------------------
C This routine saves or restores (depending on JOB) the contents of
C the Common blocks SLS001, SLSS01, which are used
C internally by one or more ODEPACK solvers.
C
C RSAV = real array of length 15 or more.
C ISAV = integer array of length 59 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, LENIL, LENISS, LENRL, LENRSS
REAL RSAV, RLS, RLSS
DIMENSION RSAV(*), ISAV(*)
COMMON /SLS001/ RLS(9), ILS(25)
COMMON /SLSS01/ RLSS(6), ILSS(34)
DATA LENRL/9/, LENIL/25/, LENRSS/6/, LENISS/34/
C
IF (JOB .EQ. 2) GO TO 100
DO 10 I = 1,LENRL
10 RSAV(I) = RLS(I)
DO 15 I = 1,LENRSS
15 RSAV(LENRL+I) = RLSS(I)
C
DO 20 I = 1,LENIL
20 ISAV(I) = ILS(I)
DO 25 I = 1,LENISS
25 ISAV(LENIL+I) = ILSS(I)
C
RETURN
C
100 CONTINUE
DO 110 I = 1,LENRL
110 RLS(I) = RSAV(I)
DO 115 I = 1,LENRSS
115 RLSS(I) = RSAV(LENRL+I)
C
DO 120 I = 1,LENIL
120 ILS(I) = ISAV(I)
DO 125 I = 1,LENISS
125 ILSS(I) = ISAV(LENIL+I)
C
RETURN
C----------------------- End of Subroutine SSRCMS ----------------------
END
*DECK SODRV
subroutine sodrv
* (n, ia,ja,a, p,ip, nsp,isp, path, flag)
c 5/2/83
c***********************************************************************
c sodrv -- driver for sparse matrix reordering routines
c***********************************************************************
c
c description
c
c sodrv 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.. sodrv 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 sodrv module have been made in
c the coding in subroutine mdi, and in the initial comments in
c subroutines sodrv and md.
c
c if only the nonzero entries in the upper triangle of m are being
c stored, then sodrv 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 sodrv
c 11n+1 illegal path specification
c
c
c conversion from real to double precision
c
c change the real declarations in sodrv and ssro to double precision
c declarations.
c
c-----------------------------------------------------------------------
c
integer ia(*), ja(*), p(*), ip(*), isp(*), path, flag,
* v, l, head, tmp, q
real a(*)
c... 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 ssro
* (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 ssro
* (n, ip, ia,ja,a, q, r, dflag)
c***********************************************************************
c ssro -- 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 ssro 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(*)
real a(*), ak
c... 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 SCDRV
subroutine scdrv
* (n, r,c,ic, ia,ja,a, b, z, nsp,isp,rsp,esp, path, flag)
c*** subroutine scdrv
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 (snroc) 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 snroc 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 snsfc -- the matrix m is processed symbolically to determine where
c fillin will occur during the numeric factorization.
c snnfc -- 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 snnsc -- the linear system mx = b is solved using the ldu
c or factorization from snnfc.
c snntc -- 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, snsfc need be done only once (for the first
c system). then snnfc is done once for each additional system. for
c several systems with the same coefficient matrix, snsfc and snnfc
c need be done only once (for the first system). then snnsc or snntc
c is done once for each additional right-hand side.
c
c nv - path - path specification. values and their meanings are --
c - 1 perform snroc, snsfc, and snnfc.
c - 2 perform snnfc only (snsfc is assumed to have been
c - done in a manner compatible with the storage
c - allocation used in the driver).
c - 3 perform snnsc only (snsfc and snnfc are assumed
c - to have been done in a manner compatible with
c - the storage allocation used in the driver).
c - 4 perform snntc only (snsfc and snnfc are assumed
c - to have been done in a manner compatible with
c - the storage allocation used in the driver).
c - 5 perform snroc and snsfc.
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 snsfc -- row = k
c - 4n+1 insufficient storage in snnfc
c - 5n+k null pivot -- row = k
c - 6n+k insufficient storage in snsfc -- row = k
c - 7n+1 insufficient storage in snnfc
c - 8n+k zero pivot -- row = k
c - 10n+1 insufficient storage in scdrv
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 (snsfc), 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 (snnfc)).
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 scdrv, as indicated below.
c (3) change e0 to d0 in the constants in statement number 10
c in subroutine snnfc and the line following that.
c
integer r(*), c(*), ic(*), ia(*), ja(*), isp(*), esp, path,
* flag, d, u, q, row, tmp, ar, umax
real a(*), b(*), z(*), rsp(*)
c 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/1/
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 snsfc 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 snroc
* (n, ic, ia,ja,a, isp(il), rsp(ar), isp(iu), flag)
if (flag.ne.0) go to 100
c
3 call snsfc
* (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 snnfc
* (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 snnsc
* (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 snntc
* (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 snroc, snsfc, snnfc, or snnsc
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 snroc (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 snroc should be used to reorder the coefficient
c matrix. the calling sequence is --
c ( (matrix ordering))
c (snroc (matrix reordering))
c snsfc (symbolic factorization to determine where fillin will
c occur during numeric factorization)
c snnfc (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 snnsc (solution of linear system for additional right-hand
c side using ldu factorization from snnfc)
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 snsfc, 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 snnfc, 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 snnsc or snntc, 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 snsfc).
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 snsfc).
c fra - z - solution x.
c - size = n.
c
c ----------------------------------------------------------------
c
c*** subroutine snroc
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
real a(*), ar(*)
c 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 snsfc
* (n, r, ic, ia,ja, jlmax,il,jl,ijl, jumax,iu,ju,iju,
* q, ira,jra, irac, irl,jrl, iru,jru, flag)
c*** subroutine snsfc
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 snnfc
* (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 snnfc
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
real a(*), l(*), d(*), u(*), z(*), b(*), row(*)
real tmp(*), lki, sum, dk
c double precision a(*), l(*), d(*), u(*), z(*), b(*), row(*)
c 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. 0.0e0) go to 108
dk = 1.0e0 / 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 snnsc
* (n, r, c, il, jl, ijl, l, d, iu, ju, iju, u, z, b, tmp)
c*** subroutine snnsc
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(*)
real l(*), d(*), u(*), b(*), z(*), tmp(*), tmpk, sum
c 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 snntc
* (n, r, c, il, jl, ijl, l, d, iu, ju, iju, u, z, b, tmp)
c*** subroutine snntc
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(*)
real l(*), d(*), u(*), b(*), z(*), tmp(*), tmpk,sum
c 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 SSTODA
SUBROUTINE SSTODA (NEQ, Y, YH, NYH, YH1, EWT, SAVF, ACOR,
1 WM, IWM, F, JAC, PJAC, SLVS)
EXTERNAL F, JAC, PJAC, SLVS
INTEGER NEQ, NYH, IWM
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER JTYP, MUSED, MXORDN, MXORDS
INTEGER I, I1, ICOUNT, IRFLAG, IREDO, IRET, J, JB, M, NCF, NEWQ
INTEGER LM1, LM1P1, LM2, LM2P1, NQM1, NQM2
INTEGER IALTH, IPUP, LMAX, NQNYH, NSLP
REAL Y, YH, YH1, EWT, SAVF, ACOR, WM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL PDNORM
REAL CONIT, CRATE, EL(13), ELCO(13,12), HOLD, RMAX,
1 TESCO(3,12)
REAL DCON, DDN, DEL, DELP, DSM, DUP, EXDN, EXSM, EXUP,
1 R, RH, RHDN, RHSM, RHUP, TOLD, SMNORM
REAL ALPHA, CM1(12),CM2(5), DM1,DM2, EXM1,EXM2, PDEST,
1 PDLAST, PDH, PNORM, RATE, RATIO, RH1, RH1IT, RH2, RM, SM1(12)
DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*),
1 ACOR(*), WM(*), IWM(*)
SAVE CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO,
1 IALTH, IPUP, LMAX, NQNYH, NSLP
SAVE CM1, CM2, PDEST, PDLAST, RATIO, SM1, ICOUNT, IRFLAG
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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
COMMON /SLSA01/ PDNORM, JTYP, MUSED, MXORDN, MXORDS
DATA SM1/0.5E0, 0.575E0, 0.55E0, 0.45E0, 0.35E0, 0.25E0,
1 0.20E0, 0.15E0, 0.10E0, 0.075E0, 0.050E0, 0.025E0/
C-----------------------------------------------------------------------
C SSTODA performs one step of the integration of an initial value
C problem for a system of ordinary differential equations.
C Note: SSTODA 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 SSTODA 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 SSTODA.
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 SLSODA/SLSODAR argument JT is
C communicated here as JTYP, but is not used in SSTODA
C except to load MITER following a method switch.)
C MITER may be reset by SSTODA.
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 = 0.0E0
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 SCFODE is called to get the needed coefficients for both methods.
