*DECK SVODPK
SUBROUTINE SVODPK (F, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL, ITASK,
1 ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JAC, PSOL, MF, RPAR, IPAR)
EXTERNAL F, JAC, PSOL
REAL Y, T, TOUT, RTOL, ATOL, RWORK, RPAR
INTEGER NEQ, ITOL, ITASK, ISTATE, IOPT, LRW, IWORK, LIW, MF, IPAR
DIMENSION Y(*), RTOL(*), ATOL(*), RWORK(LRW), IWORK(LIW),
1 RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C This is the 23 April 2002 version of
C SVODPK: Variable-coefficient Ordinary Differential equation solver
C with the Preconditioned Krylov method GMRES for the solution
C of linear systems.
C
C This version is in single precision.
C
C SVODPK solves the initial value problem for stiff or nonstiff
C systems of first order ODEs,
C dy/dt = f(t,y) , or, in component form,
C dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(neq)) (i = 1,...,neq).
C SVODPK is a package based on the VODE and LSODPK packages, and on
C the October 23, 1978 version of the ODEPACK user interface standard,
C with minor modifications.
C-----------------------------------------------------------------------
C Authors:
C Alan C. Hindmarsh and Peter N. Brown
C Center for Applied Scientific Computing, L-561
C Lawrence Livermore National Laboratory
C Livermore, CA 94551
C and
C George D. Byrne
C Illinois Institute of Technology
C Chicago, IL 60616
C-----------------------------------------------------------------------
C References:
C 1. P. N. Brown, G. D. Byrne, and A. C. Hindmarsh, "VODE, A Variable-
C Coefficient ODE Solver," SIAM J. Sci. Stat. Comput., 10 (1989),
C pp., 1038-1051. Also LLNL report UCRL-98412, June 1988.
C 2. P. N. Brown and A. C. Hindmarsh, "Reduced Storage Matrix Methods
C in Stiff ODE Systems," J. Appl. Math. & Comp., 31 (1989), pp.40-91.
C Also LLNL report UCRL-95088, Rev. 1, June 1987.
C 3. G. D. Byrne, "Pragmatic Experiments with Krylov Methods in the
C Stiff ODE Setting," Computational Ordinary Differential Equations,
C J. Cash and I. Gladwell, eds., Oxford Univ. Press, Oxford, 1992,
C pp. 323-356.
C-----------------------------------------------------------------------
C Introduction.
C
C This is a modification of the VODE package which incorporates
C the preconditioned Krylov subspace iterative method SPIGMR for the
C linear algebraic systems that arise in the case of stiff systems.
C SPIGMR denotes a scaled preconditioned incomplete version of the
C GMRES (Generalized Minimum Residual) method.
C
C The linear systems that are solved have the form
C A * x = b , where A = I - hrl1 * (df/dy) .
C here hrl1 is a scalar, I is the identity matrix, and df/dy is the
C Jacobian matrix of partial derivatives of f with respect to y
C (an NEQ by NEQ matrix).
C
C The particular Krylov method is chosen by setting the second digit,
C MITER, in the method flag MF.
C Currently, the values of MITER have the following meanings:
C
C 1 means SPIGMR, a scaled, preconditioned, incomplete version
C of GMRES, a generalized minimum residual method.
C This is the best choice in general.
C
C 9 means that only a user-supplied matrix P (approximating A)
C will be used, with no Krylov iteration done internally to
C SVODPK. This option allows the user to provide the
C complete linear system solution algorithm, if desired.
C
C The user can apply preconditioning to the linear system A*x = b,
C by means of arbitrary matrices (the preconditioners).
C
C In the case of SPIGMR, one can apply left and right
C preconditioners P1 and P2, and the basic iterative method is then
C applied to the matrix (P1-inverse)*A*(P2-inverse) instead of to the
C matrix A. The product P1*P2 should be an approximation to A
C such that linear systems with P1 or P2 are easier to solve than with
C A alone. Preconditioning from the left only or right only means using
C P2 = I or P1 = I, respectively.