C-----------------------------------------------------------------------
LMAX = MAXORD + 1
NQ = 1
L = 2
IALTH = 2
RMAX = 10000.0E0
RC = 0.0E0
EL0 = 1.0E0
CRATE = 0.7E0
HOLD = H
NSLP = 0
IPUP = MITER
IRET = 3
C Initialize switching parameters. METH = 1 is assumed initially. -----
ICOUNT = 20
IRFLAG = 0
PDEST = 0.0E0
PDLAST = 0.0E0
RATIO = 5.0E0
CALL SCFODE (2, ELCO, TESCO)
DO 10 I = 1,5
10 CM2(I) = TESCO(2,I)*ELCO(I+1,I)
CALL SCFODE (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, SCFODE 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 SCFODE (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 = 0.5E0/(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(1.0E0,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,0.000001E0)
IF (RH*PDH*1.00001E0 .LT. SM1(NQ)) GO TO 178
RH = SM1(NQ)/PDH
IRFLAG = 1
178 CONTINUE
R = 1.0E0
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-1.0E0) .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 = SMNORM (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 = 0.0E0
DEL = 0.0E0
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 = 1.0E0
NSLP = NST
CRATE = 0.7E0
IF (IERPJ .NE. 0) GO TO 430
250 DO 260 I = 1,N
260 ACOR(I) = 0.0E0
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 = SMNORM (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 = SMNORM (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. 100.0E0*PNORM*UROUND) GO TO 450
IF (M .EQ. 0 .AND. METH .EQ. 1) GO TO 405
IF (M .EQ. 0) GO TO 402
RM = 1024.0E0
IF (DEL .LE. 1024.0E0*DELP) RM = DEL/DELP
RATE = MAX(RATE,RM)
CRATE = MAX(0.2E0*CRATE,RM)
402 DCON = DEL*MIN(1.0E0,1.5E0*CRATE)/(TESCO(2,NQ)*CONIT)
IF (DCON .GT. 1.0E0) GO TO 405
PDEST = MAX(PDEST,RATE/ABS(H*EL(1)))
IF (PDEST .NE. 0.0E0) PDLAST = PDEST
GO TO 450
405 CONTINUE
M = M + 1
IF (M .EQ. MAXCOR) GO TO 410
IF (M .GE. 2 .AND. DEL .GT. 2.0E0*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 = 2.0E0
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*1.00001E0) GO TO 670
IF (NCF .EQ. MXNCF) GO TO 670
RH = 0.25E0
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 = SMNORM (N, ACOR, EWT)/TESCO(2,NQ)
IF (DSM .GT. 1.0E0) 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. 100.0E0*PNORM*UROUND .AND. PDEST .NE. 0.0E0)
1 GO TO 470
IF (IRFLAG .EQ. 0) GO TO 488
RH2 = 2.0E0
NQM2 = MIN(NQ,MXORDS)
GO TO 478
470 CONTINUE
EXSM = 1.0E0/L
RH1 = 1.0E0/(1.2E0*DSM**EXSM + 0.0000012E0)
RH1IT = 2.0E0*RH1
PDH = PDLAST*ABS(H)
IF (PDH*RH1 .GT. 0.00001E0) RH1IT = SM1(NQ)/PDH
RH1 = MIN(RH1,RH1IT)
IF (NQ .LE. MXORDS) GO TO 474
NQM2 = MXORDS
LM2 = MXORDS + 1
EXM2 = 1.0E0/LM2
LM2P1 = LM2 + 1
DM2 = SMNORM (N, YH(1,LM2P1), EWT)/CM2(MXORDS)
RH2 = 1.0E0/(1.2E0*DM2**EXM2 + 0.0000012E0)
GO TO 476
474 DM2 = DSM*(CM1(NQ)/CM2(NQ))
RH2 = 1.0E0/(1.2E0*DM2**EXSM + 0.0000012E0)
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 = 0.0E0
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 = 1.0E0/L
IF (MXORDN .GE. NQ) GO TO 484
NQM1 = MXORDN
LM1 = MXORDN + 1
EXM1 = 1.0E0/LM1
LM1P1 = LM1 + 1
DM1 = SMNORM (N, YH(1,LM1P1), EWT)/CM1(MXORDN)
RH1 = 1.0E0/(1.2E0*DM1**EXM1 + 0.0000012E0)
GO TO 486
484 DM1 = DSM*(CM2(NQ)/CM1(NQ))
RH1 = 1.0E0/(1.2E0*DM1**EXSM + 0.0000012E0)
NQM1 = NQ
EXM1 = EXSM
486 RH1IT = 2.0E0*RH1
PDH = PDNORM*ABS(H)
IF (PDH*RH1 .GT. 0.00001E0) RH1IT = SM1(NQM1)/PDH
RH1 = MIN(RH1,RH1IT)
RH2 = 1.0E0/(1.2E0*DSM**EXSM + 0.0000012E0)
IF (RH1*RATIO .LT. 5.0E0*RH2) GO TO 488
ALPHA = MAX(0.001E0,RH1)
DM1 = (ALPHA**EXM1)*DM1
IF (DM1 .LE. 1000.0E0*UROUND*PNORM) GO TO 488
C The switch test passed. Reset relevant quantities for Adams. --------
RH = RH1
ICOUNT = 20
METH = 1
MITER = 0
PDLAST = 0.0E0
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 = 2.0E0
IF (ABS(H) .LE. HMIN*1.00001E0) GO TO 660
IF (KFLAG .LE. -3) GO TO 640
IREDO = 2
RHUP = 0.0E0
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 = 0.0E0
IF (L .EQ. LMAX) GO TO 540
DO 530 I = 1,N
530 SAVF(I) = ACOR(I) - YH(I,LMAX)
DUP = SMNORM (N, SAVF, EWT)/TESCO(3,NQ)
EXUP = 1.0E0/(L+1)
RHUP = 1.0E0/(1.4E0*DUP**EXUP + 0.0000014E0)
540 EXSM = 1.0E0/L
RHSM = 1.0E0/(1.2E0*DSM**EXSM + 0.0000012E0)
RHDN = 0.0E0
IF (NQ .EQ. 1) GO TO 550
DDN = SMNORM (N, YH(1,L), EWT)/TESCO(1,NQ)
EXDN = 1.0E0/NQ
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
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,0.000001E0)
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 = 0.0E0
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. 1.0E0) RH = 1.0E0
GO TO 620
590 NEWQ = L
RH = RHUP
IF (RH .LT. 1.1E0) 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*1.00001E0 .GE. SM1(NEWQ)) GO TO 625
622 IF (KFLAG .EQ. 0 .AND. RH .LT. 1.1E0) GO TO 610
625 IF (KFLAG .LE. -2) RH = MIN(RH,0.2E0)
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 = 0.1E0
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 = 10.0E0
700 R = 1.0E0/TESCO(2,NQU)
DO 710 I = 1,N
710 ACOR(I) = ACOR(I)*R
720 HOLD = H
JSTART = 1
RETURN
C----------------------- End of Subroutine SSTODA ----------------------
END
*DECK SPRJA
SUBROUTINE SPRJA (NEQ, Y, YH, NYH, EWT, FTEM, SAVF, WM, IWM,
1 F, JAC)
EXTERNAL F, JAC
INTEGER NEQ, NYH, IWM
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER JTYP, MUSED, MXORDN, MXORDS
INTEGER I, I1, I2, IER, II, J, J1, JJ, LENP,
1 MBA, MBAND, MEB1, MEBAND, ML, ML3, MU, NP1
REAL Y, YH, EWT, FTEM, SAVF, WM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL PDNORM
REAL CON, FAC, HL0, R, R0, SRUR, YI, YJ, YJJ,
1 SMNORM, SFNORM, SBNORM
DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), FTEM(*), SAVF(*),
1 WM(*), IWM(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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
COMMON /SLSA01/ PDNORM, JTYP, MUSED, MXORDN, MXORDS
C-----------------------------------------------------------------------
C SPRJA is called by SSTODA 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 SMNORM) 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 SGEFA if MITER = 1 or 2, and by SGBFA if MITER = 4 or 5.
C
C In addition to variables described previously, communication
C with SPRJA uses the following:
C Y = array containing predicted values on entry.
C FTEM = work array of length N (ACOR in SSTODA).
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) = 0.0E0
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 = SMNORM (N, SAVF, EWT)
R0 = 1000.0E0*ABS(H)*UROUND*N*FAC
IF (R0 .EQ. 0.0E0) R0 = 1.0E0
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 = SFNORM (N, WM(3), EWT)/ABS(HL0)
C Add identity matrix. -------------------------------------------------
J = 3
NP1 = N + 1
DO 250 I = 1,N
WM(J) = WM(J) + 1.0E0
250 J = J + NP1
C Do LU decomposition on P. --------------------------------------------
CALL SGEFA (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) = 0.0E0
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 = SMNORM (N, SAVF, EWT)
R0 = 1000.0E0*ABS(H)*UROUND*N*FAC
IF (R0 .EQ. 0.0E0) R0 = 1.0E0
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 = SBNORM (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) + 1.0E0
580 II = II + MEBAND
C Do LU decomposition of P. --------------------------------------------
CALL SGBFA (WM(3), MEBAND, N, ML, MU, IWM(21), IER)
IF (IER .NE. 0) IERPJ = 1
RETURN
C----------------------- End of Subroutine SPRJA -----------------------
END
*DECK SMNORM
REAL FUNCTION SMNORM (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 SMNORM = MAX(i=1,...,N) ABS(V(i))*W(i)
C-----------------------------------------------------------------------
INTEGER N, I
REAL V, W, VM
DIMENSION V(N), W(N)
VM = 0.0E0
DO 10 I = 1,N
10 VM = MAX(VM,ABS(V(I))*W(I))
SMNORM = VM
RETURN
C----------------------- End of Function SMNORM ------------------------
END
*DECK SFNORM
REAL FUNCTION SFNORM (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 SFNORM = MAX(i=1,...,N) ( W(i) * Sum(j=1,...,N) ABS(a(i,j))/W(j) )
C-----------------------------------------------------------------------
INTEGER N, I, J
REAL A, W, AN, SUM
DIMENSION A(N,N), W(N)
AN = 0.0E0
DO 20 I = 1,N
SUM = 0.0E0
DO 10 J = 1,N
10 SUM = SUM + ABS(A(I,J))/W(J)
AN = MAX(AN,SUM*W(I))
20 CONTINUE
SFNORM = AN
RETURN
C----------------------- End of Function SFNORM ------------------------
END
*DECK SBNORM
REAL FUNCTION SBNORM (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 SBNORM = 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
REAL A, W
REAL AN, SUM
DIMENSION A(NRA,N), W(N)
AN = 0.0E0
DO 20 I = 1,N
SUM = 0.0E0
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
SBNORM = AN
RETURN
C----------------------- End of Function SBNORM ------------------------
END
*DECK SSRCMA
SUBROUTINE SSRCMA (RSAV, ISAV, JOB)
C-----------------------------------------------------------------------
C This routine saves or restores (depending on JOB) the contents of
C the Common blocks SLS001, SLSA01, which are used
C internally by one or more ODEPACK solvers.
C
C RSAV = real array of length 10 or more.