C
C If the Jacobian J = df/dy splits in a natural way into a sum
C J = J1 + J2, then one possible choice of preconditioners is
C P1 = I - hrl1 * J1 and P2 = I - hrl1 * J2
C provided each of these is easy to solve (or to approximately solve).
C
C NOTE: To achieve an efficient solution, the preconditioned Krylov
C methods in SVODPK generally require a thoughtful choice of
C preconditioners. If the ODE system produces linear systems that are
C not amenable to solution by such iterative methods, the cost can be
C higher than with a solver that uses sparse direct methods. However,
C for many systems, careful use of SVODPK can be highly effective.
C
C See Ref. 2 for more details on the methods and applications.
C-----------------------------------------------------------------------
C Summary of usage.
C
C Communication between the user and the SVODPK package, for normal
C situations, is summarized here. This summary describes only a subset
C of the full set of options available. See full description (below)
C for details, including optional communication, nonstandard options,
C and instructions for special situations. See also the example
C program embedded in the comments below.
C
C A. First provide a subroutine of the form
C SUBROUTINE F (NEQ, T, Y, YDOT, RPAR, IPAR)
C DIMENSION Y(NEQ), YDOT(NEQ), RPAR(*), IPAR(*)
C which supplies the vector function f by loading YDOT(i) with f(i).
C
C B. Next determine (or guess) whether or not the problem is stiff.
C Stiffness occurs when the Jacobian matrix df/dy has an eigenvalue
C whose real part is negative and large in magnitude, compared to the
C reciprocal of the t span of interest. If the problem is nonstiff,
C use method flag MF = 10. If it is stiff, MF should be 21.
C
C The following four parameters must also be set.
C IWORK(1) = LWP = length of real array WP for preconditioning.
C IWORK(2) = LIWP = length of integer array IWP for preconditioning.
C IWORK(3) = JPRE = preconditioner type flag:
C = 0 for no preconditioning (P1 = P2 = I)
C = 1 for left-only preconditioning (P2 = I)
C = 2 for right-only preconditioning (P1 = I)
C = 3 for two-sided preconditioning
C IWORK(4) = JACFLG = flag for whether JAC is called.
C = 0 if JAC is not to be called,
C = 1 if JAC is to be called.
C Use JACFLG = 1 if JAC computes any nonconstant data for use in
C preconditioning, such as Jacobian elements. See next paragraph.
C The arrays WP and IWP are work arrays under the user's control,
C for use in the routines that perform preconditioning operations.
C
C C. If the problem is stiff, you must supply two routines that deal
C with the preconditioning of the linear systems to be solved.
C These are as follows:
C
C SUBROUTINE JAC (F, NEQ, T, Y, YSV, REWT, FTY, V, HRL1, WP, IWP,
C 1 IER, RPAR, IPAR)
C DIMENSION Y(NEQ), YSV(NEQ), REWT(NEQ), FTY(NEQ), V(NEQ),
C 1 WP(*), IWP(*), RPAR(*), IPAR(*)
C
C This routine is optional, and is to evaluate and preprocess
C any parts of the Jacobian matrix df/dy involved in the
C preconditioners P1 and P2.
C The Y and FTY arrays contain the current values of y and f(t,y),
C respectively, and YSV also contains the current value of y.
C The array V is work space of length NEQ.
C JAC must multiply all computed Jacobian elements by the scalar
C -hrl1, add the identity matrix I, and do any factorization
C operations called for, in preparation for solving linear systems
C with a coefficient matrix of P1 or P2. The matrix P1*P2 should
C be an approximation to I - hrl1 * (df/dy), where hrl1 is a
C scalar stored in HRL1.
C JAC should return IER = 0 if successful, and IER .ne. 0 if not.
C (If IER .ne. 0, a smaller time step will be tried.)
C
C SUBROUTINE PSOL (NEQ, T, Y, FTY, WK, HRL1, WP, IWP, B, LR,
C 1 IER, RPAR, IPAR)
C DIMENSION Y(NEQ), FTY(NEQ), WK(NEQ), WP(*), IWP(*), B(NEQ),
C 1 RPAR(*), IPAR(*)
C
C This routine must solve a linear system with b (stored in B)
C as right-hand side and one of the preconditioning matrices, P1 or
C P2, as coefficient matrix, and return the solution vector in B.
C LR is a flag concerning left vs. right preconditioning, input
C to PSOL. PSOL is to use P1 if LR = 1, and P2 if LR = 2.
C
C PSOL can use data generated in the JAC routine and stored in
C WP and IWP. WK is a work array of length NEQ.
C The argument HRL1 is the current value of the scalar appearing
C in the linear system. If the old value, at the time of the last
C JAC call, is needed, it must have been saved by JAC in WP.
C on return, PSOL should set the error flag IER as follows:
C IER = 0 if PSOL was successful,
C IER .gt. 0 if a recoverable error occurred, meaning that the
C time step will be retried,
C IER .lt. 0 if an unrecoverable error occurred, meaning that the
C solver is to stop immediately.
C
C D. Write a main program which calls subroutine SVODPK once for
C each point at which answers are desired. This should also provide
C for possible use of logical unit 6 for output of error messages
C by SVODPK. on the first call to SVODPK, supply arguments as follows:
C F = name of subroutine for right-hand side vector f.
C This name must be declared EXTERNAL in calling program.
C NEQ = number of first order ODEs.
C Y = array of initial values, of length NEQ.
C T = the initial value of the independent variable.
C TOUT = first point where output is desired (.ne. T).
C ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
C RTOL = relative tolerance parameter (scalar).
C ATOL = absolute tolerance parameter (scalar or array).
C The estimated local error in Y(i) will be controlled so as
C to be roughly less (in magnitude) than
C EWT(i) = RTOL*abs(Y(i)) + ATOL if ITOL = 1, or
C EWT(i) = RTOL*abs(Y(i)) + ATOL(i) if ITOL = 2.
C Thus the local error test passes if, in each component,
C either the absolute error is less than ATOL (or ATOL(i)),
C or the relative error is less than RTOL.
C Use RTOL = 0.0 for pure absolute error control, and
C use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
C control. Caution: Actual (global) errors may exceed these
C local tolerances, so choose them conservatively.
C ITASK = 1 for normal computation of output values of Y at t = TOUT.
C ISTATE = integer flag (input and output). Set ISTATE = 1.
C IOPT = 0 to indicate no optional input used.
C RWORK = real work array of length at least:
C 20 + 16*NEQ for MF = 10,
C 61 + 17*NEQ + LWP for MF = 21.
C LRW = declared length of RWORK (in user's DIMENSION statement).
C IWORK = integer work array of length at least:
C 30 for MF = 10,
C 30 + LIWP for MF = 21.
C LIW = declared length of IWORK (in user's DIMENSION statement).
C JAC,PSOL = names of subroutines for preconditioning. These names
C must be declared EXTERNAL in the user's calling program.
C MF = method flag. Standard values are:
C 10 for nonstiff (Adams) method.
C 21 for stiff (BDF) method, with SPIGMR.
C
C RPAR, IPAR User-specified arrays used to communicate real and integer
C parameters (respectively) to user-supplied subroutines.
C to user-supplied subroutines. If RPAR is a vector, then
C it must be dimensioned in the user's main program. If it
C is unused or a scalar, then it need not be dimensioned.
C
C IPAR User-specified array used to communicate integer parameter
C to user-supplied subroutines. The comments on dimensioning
C RPAR apply to IPAR.
C
C Note that the user's main (calling) program must declare arrays
C Y, RWORK, IWORK, and possibly ATOL, RPAR, and IPAR.
C
C E. The output from the first call (or any call) is:
C Y = array of computed values of y(t) vector.