C ISAV = integer array of length 29 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, LENILA
REAL RSAV
REAL RLS, RLSA
DIMENSION RSAV(*), ISAV(*)
COMMON /SLS001/ RLS(9), ILS(25)
COMMON /SLSA01/ RLSA, ILSA(4)
DATA LENRLS/9/, LENILS/25/, LENILA/4/
C
IF (JOB .EQ. 2) GO TO 100
DO 10 I = 1,LENRLS
10 RSAV(I) = RLS(I)
RSAV(LENRLS+1) = RLSA
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)
RLSA = RSAV(LENRLS+1)
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 SSRCMA ----------------------
END
*DECK SRCHEK
SUBROUTINE SRCHEK (JOB, G, NEQ, Y, YH,NYH, G0, G1, GX, JROOT, IRT)
EXTERNAL G
INTEGER JOB, NEQ, NYH, JROOT, IRT
REAL Y, YH, G0, G1, GX
DIMENSION NEQ(*), Y(*), YH(NYH,*), G0(*), G1(*), GX(*), JROOT(*)
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NJE, NQU
INTEGER IRFND, ITASKC, NGC, NGE
INTEGER I, IFLAG, JFLAG
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL T0, TLAST, TOUTC
REAL HMING, T1, TEMP1, TEMP2, X
LOGICAL ZROOT
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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
COMMON /SLSR01/ T0, TLAST, TOUTC, IRFND, ITASKC, NGC, NGE
C-----------------------------------------------------------------------
C This routine checks for the presence of a root in the
C vicinity of the current T, in a manner depending on the
C input flag JOB. It calls Subroutine SROOTS to locate the root
C as precisely as possible.
C
C In addition to variables described previously, SRCHEK
C uses the following for communication:
C JOB = integer flag indicating type of call:
C JOB = 1 means the problem is being initialized, and SRCHEK
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 SRCHEK 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 SRCHEK
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 SRCHEK, 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-----------------------------------------------------------------------
C
IRT = 0
DO 10 I = 1,NGC
10 JROOT(I) = 0
HMING = (ABS(TN) + ABS(H))*UROUND*100.0E0
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. 0.0E0) ZROOT = .TRUE.
IF (.NOT. ZROOT) GO TO 190
C g has a zero at T. Look at g at T + (small increment). --------------
TEMP1 = SIGN(HMING,H)
T0 = T0 + TEMP1
TEMP2 = TEMP1/H
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. 0.0E0) 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 SINTDY (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. 0.0E0) 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. 0.0E0) 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 SINTDY (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. 0.0E0) 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. 0.0E0) GO TO 310
T1 = TOUTC
IF ((T1 - T0)*H .LE. 0.0E0) GO TO 390
CALL SINTDY (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 SROOTS to search for root in interval from T0 to T1. ------------
JFLAG = 0
350 CONTINUE
CALL SROOTS (NGC, HMING, JFLAG, T0, T1, G0, G1, GX, X, JROOT)
IF (JFLAG .GT. 1) GO TO 360
CALL SINTDY (X, 0, YH, NYH, Y, IFLAG)
CALL G (NEQ, X, Y, NGC, GX)
NGE = NGE + 1
GO TO 350
360 T0 = X
CALL SCOPY (NGC, GX, 1, G0, 1)
IF (JFLAG .EQ. 4) GO TO 390
C Found a root. Interpolate to X and return. --------------------------
CALL SINTDY (X, 0, YH, NYH, Y, IFLAG)
IRT = 1
RETURN
C
390 CONTINUE
RETURN
C----------------------- End of Subroutine SRCHEK ----------------------
END
*DECK SROOTS
SUBROUTINE SROOTS (NG, HMIN, JFLAG, X0, X1, G0, G1, GX, X, JROOT)
INTEGER NG, JFLAG, JROOT
REAL HMIN, X0, X1, G0, G1, GX, X
DIMENSION G0(NG), G1(NG), GX(NG), JROOT(NG)
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 SROOTS.
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 SROOTS needs a value of g(x). Set GX = g(X)
C and call SROOTS 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, IMAX, IMXOLD, LAST, NXLAST
REAL ALPHA, T2, TMAX, X2, ZERO
LOGICAL ZROOT, SGNCHG, XROOT
SAVE ALPHA, X2, IMAX, LAST
DATA ZERO/0.0E0/
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(1.0E0,G0(I)) .EQ. SIGN(1.0E0,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 = 1.0E0
GO TO 180
160 IF (LAST .EQ. 0) GO TO 170
ALPHA = 0.5E0*ALPHA
GO TO 180
170 ALPHA = 2.0E0*ALPHA
180 X2 = X1 - (X1-X0)*G1(IMAX)/(G1(IMAX) - ALPHA*G0(IMAX))
IF ((ABS(X2-X0) .LT. HMIN) .AND.
1 (ABS(X1-X0) .GT. 10.0E0*HMIN)) X2 = X0 + 0.1E0*(X1-X0)
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(1.0E0,G0(I)) .EQ. SIGN(1.0E0,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 SCOPY (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 SCOPY (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 SCOPY (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 SCOPY (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(1.0E0,G0(I)) .NE. SIGN(1.0E0,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 SCOPY (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 SCOPY (NG, G1, 1, GX, 1)
X = X1
JFLAG = 4
RETURN
C----------------------- End of Subroutine SROOTS ----------------------
END
*DECK SSRCAR
SUBROUTINE SSRCAR (RSAV, ISAV, JOB)
C-----------------------------------------------------------------------
C This routine saves or restores (depending on JOB) the contents of
C the Common blocks SLS001, SLSA01, SLSR01, which are used
C internally by one or more ODEPACK solvers.
C
C RSAV = real array of length 13 or more.
C ISAV = integer array of length 33 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, LENILA, LENRLR, LENILR
REAL RSAV
REAL RLS, RLSA, RLSR
DIMENSION RSAV(*), ISAV(*)
COMMON /SLS001/ RLS(9), ILS(25)
COMMON /SLSA01/ RLSA, ILSA(4)
COMMON /SLSR01/ RLSR(3), ILSR(4)
DATA LENRLS/9/, LENILS/25/, LENILA/4/
DATA LENRLR/3/, LENILR/4/
C
IF (JOB .EQ. 2) GO TO 100
DO 10 I = 1,LENRLS
10 RSAV(I) = RLS(I)
RSAV(LENRLS+1) = RLSA
IOFF = LENRLS + 1
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)
RLSA = RSAV(LENRLS+1)
IOFF = LENRLS + 1
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 SSRCAR ----------------------
END
*DECK SSTODPK
SUBROUTINE SSTODPK (NEQ, Y, YH, NYH, YH1, EWT, SAVF, SAVX, ACOR,
1 WM, IWM, F, JAC, PSOL)
EXTERNAL F, JAC, PSOL
INTEGER NEQ, NYH, IWM
REAL Y, YH, YH1, EWT, SAVF, SAVX, ACOR, WM
DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*),
1 SAVX(*), ACOR(*), WM(*), IWM(*)
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NPE, NQU
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
1 NNI, NLI, NPS, NCFN, NCFL
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL DELT, EPCON, SQRTN, RSQRTN
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NPE, NQU
COMMON /SLPK01/ DELT, EPCON, SQRTN, RSQRTN,
1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
2 NNI, NLI, NPS, NCFN, NCFL
C-----------------------------------------------------------------------
C SSTODPK 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 SSTODPK
C from Subroutine SSTODE:
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 /SLPK01/ 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 SPKSET (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 SSOLPK and includes NEQ, SAVX, EWT, F, and JAC.
C SAVX was added because SSOLPK now needs Y and SAVF undisturbed.
C 9. The nonlinear convergence failure count NCFN is set at 430.
C-----------------------------------------------------------------------
C Note: SSTODPK 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 SSTODPK 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 SPKSET 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 SPKSET or SSOLPK.
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 SPKSET 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
INTEGER IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
REAL DCON, DDN, DEL, DELP, DSM, DUP, EXDN, EXSM, EXUP,
1 R, RH, RHDN, RHSM, RHUP, TOLD, SVNORM
REAL CONIT, CRATE, EL(13), ELCO(13,12), HOLD, RMAX,
1 TESCO(3,12)
SAVE CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO,
1 IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
C
KFLAG = 0
TOLD = TN
NCF = 0
IERPJ = 0
IERSL = 0
JCUR = 0
ICF = 0
DELP = 0.0E0
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 = 10000.0E0
RC = 0.0E0
EL0 = 1.0E0
CRATE = 0.7E0
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, SCFODE 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 SCFODE (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 = 0.5E0/(NQ+2)
EPCON = CONIT*TESCO(2,NQ)
DDN = SVNORM (N, SAVF, EWT)/TESCO(1,L)
EXDN = 1.0E0/L
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
RH = MIN(RHDN,1.0E0)
IREDO = 3
IF (H .EQ. HOLD) GO TO 170
RH = MIN(RH,ABS(H/HOLD))
H = HOLD
GO TO 175
C-----------------------------------------------------------------------
C SCFODE 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 SCFODE (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 = 0.5E0/(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(1.0E0,ABS(H)*HMXI*RH)
R = 1.0E0
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 SPKSET.
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 = 0.7E0
GO TO 205
202 IF (ABS(RC-1.0E0) .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, SPKSET 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 SPKSET (NEQ, Y, YH1, EWT, ACOR, SAVF, WM, IWM, F, JAC)
IPUP = 0
RC = 1.0E0
NSLP = NST
CRATE = 0.7E0
IF (IERPJ .NE. 0) GO TO 430
250 DO 260 I = 1,N
260 ACOR(I) = 0.0E0
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 = SVNORM (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 SSOLPK (NEQ, Y, SAVF, SAVX, EWT, WM, IWM, F, PSOL)
IF (IERSL .LT. 0) GO TO 430
IF (IERSL .GT. 0) GO TO 410
DEL = SVNORM (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(0.2E0*CRATE,DEL/DELP)
DCON = DEL*MIN(1.0E0,1.5E0*CRATE)/EPCON
IF (DCON .LE. 1.0E0) GO TO 450
M = M + 1
IF (M .EQ. MAXCOR) GO TO 410
IF (M .GE. 2 .AND. DEL .GT. 2.0E0*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, SPKSET 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 = 2.0E0
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*1.00001E0) GO TO 670
IF (NCF .EQ. MXNCF) GO TO 670
RH = 0.5E0
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 = SVNORM (N, ACOR, EWT)/TESCO(2,NQ)
IF (DSM .GT. 1.0E0) 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 = 2.0E0
IF (ABS(H) .LE. HMIN*1.00001E0) GO TO 660
IF (KFLAG .LE. -3) GO TO 640
IREDO = 2
RHUP = 0.0E0
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 = 0.0E0
IF (L .EQ. LMAX) GO TO 540
DO 530 I = 1,N
530 SAVF(I) = ACOR(I) - YH(I,LMAX)
DUP = SVNORM (N, SAVF, EWT)/TESCO(3,NQ)
EXUP = 1.0E0/(L+1)
RHUP = 1.0E0/(1.4E0*DUP**EXUP + 0.0000014E0)
540 EXSM = 1.0E0/L
RHSM = 1.0E0/(1.2E0*DSM**EXSM + 0.0000012E0)
RHDN = 0.0E0
IF (NQ .EQ. 1) GO TO 560
DDN = SVNORM (N, YH(1,L), EWT)/TESCO(1,NQ)
EXDN = 1.0E0/NQ
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
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. 1.0E0) RH = 1.0E0
GO TO 620
590 NEWQ = L
RH = RHUP
IF (RH .LT. 1.1E0) 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. 1.1E0)) GO TO 610
IF (KFLAG .LE. -2) RH = MIN(RH,0.2E0)
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 = 0.1E0
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 = 10.0E0
700 R = 1.0E0/TESCO(2,NQU)
DO 710 I = 1,N
710 ACOR(I) = ACOR(I)*R
720 HOLD = H
JSTART = 1
RETURN
C----------------------- End of Subroutine SSTODPK ---------------------
END
*DECK SPKSET
SUBROUTINE SPKSET (NEQ, Y, YSV, EWT, FTEM, SAVF, WM, IWM, F, JAC)
EXTERNAL F, JAC
INTEGER NEQ, IWM
REAL Y, YSV, EWT, FTEM, SAVF, WM
DIMENSION NEQ(*), Y(*), YSV(*), EWT(*), FTEM(*), SAVF(*),
1 WM(*), IWM(*)
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NPE, NQU
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
1 NNI, NLI, NPS, NCFN, NCFL
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL DELT, EPCON, SQRTN, RSQRTN
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NPE, NQU
COMMON /SLPK01/ DELT, EPCON, SQRTN, RSQRTN,
1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
2 NNI, NLI, NPS, NCFN, NCFL
C-----------------------------------------------------------------------
C SPKSET is called by SSTODPK 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 SPKSET uses the following:
C Y = array containing predicted values on entry.
C YSV = array containing predicted y, to be saved (YH1 in SSTODPK).
C FTEM = work array of length N (ACOR in SSTODPK).
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, NPE.
C-----------------------------------------------------------------------
INTEGER IER
REAL 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)
NPE = NPE + 1
IF (IER .EQ. 0) RETURN
IERPJ = 1
RETURN
C----------------------- End of Subroutine SPKSET ----------------------
END
*DECK SSOLPK
SUBROUTINE SSOLPK (NEQ, Y, SAVF, X, EWT, WM, IWM, F, PSOL)
EXTERNAL F, PSOL
INTEGER NEQ, IWM
REAL Y, SAVF, X, EWT, WM
DIMENSION NEQ(*), Y(*), SAVF(*), X(*), EWT(*), WM(*), IWM(*)
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NPE, NQU
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
1 NNI, NLI, NPS, NCFN, NCFL
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL DELT, EPCON, SQRTN, RSQRTN
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NPE, NQU
COMMON /SLPK01/ 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 SSPIOM, SSPIGMR, SPCG, SPCGS, or
C SUSOL, 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 SSOLPK 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
REAL 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 SCOPY (N, X, 1, WM(LB), 1)
CALL SSCAL (N, RSQRTN, EWT, 1)
CALL SSPIOM (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 SSCAL (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 SCOPY (N, X, 1, WM(LB), 1)
CALL SSCAL (N, RSQRTN, EWT, 1)
CALL SSPIGMR (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 SSCAL (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 SPCG 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 SCOPY (N, X, 1, WM(LR), 1)
CALL SPCG (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 SPCGS 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 SCOPY (N, X, 1, WM(LR), 1)
CALL SPCGS (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 SUSOL, which interfaces to PSOL, to solve the linear system
C (no Krylov iteration).
C-----------------------------------------------------------------------
900 CONTINUE
LB = 1
LWK = LB + N
CALL SCOPY (N, X, 1, WM(LB), 1)
CALL SUSOL (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 SSOLPK ----------------------
END
*DECK SSPIOM
SUBROUTINE SSPIOM (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
REAL 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 SSPIOM.)
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 SATV 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 SHEFA and used in SHESL.
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
REAL BNRM, BNRM0, PROD, RHO, SNORMW, SNRM2, 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 = SNRM2 (N, V, 1)
BNRM = BNRM0
IF (BNRM0 .GT. DELTA) GO TO 30
IF (MNEWT .GT. 0) GO TO 20
CALL SCOPY (N, B, 1, X, 1)
RETURN
20 DO 25 I = 1,N
25 X(I) = 0.0E0
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 = SNRM2(N, V, 1)
DELTA = DELTA*(BNRM/BNRM0)
55 TEM = 1.0E0/BNRM
CALL SSCAL (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) = 0.0E0
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 = 1.0E0
DO 90 LL = 1,MAXL
LIOM = LL
C-----------------------------------------------------------------------
C Call routine SATV to compute VNEW = Abar*v(l), where Abar is
C the matrix A with scaling and inverse preconditioner factors applied.
C Call routine SORTHOG to orthogonalize the new vector vnew = V(*,l+1).
C Call routine SHEFA to update the factors of HES.
C-----------------------------------------------------------------------
CALL SATV (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 SORTHOG (V(1,LL+1), V, HES, N, LL, MAXL, KMP, SNORMW)
CALL SHEFA (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. 0.0E0) 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 = 1.0E0/SNORMW
CALL SSCAL (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. 1.0E0) 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) = 0.0E0
B(1) = BNRM
CALL SHESL (HES, MAXL, LL, IPVT, B)
DO 220 K = 1,N
220 X(K) = 0.0E0
DO 230 I = 1,LL
CALL SAXPY (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 SSPIOM ----------------------
END
*DECK SATV
SUBROUTINE SATV (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
REAL Y, SAVF, V, WGHT, FTEM, Z, VTEM, WP, HL0
DIMENSION NEQ(*), Y(*), SAVF(*), V(*), WGHT(*), FTEM(*), Z(*),
1 VTEM(*), WP(*), IWP(*)
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NPE, NQU
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NPE, 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
REAL FAC, RNORM, SNRM2, 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 SCOPY (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 = SNRM2 (N, Z, 1)
RNORM = 1.0E0/TEMPN
C Save Y in Z and increment Y by VTEM/norm.
CALL SCOPY (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 SCOPY (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 SATV ------------------------
END
*DECK SORTHOG
SUBROUTINE SORTHOG (VNEW, V, HES, N, LL, LDHES, KMP, SNORMW)
INTEGER N, LL, LDHES, KMP
REAL 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
REAL ARG, SDOT, SNRM2, SUMDSQ, TEM, VNRM
C
C Get norm of unaltered VNEW for later use. ----------------------------
VNRM = SNRM2 (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) = SDOT (N, V(1,I), 1, VNEW, 1)
TEM = -HES(I,LL)
CALL SAXPY (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 = SNRM2 (N, VNEW, 1)
IF (VNRM + 0.001E0*SNORMW .NE. VNRM) RETURN
SUMDSQ = 0.0E0
DO 30 I = I0,LL
TEM = -SDOT (N, V(1,I), 1, VNEW, 1)
IF (HES(I,LL) + 0.001E0*TEM .EQ. HES(I,LL)) GO TO 30
HES(I,LL) = HES(I,LL) - TEM
CALL SAXPY (N, TEM, V(1,I), 1, VNEW, 1)
SUMDSQ = SUMDSQ + TEM**2
30 CONTINUE
IF (SUMDSQ .EQ. 0.0E0) RETURN
ARG = MAX(0.0E0,SNORMW**2 - SUMDSQ)
SNORMW = SQRT(ARG)
C
RETURN
C----------------------- End of Subroutine SORTHOG ---------------------
END
*DECK SSPIGMR
SUBROUTINE SSPIGMR (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
REAL 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 SSPIGMR.)
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 SATV 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 SHEQR and used in SHELS.
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
REAL BNRM,BNRM0,C,DLNRM,PROD,RHO,S,SNORMW,SNRM2,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 = SNRM2 (N, V, 1)
BNRM = BNRM0
IF (BNRM0 .GT. DELTA) GO TO 30
IF (MNEWT .GT. 0) GO TO 20
CALL SCOPY (N, B, 1, X, 1)
RETURN
20 DO 25 I = 1,N
25 X(I) = 0.0E0
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 = SNRM2 (N, V, 1)
DELTA = DELTA*(BNRM/BNRM0)
55 TEM = 1.0E0/BNRM
CALL SSCAL (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) = 0.0E0
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 = 1.0E0
DO 90 LL = 1,MAXL
LGMR = LL
C-----------------------------------------------------------------------
C Call routine SATV to compute VNEW = Abar*v(ll), where Abar is
C the matrix A with scaling and inverse preconditioner factors applied.
C Call routine SORTHOG to orthogonalize the new vector VNEW = V(*,LL+1).
C Call routine SHEQR to update the factors of HES.
C-----------------------------------------------------------------------
CALL SATV (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 SORTHOG (V(1,LL+1), V, HES, N, LL, MAXLP1, KMP, SNORMW)
HES(LL+1,LL) = SNORMW
CALL SHEQR (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 SCOPY (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 = SNRM2 (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 = 1.0E0/SNORMW
CALL SSCAL (N, TEM, V(1,LL+1), 1)
90 CONTINUE
100 CONTINUE
IF (RHO .LE. 1.0E0) 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) = 0.0E0
B(1) = BNRM
CALL SHELS (HES, MAXLP1, LL, Q, B)
DO 220 K = 1,N
220 X(K) = 0.0E0
DO 230 I = 1,LL
CALL SAXPY (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 SSPIGMR ---------------------
END
*DECK SPCG
SUBROUTINE SPCG (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
REAL 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 SATP.