C T = corresponding value of independent variable (normally TOUT).
C ISTATE = 2 if SVODPK was successful, negative otherwise.
C -1 means excess work done on this call (perhaps wrong MF).
C -2 means excess accuracy requested (tolerances too small).
C -3 means illegal input detected (see printed message).
C -4 means repeated error test failures (check all input).
C -5 means repeated convergence failures (perhaps bad JAC
C or PSOL routine supplied or wrong choice of MF or
C tolerances, or this solver is inappropriate).
C -6 means error weight became zero during problem. (Solution
C component i vanished, and ATOL or ATOL(i) = 0.)
C -7 means an unrecoverable error occurred in JAC or PSOL.
C
C F. To continue the integration after a successful return, simply
C reset TOUT and call SVODPK again. No other parameters need be reset.
C
C-----------------------------------------------------------------------
C Example problem.
C An ODE system is generated from the following 2-species diurnal
C kinetics advection-diffusion PDE system in 2 space dimensions:
C
C dc(i)/dt = Kh*(d/dx)**2 c(i) + V*dc(i)/dx + (d/dz)(Kv(z)*dc(i)/dz)
C + Ri(c1,c2,t) for i = 1,2, where
C R1(c1,c2,t) = -q1*c1*c3 - q2*c1*c2 + 2*q3(t)*c3 + q4(t)*c2 ,
C R2(c1,c2,t) = q1*c1*c3 - q2*c1*c2 - q4(t)*c2 ,
C Kv(z) = Kv0*exp(z/5) ,
C Kh, V, Kv0, q1, q2, and c3 are constants, and q3(t) and q4(t)
C vary diurnally. The problem is posed on the square
C 0 .le. x .le. 20, 30 .le. z .le. 50 (all in km),
C with homogeneous Neumann boundary conditions, and for time t in
C 0 .le. t .le. 86400 sec (1 day).
C The PDE system is treated by central differences on a uniform
C 10 x 10 mesh, with simple polynomial initial profiles.
C The problem is solved with SVODPK, with the BDF/GMRES method and
C the block-diagonal part of the Jacobian as a left preconditioner.
C-----------------------------------------------------------------------
C EXTERNAL FEX, JACBD, SOLBD
C REAL Q1,Q2,Q3,Q4, A3,A4, OM, C3, DZ, HDCO,VDCO,HACO
C COMMON /PCOM/ Q1,Q2,Q3,Q4,A3,A4,OM,C3,DZ,HDCO,VDCO,HACO,MX,MZ,MM
C REAL ATOL, AVDIM, CX, CZ, DKH, DKV0, DX, FLOOR,
C 1 HALFDA, PI, RPAR, RTOL, RWORK, T, TOUT, TWOHR, VEL, X, Y, Z
C DIMENSION Y(2,10,10), RWORK(3861), IWORK(230)
C DATA DKH/4.0E-6/, VEL/0.001E0/, DKV0/1.0E-8/, HALFDA/4.32E4/,
C 1 PI/3.1415926535898E0/, TWOHR/7200.0E0/, RTOL/1.0E-5/,
C 2 FLOOR/100.0E0/, LRW/3861/, LIW/230/, MF/21/, JPRE/1/, JACFLG/1/
C
C Load Common block of problem parameters.
C MX = 10
C MZ = 10
C MM = MX*MZ
C Q1 = 1.63E-16
C Q2 = 4.66E-16
C A3 = 22.62E0
C A4 = 7.601E0
C OM = PI/HALFDA
C C3 = 3.7E16
C DX = 20.0E0/(MX - 1.0E0)
C DZ = 20.0E0/(MZ - 1.0E0)
C HDCO = DKH/DX**2
C HACO = VEL/(2.0E0*DX)
C VDCO = (1.0E0/DZ**2)*DKV0
C Set other input arguments.