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
REAL ALPHA,BETA,BNRM,PTW,RNRM,SDOT,SVNORM,ZTR,ZTR0
C
IFLAG = 0
NPSL = 0
LPCG = 0
DO 10 I = 1,N
10 X(I) = 0.0E0
BNRM = SVNORM (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 SCOPY (N, R, 1, X, 1)
RETURN
C
20 ZTR = 0.0E0
C Loop point for PCG iterations. ---------------------------------------
30 CONTINUE
LPCG = LPCG + 1
CALL SCOPY (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 = SDOT (N, Z, 1, R, 1)
IF (LPCG .NE. 1) GO TO 50
CALL SCOPY (N, Z, 1, P, 1)
GO TO 70
50 CONTINUE
IF (ZTR0 .EQ. 0.0E0) GO TO 200
BETA = ZTR/ZTR0
DO 60 I = 1,N
60 P(I) = Z(I) + BETA*P(I)
70 CONTINUE
C-----------------------------------------------------------------------
C Call SATP to compute A*p and return the answer in W.
C-----------------------------------------------------------------------
CALL SATP (NEQ, Y, SAVF, P, WGHT, HL0, WK, F, W)
C
PTW = SDOT (N, P, 1, W, 1)
IF (PTW .EQ. 0.0E0) GO TO 200
ALPHA = ZTR/PTW
CALL SAXPY (N, ALPHA, P, 1, X, 1)
ALPHA = -ALPHA
CALL SAXPY (N, ALPHA, W, 1, R, 1)
RNRM = SVNORM (N, R, WGHT)
IF (RNRM .LE. DELTA) RETURN
IF (LPCG .LT. MAXL) GO TO 30
IFLAG = 2
IF (RNRM .LE. 1.0E0) 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 SPCG ------------------------
END
*DECK SPCGS
SUBROUTINE SPCGS (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
REAL 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 SATP.
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
REAL ALPHA, BETA, BNRM, PTW, RNRM, SVNORM, ZTR, ZTR0
C
IFLAG = 0
NPSL = 0
LPCG = 0
DO 10 I = 1,N
10 X(I) = 0.0E0
BNRM = SVNORM (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 SCOPY (N, R, 1, X, 1)
RETURN
C
20 ZTR = 0.0E0
C Loop point for PCG iterations. ---------------------------------------
30 CONTINUE
LPCG = LPCG + 1
CALL SCOPY (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 = 0.0E0
DO 45 I = 1,N
45 ZTR = ZTR + Z(I)*R(I)*WGHT(I)**2
IF (LPCG .NE. 1) GO TO 50
CALL SCOPY (N, Z, 1, P, 1)
GO TO 70
50 CONTINUE
IF (ZTR0 .EQ. 0.0E0) GO TO 200
BETA = ZTR/ZTR0
DO 60 I = 1,N
60 P(I) = Z(I) + BETA*P(I)
70 CONTINUE
C-----------------------------------------------------------------------
C Call SATP to compute A*p and return the answer in W.
C-----------------------------------------------------------------------
CALL SATP (NEQ, Y, SAVF, P, WGHT, HL0, WK, F, W)
C
PTW = 0.0E0
DO 80 I = 1,N
80 PTW = PTW + P(I)*W(I)*WGHT(I)**2
IF (PTW .EQ. 0.0E0) GO TO 200
ALPHA = ZTR/PTW
CALL SAXPY (N, ALPHA, P, 1, X, 1)
ALPHA = -ALPHA
CALL SAXPY (N, ALPHA, W, 1, R, 1)
RNRM = SVNORM (N, R, WGHT)
IF (RNRM .LE. DELTA) RETURN
IF (LPCG .LT. MAXL) GO TO 30
IFLAG = 2
IF (RNRM .LE. 1.0E0) 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 SPCGS -----------------------
END
*DECK SATP
SUBROUTINE SATP (NEQ, Y, SAVF, P, WGHT, HL0, WK, F, W)
EXTERNAL F
INTEGER NEQ
REAL Y, SAVF, P, WGHT, HL0, WK, W
DIMENSION NEQ(*), Y(*), SAVF(*), P(*), WGHT(*), WK(*), W(*)
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NPE, NQU
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NPE, 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
REAL FAC, PNRM, RPNRM, SVNORM
C
PNRM = SVNORM (N, P, WGHT)
RPNRM = 1.0E0/PNRM
CALL SCOPY (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 SCOPY (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 SATP ------------------------
END
*DECK SUSOL
SUBROUTINE SUSOL (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
REAL 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
REAL BNRM, SVNORM
C
IFLAG = 0
NPSL = 0
C-----------------------------------------------------------------------
C Test for an immediate return with X = 0 or X = b.
C-----------------------------------------------------------------------
BNRM = SVNORM (N, B, WGHT)
IF (BNRM .GT. DELTA) GO TO 30
IF (MNEWT .GT. 0) GO TO 10
CALL SCOPY (N, B, 1, X, 1)
RETURN
10 DO 20 I = 1,N
20 X(I) = 0.0E0
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 SCOPY (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 SUSOL -----------------------
END
*DECK SSRCPK
SUBROUTINE SSRCPK (RSAV, ISAV, JOB)
C-----------------------------------------------------------------------
C This routine saves or restores (depending on JOB) the contents of
C the Common blocks SLS001, SLPK01, which are used
C internally by the SLSODPK solver.
C
C RSAV = real array of length 13 or more.
C ISAV = integer array of length 38 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
REAL RSAV, RLS, RLSP
DIMENSION RSAV(*), ISAV(*)
COMMON /SLS001/ RLS(9), ILS(25)
COMMON /SLPK01/ RLSP(4), ILSP(13)
DATA LENRLS/9/, LENILS/25/, LENRLP/4/, LENILP/13/
C
IF (JOB .EQ. 2) GO TO 100
CALL SCOPY (LENRLS, RLS, 1, RSAV, 1)
CALL SCOPY (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 SCOPY (LENRLS, RSAV, 1, RLS, 1)
CALL SCOPY (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 SSRCPK ----------------------
END
*DECK SHEFA
SUBROUTINE SHEFA (A, LDA, N, IPVT, INFO, JOB)
INTEGER LDA, N, IPVT(*), INFO, JOB
REAL A(LDA,*)
C-----------------------------------------------------------------------
C This routine is a modification of the LINPACK routine SGEFA 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 SHEFA factors an upper Hessenberg matrix by elimination.
C
C On entry
C
C A REAL(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 SHESL 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: SAXPY, ISAMAX
C-----------------------------------------------------------------------
INTEGER ISAMAX, J, K, KM1, KP1, L, NM1
REAL 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 = ISAMAX (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. 0.0E0) 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 = -1.0E0/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 SAXPY (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. 0.0E0) 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 = ISAMAX (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. 0.0E0) 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 = -1.0E0/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. 0.0E0) INFO = N
RETURN
C----------------------- End of Subroutine SHEFA -----------------------
END
*DECK SHESL
SUBROUTINE SHESL (A, LDA, N, IPVT, B)
INTEGER LDA, N, IPVT(*)
REAL A(LDA,*), B(*)
C-----------------------------------------------------------------------
C This is essentially the LINPACK routine SGESL except for changes
C due to the fact that A is an upper Hessenberg matrix.
C-----------------------------------------------------------------------
C SHESL solves the real system A * x = b
C using the factors computed by SHEFA.
C
C On entry
C
C A REAL(LDA, N)
C the output from SHEFA.
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 SHEFA.
C
C B REAL(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: SAXPY
C-----------------------------------------------------------------------
INTEGER K, KB, L, NM1
REAL 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 SAXPY (K-1, T, A(1,K), 1, B(1), 1)
40 CONTINUE
RETURN
C----------------------- End of Subroutine SHESL -----------------------
END
*DECK SHEQR
SUBROUTINE SHEQR (A, LDA, N, Q, INFO, IJOB)
INTEGER LDA, N, INFO, IJOB
REAL 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 SHEQR decomposes an upper Hessenberg matrix by using Givens
C rotations.
C
C On entry
C
C A REAL(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 REAL(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 SHELS 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
REAL 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. 0.0E0) GO TO 30
C = 1.0E0
S = 0.0E0
GO TO 50
30 CONTINUE
IF (ABS(T2) .LT. ABS(T1)) GO TO 40
T = T1/T2
S = -1.0E0/SQRT(1.0E0+T*T)
C = -S*T
GO TO 50
40 CONTINUE
T = T2/T1
C = 1.0E0/SQRT(1.0E0+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. 0.0E0) 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. 0.0E0) GO TO 110
C = 1.0E0
S = 0.0E0
GO TO 130
110 CONTINUE
IF (ABS(T2) .LT. ABS(T1)) GO TO 120
T = T1/T2
S = -1.0E0/SQRT(1.0E0+T*T)
C = -S*T
GO TO 130
120 CONTINUE
T = T2/T1
C = 1.0E0/SQRT(1.0E0+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. 0.0E0) INFO = N
RETURN
C----------------------- End of Subroutine SHEQR -----------------------
END
*DECK SHELS
SUBROUTINE SHELS (A, LDA, N, Q, B)
INTEGER LDA, N
REAL A(LDA,*), B(*), Q(*)
C-----------------------------------------------------------------------
C This is part of the LINPACK routine SGESL with changes
C due to the fact that A is an upper Hessenberg matrix.
C-----------------------------------------------------------------------
C SHELS solves the least squares problem
C
C min (b-A*x, b-A*x)
C
C using the factors computed by SHEQR.
C
C On entry
C
C A REAL(LDA, N)
C the output from SHEQR 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 REAL(2*N)
C The coefficients of the N givens rotations
C used in the QR factorization of A.
C
C B REAL(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: SAXPY
C-----------------------------------------------------------------------
INTEGER IQ, K, KB, KP1
REAL 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 SAXPY (K-1, T, A(1,K), 1, B(1), 1)
40 CONTINUE
RETURN
C----------------------- End of Subroutine SHELS -----------------------
END
*DECK SLHIN
SUBROUTINE SLHIN (NEQ, N, T0, Y0, YDOT, F, TOUT, UROUND,
1 EWT, ITOL, ATOL, Y, TEMP, H0, NITER, IER)
EXTERNAL F
REAL 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 SLHIN: F, SCOPY
C Function routines called by SLHIN: SVNORM
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 SLHIN 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
REAL AFI, ATOLI, DELYI, HALF, HG, HLB, HNEW, HRAT,
1 HUB, HUN, PT1, T1, TDIST, TROUND, TWO, SVNORM, 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 /0.5E0/, HUN /100.0E0/, PT1 /0.1E0/, TWO /2.0E0/
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 = SVNORM (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 SCOPY (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 SLHIN -----------------------
END
*DECK SSTOKA
SUBROUTINE SSTOKA (NEQ, Y, YH, NYH, YH1, EWT, SAVF, SAVX, ACOR,
1 WM, IWM, F, JAC, PSOL)
EXTERNAL F, JAC, PSOL
INTEGER NEQ, NYH, IWM
REAL Y, YH, YH1, EWT, SAVF, SAVX, ACOR, WM
DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*),
1 SAVX(*), ACOR(*), WM(*), IWM(*)
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NPE, NQU
INTEGER NEWT, NSFI, NSLJ, NJEV
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
1 NNI, NLI, NPS, NCFN, NCFL
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL STIFR
REAL DELT, EPCON, SQRTN, RSQRTN
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NPE, NQU
COMMON /SLS002/ STIFR, NEWT, NSFI, NSLJ, NJEV
COMMON /SLPK01/ DELT, EPCON, SQRTN, RSQRTN,
1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
2 NNI, NLI, NPS, NCFN, NCFL
C-----------------------------------------------------------------------
C SSTOKA 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 SSTODPK in the SLSODPK
C package by the addition of automatic functional/Newton iteration
C switching and logic for re-use of Jacobian data.
C-----------------------------------------------------------------------
C Note: SSTOKA 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 SSTOKA 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 SSETPK 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 SSETPK or SSOLPK.
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 SSETPK 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
INTEGER IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
REAL DCON, DDN, DEL, DELP, DRC, DSM, DUP, EXDN, EXSM,
1 EXUP, SFNORM, R, RH, RHDN, RHSM, RHUP, ROC, STIFF, TOLD, SVNORM
REAL CONIT, CRATE, EL(13), ELCO(13,12), HOLD, RMAX,
1 TESCO(3,12)
SAVE CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO,
1 IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
C
KFLAG = 0
TOLD = TN
NCF = 0
IERPJ = 0
IERSL = 0
JCUR = 0
ICF = 0
DELP = 0.0E0
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 = 10000.0E0
RC = 0.0E0
EL0 = 1.0E0
CRATE = 0.7E0
HOLD = H
MEO = METH
NSLP = 0
NSLJ = 0
IPUP = 0
IRET = 3
NEWT = 0
STIFR = 0.0E0
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, SCFODE 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 SCFODE (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 = 0.5E0/(NQ+2)
EPCON = CONIT*TESCO(2,NQ)
DDN = SVNORM (N, SAVF, EWT)/TESCO(1,L)
EXDN = 1.0E0/L
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
RH = MIN(RHDN,1.0E0)
IREDO = 3
IF (H .EQ. HOLD) GO TO 170
RH = MIN(RH,ABS(H/HOLD))
H = HOLD
GO TO 175
C-----------------------------------------------------------------------
C SCFODE 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 SCFODE (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 = 0.5E0/(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(1.0E0,ABS(H)*HMXI*RH)
R = 1.0E0
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 SSETPK.
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 = 0.0E0
IPUP = 0
CRATE = 0.7E0
ELSE
DRC = ABS(RC - 1.0E0)
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 = 0.0E0
ROC = 0.05E0
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, SSETPK 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. 0.2E0) JOK = -1
IF (ICF .EQ. 2) JOK = -1
IF (JOK .EQ. -1) THEN
NSLJ = NST
NJEV = NJEV + 1
ENDIF
CALL SSETPK (NEQ, Y, YH1, EWT, ACOR, SAVF, JOK, WM, IWM, F, JAC)
IPUP = 0
RC = 1.0E0
DRC = 0.0E0
NSLP = NST
CRATE = 0.7E0
IF (IERPJ .NE. 0) GO TO 430
250 DO 260 I = 1,N
260 ACOR(I) = 0.0E0
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 = SVNORM (N, Y, EWT)
DO 300 I = 1,N
Y(I) = YH(I,1) + EL(1)*SAVF(I)
300 ACOR(I) = SAVF(I)
STIFF = 1.0E0
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))
SFNORM = SVNORM (N, SAVX, EWT)
CALL SSOLPK (NEQ, Y, SAVF, SAVX, EWT, WM, IWM, F, PSOL)
IF (IERSL .LT. 0) GO TO 430
IF (IERSL .GT. 0) GO TO 410
DEL = SVNORM (N, SAVX, EWT)
IF (DEL .GT. 1.0E-8) STIFF = MAX(STIFF, SFNORM/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(0.05E0, DEL/DELP)
CRATE = MAX(0.2E0*CRATE,ROC)
ENDIF
DCON = DEL*MIN(1.0E0,1.5E0*CRATE)/EPCON
IF (DCON .LE. 1.0E0) GO TO 450
M = M + 1
IF (M .EQ. MAXCOR) GO TO 410
IF (M .GE. 2 .AND. DEL .GT. 2.0E0*DELP) GO TO 410
IF (ROC .GT. 10.0E0) GO TO 410
IF (ROC .GT. 0.8E0) 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 = 1023.0E0
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 = 2.0E0
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*1.00001E0) GO TO 670
IF (NCF .EQ. MXNCF) GO TO 670
RH = 0.5E0
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 = 0.5E0*(STIFR + STIFF)
IF (M .EQ. 0) DSM = DEL/TESCO(2,NQ)
IF (M .GT. 0) DSM = SVNORM (N, ACOR, EWT)/TESCO(2,NQ)
IF (DSM .GT. 1.0E0) 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. 1.5E0) 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 = 2.0E0
IF (ABS(H) .LE. HMIN*1.00001E0) GO TO 660
IF (KFLAG .LE. -3) GO TO 640
IREDO = 2
RHUP = 0.0E0
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 = 0.0E0
IF (L .EQ. LMAX) GO TO 540
DO 530 I = 1,N
530 SAVF(I) = ACOR(I) - YH(I,LMAX)
DUP = SVNORM (N, SAVF, EWT)/TESCO(3,NQ)
EXUP = 1.0E0/(L+1)
RHUP = 1.0E0/(1.4E0*DUP**EXUP + 0.0000014E0)
540 EXSM = 1.0E0/L
RHSM = 1.0E0/(1.2E0*DSM**EXSM + 0.0000012E0)
RHDN = 0.0E0
IF (NQ .EQ. 1) GO TO 560
DDN = SVNORM (N, YH(1,L), EWT)/TESCO(1,NQ)
EXDN = 1.0E0/NQ
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
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. 1.0E0) RH = 1.0E0
GO TO 620
590 NEWQ = L
RH = RHUP
IF (RH .LT. 1.1E0) 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. 1.1E0)) GO TO 610
IF (KFLAG .LE. -2) RH = MIN(RH,0.2E0)
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 = 0.1E0
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 = 10.0E0
700 R = 1.0E0/TESCO(2,NQU)
DO 710 I = 1,N
710 ACOR(I) = ACOR(I)*R
720 HOLD = H
JSTART = 1
RETURN
C----------------------- End of Subroutine SSTOKA ----------------------
END
*DECK SSETPK
SUBROUTINE SSETPK (NEQ, Y, YSV, EWT, FTEM, SAVF, JOK, WM, IWM,
1 F, JAC)
EXTERNAL F, JAC
INTEGER NEQ, JOK, IWM
REAL Y, YSV, EWT, FTEM, SAVF, WM
DIMENSION NEQ(*), Y(*), YSV(*), EWT(*), FTEM(*), SAVF(*),
1 WM(*), IWM(*)
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NFE, NPE, NQU
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
1 NNI, NLI, NPS, NCFN, NCFL
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL DELT, EPCON, SQRTN, RSQRTN
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NPE, NQU
COMMON /SLPK01/ DELT, EPCON, SQRTN, RSQRTN,
1 JPRE, JACFLG, LOCWP, LOCIWP, LSAVX, KMP, MAXL, MNEWT,
2 NNI, NLI, NPS, NCFN, NCFL
C-----------------------------------------------------------------------
C SSETPK is called by SSTOKA 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 SSETPK uses the following:
C Y = array containing predicted values on entry.
C YSV = array containing predicted y, to be saved (YH1 in SSTOKA).
C FTEM = work array of length N (ACOR in SSTOKA).
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, NPE.
C-----------------------------------------------------------------------
INTEGER IER
REAL 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)
NPE = NPE + 1
IF (IER .EQ. 0) RETURN
IERPJ = 1
RETURN
C----------------------- End of Subroutine SSETPK ----------------------
END
*DECK SSRCKR
SUBROUTINE SSRCKR (RSAV, ISAV, JOB)
C-----------------------------------------------------------------------
C This routine saves or restores (depending on JOB) the contents of
C the Common blocks SLS001, SLS002, SLSR01, SLPK01, which
C are used internally by the SLSODKR solver.
C
C RSAV = real array of length 17 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, ILS2, ILSR, ILSP
INTEGER I, IOFF, LENILP, LENRLP, LENILS, LENRLS, LENILR, LENRLR
REAL RSAV, RLS, RLS2, RLSR, RLSP
DIMENSION RSAV(*), ISAV(*)
COMMON /SLS001/ RLS(9), ILS(25)
COMMON /SLS002/ RLS2, ILS2(4)
COMMON /SLSR01/ RLSR(3), ILSR(4)
COMMON /SLPK01/ RLSP(4), ILSP(13)
DATA LENRLS/9/, LENILS/25/, LENRLP/4/, LENILP/13/
DATA LENRLR/3/, LENILR/4/
C
IF (JOB .EQ. 2) GO TO 100
CALL SCOPY (LENRLS, RLS, 1, RSAV, 1)
RSAV(LENRLS+1) = RLS2
CALL SCOPY (LENRLR, RLSR, 1, RSAV(LENRLS+2), 1)
CALL SCOPY (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 SCOPY (LENRLS, RSAV, 1, RLS, 1)
RLS2 = RSAV(LENRLS+1)
CALL SCOPY (LENRLR, RSAV(LENRLS+2), 1, RLSR, 1)
CALL SCOPY (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 SSRCKR ----------------------
END
*DECK SAINVG
SUBROUTINE SAINVG (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
REAL 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 SLSODI for
C initialization only, when ISTATE = 0 .
C SAINVG returns an error flag IER:
C IER = 0 means SAINVG 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) = 0.0E0
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 SGEFA ( PW, NEQ, NEQ, IPVT, IER )
IF (IER .EQ. 0) GO TO 20
IER = -IER
RETURN
20 CALL SGESL ( 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) = 0.0E0
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 SGBFA ( PW, NROWPW, NEQ, ML, MU, IPVT, IER )
IF (IER .EQ. 0) GO TO 120
IER = -IER
RETURN
120 CALL SGBSL ( PW, NROWPW, NEQ, ML, MU, IPVT, YDOT, 0 )
RETURN
C----------------------- End of Subroutine SAINVG ----------------------
END
*DECK SSTODI
SUBROUTINE SSTODI (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
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NRE, NJE, NQU
INTEGER I, I1, IREDO, IRES, IRET, J, JB, KGO, M, NCF, NEWQ
INTEGER IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
REAL Y, YH, YH1, EWT, SAVF, SAVR, ACOR, WM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL DCON, DDN, DEL, DELP, DSM, DUP,
1 ELJH, EL1H, EXDN, EXSM, EXUP,
2 R, RH, RHDN, RHSM, RHUP, TOLD, SVNORM
REAL CONIT, CRATE, EL(13), ELCO(13,12), HOLD, RMAX,
1 TESCO(3,12)
DIMENSION NEQ(*), Y(*), YH(NYH,*), YH1(*), EWT(*), SAVF(*),
1 SAVR(*), ACOR(*), WM(*), IWM(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NRE, NJE, NQU
SAVE CONIT, CRATE, EL, ELCO, HOLD, RMAX, TESCO,
1 IALTH, IPUP, LMAX, MEO, NQNYH, NSLP
C-----------------------------------------------------------------------
C SSTODI performs one step of the integration of an initial value
C problem for a system of Ordinary Differential Equations.
C Note: SSTODI 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 SSTODI 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 = 0.0E0
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 = 10000.0E0
RC = 0.0E0
EL0 = 1.0E0
CRATE = 0.7E0
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, SCFODE 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 SCFODE (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 = 0.5E0/(NQ+2)
DDN = SVNORM (N, SAVF, EWT)/TESCO(1,L)
EXDN = 1.0E0/L
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
RH = MIN(RHDN,1.0E0)
IREDO = 3
IF (H .EQ. HOLD) GO TO 170
RH = MIN(RH,ABS(H/HOLD))
H = HOLD
GO TO 175
C-----------------------------------------------------------------------
C SCFODE 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 SCFODE (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 = 0.5E0/(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(1.0E0,ABS(H)*HMXI*RH)
R = 1.0E0
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-1.0E0) .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 = 1.0E0
NSLP = NST
CRATE = 0.7E0
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 )
NRE = NRE + 1
KGO = ABS(IRES)
GO TO ( 250, 435, 430 ) , KGO
250 DO 260 I = 1,N
260 ACOR(I) = 0.0E0
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 = SVNORM (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(0.2E0*CRATE,DEL/DELP)
DCON = DEL*MIN(1.0E0,1.5E0*CRATE)/(TESCO(2,NQ)*CONIT)
IF (DCON .LE. 1.0E0) GO TO 460
M = M + 1
IF (M .EQ. MAXCOR) GO TO 410
IF (M .GE. 2 .AND. DEL .GT. 2.0E0*DELP) GO TO 410
DELP = DEL
IRES = 1
CALL RES ( NEQ, TN, Y, SAVF, SAVR, IRES )
NRE = NRE + 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 = 2.0E0
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*1.00001E0) GO TO 450
IF (NCF .EQ. MXNCF) GO TO 450
RH = 0.25E0
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) * SVNORM (N, ACOR, EWT)/TESCO(2,NQ)
IF (DSM .GT. 1.0E0) 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 = 2.0E0
IF (ABS(H) .LE. HMIN*1.00001E0) GO TO 660
IF (KFLAG .LE. -7) GO TO 660
IREDO = 2
RHUP = 0.0E0
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 = 0.0E0
IF (L .EQ. LMAX) GO TO 540
DO 530 I = 1,N
530 SAVF(I) = ACOR(I) - YH(I,LMAX)
DUP = ABS(H) * SVNORM (N, SAVF, EWT)/TESCO(3,NQ)
EXUP = 1.0E0/(L+1)
RHUP = 1.0E0/(1.4E0*DUP**EXUP + 0.0000014E0)
540 EXSM = 1.0E0/L
RHSM = 1.0E0/(1.2E0*DSM**EXSM + 0.0000012E0)
RHDN = 0.0E0
IF (NQ .EQ. 1) GO TO 560
DDN = SVNORM (N, YH(1,L), EWT)/TESCO(1,NQ)
EXDN = 1.0E0/NQ
RHDN = 1.0E0/(1.3E0*DDN**EXDN + 0.0000013E0)
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. 1.0E0) RH = 1.0E0
GO TO 620
590 NEWQ = L
RH = RHUP
IF (RH .LT. 1.1E0) 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. 1.1E0)) GO TO 610
IF (KFLAG .LE. -2) RH = MIN(RH,0.1E0)
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 = 10.0E0
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 SSTODI ----------------------
END
*DECK SPREPJI
SUBROUTINE SPREPJI (NEQ, Y, YH, NYH, EWT, RTEM, SAVR, S, WM, IWM,
1 RES, JAC, ADDA)
EXTERNAL RES, JAC, ADDA
INTEGER NEQ, NYH, IWM
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NRE, NJE, NQU
INTEGER I, I1, I2, IER, II, IRES, J, J1, JJ, LENP,
1 MBA, MBAND, MEB1, MEBAND, ML, ML3, MU
REAL Y, YH, EWT, RTEM, SAVR, S, WM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL CON, FAC, HL0, R, SRUR, YI, YJ, YJJ
DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), RTEM(*),
1 S(*), SAVR(*), WM(*), IWM(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NRE, NJE, NQU
C-----------------------------------------------------------------------
C SPREPJI is called by SSTODI 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 SGEFA if MITER = 1 or 2, and by
C SGBFA if MITER = 4 or 5.
C
C In addition to variables described previously, communication
C with SPREPJI uses the following:
C Y = array containing predicted values on entry.
C RTEM = work array of length N (ACOR in SSTODI).
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 SSTODI).
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, NRE, 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)
NRE = NRE + 1
IF (IRES .GT. 1) GO TO 600
LENP = N*N
DO 110 I = 1,LENP
110 WM(I+2) = 0.0E0
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)
NRE = NRE + 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),0.01E0/EWT(J))
Y(J) = Y(J) + R
FAC = -HL0/R
CALL RES ( NEQ, TN, Y, S, RTEM, IRES )
NRE = NRE + 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)
NRE = NRE + 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 SGEFA (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)
NRE = NRE + 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) = 0.0E0
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)
NRE = NRE + 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),0.01E0/EWT(I))
530 Y(I) = Y(I) + R
CALL RES ( NEQ, TN, Y, S, RTEM, IRES)
NRE = NRE + 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),0.01E0/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)
NRE = NRE + 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 SGBFA (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 SPREPJI ---------------------
END
*DECK SAIGBT
SUBROUTINE SAIGBT (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
REAL 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 SLSOIBT for
C initialization only, when ISTATE = 0 .
C SAIGBT returns an error flag IER:
C IER = 0 means SAIGBT 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) = 0.0E0
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 SDECBT (MB, NB, PW, PW(LPB), PW(LPC), IPVT, IER)
IF (IER .EQ. 0) GO TO 20
IER = -IER
RETURN
20 CALL SSOLBT (MB, NB, PW, PW(LPB), PW(LPC), YDOT, IPVT)
RETURN
C----------------------- End of Subroutine SAIGBT ----------------------
END
*DECK SPJIBT
SUBROUTINE SPJIBT (NEQ, Y, YH, NYH, EWT, RTEM, SAVR, S, WM, IWM,
1 RES, JAC, ADDA)
EXTERNAL RES, JAC, ADDA
INTEGER NEQ, NYH, IWM
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NRE, 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
REAL Y, YH, EWT, RTEM, SAVR, S, WM
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL CON, FAC, HL0, R, SRUR
DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), RTEM(*),
1 S(*), SAVR(*), WM(*), IWM(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NRE, NJE, NQU
C-----------------------------------------------------------------------
C SPJIBT is called by SSTODI 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 SDECBT 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 SPJIBT uses the following:
C Y = array containing predicted values on entry.
C RTEM = work array of length N (ACOR in SSTODI).
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 SSTODI).
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, NRE, 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)
NRE = NRE + 1
IF (IRES .GT. 1) GO TO 600
DO 110 I = 1,LENP
110 WM(I+2) = 0.0E0
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)
NRE = NRE + 1
IF (IRES .GT. 1) GO TO 600
MWID = 3*MB
SRUR = WM(1)
DO 205 I = 1,LENP
205 WM(2+I) = 0.0E0
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)),0.01E0/EWT(I))
Y(I) = Y(I) + R
210 CONTINUE
CALL RES (NEQ, TN, Y, S, RTEM, IRES)
NRE = NRE + 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)),0.01E0/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)
NRE = NRE + 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 SDECBT (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 SPJIBT ----------------------
END
*DECK SSLSBT
SUBROUTINE SSLSBT (WM, IWM, X, TEM)
INTEGER IWM
INTEGER LBLOX, LPB, LPC, MB, NB
REAL WM, X, TEM
DIMENSION WM(*), IWM(*), X(*), TEM(*)
C-----------------------------------------------------------------------
C This routine acts as an interface between the core integrator
C routine and the SSOLBT routine for the solution of the linear system
C arising from chord iteration.
C Communication with SSLSBT 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 SSOLBT (MB, NB, WM(3), WM(LPB), WM(LPC), X, IWM(21))
RETURN
C----------------------- End of Subroutine SSLSBT ----------------------
END
*DECK SDECBT
SUBROUTINE SDECBT (M, N, A, B, C, IP, IER)
INTEGER M, N, IP(M,N), IER
REAL 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 SDECBT uses block Gauss elimination and Subroutines SGEFA and SGESL
C for solution of blocks. Partial pivoting is done within
C block-rows only.
C
C Note: this version uses LINPACK routines SGEFA/SGESL instead of
C of dec/sol for solution of blocks, and it uses the BLAS routine SDOT
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 SSOLBT to solve the associated linear system.
C
C External routines required: SGEFA and SGESL (from LINPACK) and
C SDOT (from the BLAS, or Basic Linear Algebra package).
C-----------------------------------------------------------------------
INTEGER NM1, NM2, KM1, I, J, K
REAL DP, SDOT
IF (M .LT. 1 .OR. N .LT. 4) GO TO 210
NM1 = N - 1
NM2 = N - 2
C Process the first block-row. -----------------------------------------
CALL SGEFA (A, M, M, IP, IER)
K = 1
IF (IER .NE. 0) GO TO 200
DO 10 J = 1,M
CALL SGESL (A, M, M, IP, B(1,J,1), 0)
CALL SGESL (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 = SDOT (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 = SDOT (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 SGEFA (A(1,1,K), M, M, IP(1,K), IER)
IF (IER .NE. 0) GO TO 200
DO 80 J = 1,M
80 CALL SGESL (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 = SDOT (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 = SDOT (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 SGEFA (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 SDECBT ----------------------
END
*DECK SSOLBT
SUBROUTINE SSOLBT (M, N, A, B, C, Y, IP)
INTEGER M, N, IP(M,N)
REAL 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 SDECBT.
C M, N, A,B,C, and IP must not have been changed since call to SDECBT.
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 SDECBT.
C IP = M by N integer array of pivot information from SDECBT.
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: SGESL (LINPACK) and SDOT (BLAS).
C-----------------------------------------------------------------------
C
INTEGER NM1, NM2, I, K, KB, KM1, KP1
REAL DP, SDOT
NM1 = N - 1
NM2 = N - 2
C Forward solution sweep. ----------------------------------------------
CALL SGESL (A, M, M, IP, Y, 0)
DO 30 K = 2,NM1
KM1 = K - 1
DO 20 I = 1,M
DP = SDOT (M, C(I,1,K), M, Y(1,KM1), 1)
Y(I,K) = Y(I,K) - DP
20 CONTINUE
CALL SGESL (A(1,1,K), M, M, IP(1,K), Y(1,K), 0)
30 CONTINUE
DO 50 I = 1,M
DP = SDOT (M, C(I,1,N), M, Y(1,NM1), 1)
1 + SDOT (M, B(I,1,N), M, Y(1,NM2), 1)
Y(I,N) = Y(I,N) - DP
50 CONTINUE
CALL SGESL (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 = SDOT (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 = SDOT (M, C(I,1,1), M, Y(1,3), 1)
Y(I,1) = Y(I,1) - DP
100 CONTINUE
RETURN
C----------------------- End of Subroutine SSOLBT ----------------------
END
*DECK SIPREPI
SUBROUTINE SIPREPI (NEQ, Y, S, RWORK, IA, JA, IC, JC, IPFLAG,
1 RES, JAC, ADDA)
EXTERNAL RES, JAC, ADDA
INTEGER NEQ, IA, JA, IC, JC, IPFLAG
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NRE, 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
INTEGER I, IMAX, LEWTN, LYHD, LYHN
REAL Y, S, RWORK
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL RLSS
DIMENSION NEQ(*), Y(*), S(*), RWORK(*), IA(*), JA(*), IC(*), JC(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
1 ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
2 LYH, LEWT, LACOR, LSAVR, LWM, LIWM, METH, MITER,
3 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NRE, NJE, NQU
COMMON /SLSS01/ 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 routine serves as an interface between the driver and
C Subroutine SPREPI. Tasks performed here are:
C * call SPREPI,
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 SPREPI error flag IPPER
C if there was trouble in Subroutine SPREPI.
C-----------------------------------------------------------------------
IPFLAG = 0
C Call SPREPI to do matrix preprocessing operations. -------------------
CALL SPREPI (NEQ, Y, S, RWORK(LYH), RWORK(LSAVR), 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 SPREPI 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 LSAVR = LYH + LENYH
LEWTN = LSAVR + 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 SIPREPI ---------------------
END
*DECK SPREPI
SUBROUTINE SPREPI (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
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NRE, 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
INTEGER I, IBR, IER, IPIL, IPIU, IPTT1, IPTT2, J, K, KNEW, KAMAX,
1 KAMIN, KCMAX, KCMIN, LDIF, LENIGP, LIWK, LJFO, MAXG, NP1, NZSUT
REAL Y, S, YH, SAVR, EWT, RTEM, WK
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL RLSS
REAL ERWT, FAC, YJ
DIMENSION NEQ(*), Y(*), S(*), YH(*), SAVR(*), EWT(*), RTEM(*),
1 IA(*), JA(*), IC(*), JC(*), WK(*), IWK(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NRE, NJE, NQU
COMMON /SLSS01/ 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 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, SPREPI 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 SODRV or SCDRV.
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 SODRV.
C = -4 other error flag from SODRV (should never occur).
C = -5 insufficient storage for SCDRV.
C = -6 other error flag from SCDRV.
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
LIWK = LENWK*LRAT
IF (MOSS .NE. 3 .AND. MOSS .NE. 4) LIWK = LIWK - N
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 = 1.0E0/EWT(I)
FAC = 1.0E0 + 1.0E0/(I + 1.0E0)
Y(I) = Y(I) + FAC*SIGN(ERWT,Y(I))
S(I) = 1.0E0 + 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.
75 WK(LIWK+I) = 0.
K = IPJAN
IWK(IPIAN) = 1
DO 95 J = 1,N
CALL ADDA (NEQ, TN, Y, J, IWK(IPIAN), IWK(IPJAN), WK(LIWK+1))
CALL JAC (NEQ, TN, Y, S, J, IWK(IPIAN), IWK(IPJAN), SAVR)
DO 90 I = 1,N
LJFO = LIWK + I
IF (WK(LJFO) .EQ. 0.) GO TO 80
WK(LJFO) = 0.
SAVR(I) = 0.
GO TO 85
80 IF (SAVR(I) .EQ. 0.) GO TO 90
SAVR(I) = 0.
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(LIWK+I) = 0.
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(LIWK+1))
YJ = Y(J)
ERWT = 1.0E0/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 = LIWK + I
IF (WK(LJFO) .EQ. 0.) GO TO 110
WK(LJFO) = 0.
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.
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.
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.) GO TO 180
SAVR(I) = 0.
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 = 1.0E0/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 SODRV(N, IWK(IPIAN), IWK(IPJAN), WK, IWK(IPR), IWK(IPIC),
1 NSP, 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) = 0.0E0
IPISP = LRAT*(IPRSP - 1) + 1
CALL SCDRV(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 SPREPI ----------------------
END
*DECK SAINVGS
SUBROUTINE SAINVGS (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
REAL T, Y, WK, TEM, YDOT
REAL RLSS
DIMENSION Y(*), WK(*), IWK(*), TEM(*), YDOT(*)
COMMON /SLSS01/ 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 SLSODIS for initialization
C only, when ISTATE = 0. The matrix A is subjected to LU
C decomposition in SCDRV. Then the system A*YDOT = g(t,y) is solved
C in SCDRV.
C In addition to variables described previously, communication
C with SAINVGS 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 SSTODI).
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 SAINVGS 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 SCDRV (should not occur here).
C = 5 if other error found in SCDRV (should not occur here).
C-----------------------------------------------------------------------
C
DO 10 I = 1,NNZ
10 WK(IBA+I) = 0.0E0
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) = 0.0E0
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) = 0.0E0
C
C Numerical factorization of matrix A. ---------------------------------
CALL SCDRV (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 SCDRV (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 SAINVGS ---------------------
END
*DECK SPRJIS
SUBROUTINE SPRJIS (NEQ, Y, YH, NYH, EWT, RTEM, SAVR, S, WK, IWK,
1 RES, JAC, ADDA)
EXTERNAL RES, JAC, ADDA
INTEGER NEQ, NYH, IWK
INTEGER ICF, IERPJ, IERSL, JCUR, JSTART, KFLAG, L,
1 LYH, LEWT, LACOR, LSAVF, LWM, LIWM, METH, MITER,
2 MAXORD, MAXCOR, MSBP, MXNCF, N, NQ, NST, NRE, 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
INTEGER I, IMUL, IRES, J, JJ, JMAX, JMIN, K, KMAX, KMIN, NG
REAL Y, YH, EWT, RTEM, SAVR, S, WK
REAL CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND
REAL RLSS
REAL CON, FAC, HL0, R, SRUR
DIMENSION NEQ(*), Y(*), YH(NYH,*), EWT(*), RTEM(*),
1 S(*), SAVR(*), WK(*), IWK(*)
COMMON /SLS001/ CCMAX, EL0, H, HMIN, HMXI, HU, RC, TN, UROUND,
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, NRE, NJE, NQU
COMMON /SLSS01/ 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 SPRJIS 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 SCDRV.
C P and its LU decomposition are stored separately in WK.
C
C In addition to variables described previously, communication
C with SPRJIS uses the following:
C Y = array containing predicted values on entry.
C RTEM = work array of length N (ACOR in SSTODI).
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 SSTODI).
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 SCDRV.
C = IRES (= 2 or 3) if RES returned IRES = 2 or 3.
C = -1 if insufficient storage for SCDRV (should not occur).
C = -2 if other error found in SCDRV (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)
NRE = NRE + 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) = 0.0E0
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)
NRE = NRE + 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)),0.01E0/EWT(JJ))
210 Y(JJ) = Y(JJ) + R
CALL RES (NEQ,TN,Y,S,RTEM,IRES)
NRE = NRE + 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)),0.01E0/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)
NRE = NRE + 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) = 0.0E0
CALL SCDRV (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 SPRJIS ----------------------
END