C ATOL = RTOL*FLOOR
C NEQ = 2*MX*MZ
C IWORK(1) = 4*MX*MZ
C IWORK(2) = NEQ
C IWORK(3) = JPRE
C IWORK(4) = JACFLG
C T = 0.0E0
C TOUT = TWOHR
C ISTATE = 1
C Set initial profiles.
C DO 20 JZ = 1,MZ
C Z = 30.0E0 + (JZ - 1.0E0)*DZ
C CZ = (0.1E0*(Z - 40.0E0))**2
C CZ = 1.0E0 - CZ + 0.5E0*CZ**2
C DO 10 JX = 1,MX
C X = (JX - 1.0E0)*DX
C CX = (0.1E0*(X - 10.0E0))**2
C CX = 1.0E0 - CX + 0.5E0*CX**2
C Y(1,JX,JZ) = 1.0E6*CX*CZ
C Y(2,JX,JZ) = 1.0E12*CX*CZ
C 10 CONTINUE
C 20 CONTINUE
C
C Loop over output points, call SVODPK, print sample solution values.
C DO 70 IOUT = 1,12
C CALL SVODPK (FEX, NEQ, Y, T, TOUT, 1, RTOL, ATOL, 1, ISTATE, 0,
C 1 RWORK, LRW, IWORK, LIW, JACBD, SOLBD, MF, RPAR, IPAR)
C WRITE(6,50) T,IWORK(11),IWORK(14),RWORK(11)
C 50 FORMAT(/' t =',E10.2,5X,'no. steps =',I5,
C 1 ' order =',I3,' stepsize =',E10.2)
C WRITE(6,60) Y(1,1,1), Y(1,5,5), Y(1,10,10),
C 1 Y(2,1,1), Y(2,5,5), Y(2,10,10)
C 60 FORMAT(' c1 (bot.left/middle/top rt.) =',3E12.3/
C 1 ' c2 (bot.left/middle/top rt.) =',3E12.3)
C IF (ISTATE .NE. 2) STOP
C TOUT = TOUT + TWOHR
C 70 CONTINUE
C
C Print final statistics.
C LENRW = IWORK(17)
C LENIW = IWORK(18)
C NST = IWORK(11)
C NFE = IWORK(12)
C NPE = IWORK(13)
C NPS = IWORK(24)
C NNI = IWORK(20)
C NLI = IWORK(23)
C AVDIM = REAL(NLI)/REAL(NNI)
C NCFN = IWORK(21)
C NCFL = IWORK(25)
C WRITE (6,80) LENRW,LENIW,NST,NFE,NPE,NPS,NNI,NLI,AVDIM,NCFN,NCFL
C 80 FORMAT(//' Final statistics:'/
C 1 ' RWORK size =',I5,5X,' IWORK size =',I4/
C 2 ' Number of steps =',I5,5X,'Number of f evals. =',I5/
C 3 ' Number of prec. evals. =',I5,5X,'Number of prec. solves =',I5/
C 4 ' Number of nonl. iters. =',I5,5X,'Number of lin. iters. =',I5/
C 5 ' Average Krylov subspace dimension (NLI/NNI) =',F8.4/
C 6 ' Number of conv. failures: nonlinear =',I3,' linear =',I3)
C STOP
C END
C
C SUBROUTINE FEX (NEQ, T, Y, YDOT, RPAR, IPAR)
C REAL T, Y, YDOT, RPAR
C DIMENSION Y(2,*), YDOT(2,*)
C REAL Q1,Q2,Q3,Q4, A3,A4, OM, C3, DZ, HDCO,VDCO,HACO
C COMMON /PCOM/ Q1,Q2,Q3,Q4,A3,A4,OM,C3,DZ,HDCO,VDCO,HACO,MX,MZ,MM
C REAL C1, C2, C1DN, C2DN, C1UP, C2UP, C1LT, C2LT,
C 1 C1RT, C2RT, CZDN, CZUP, HORD1, HORD2, HORAD1, HORAD2,
C 2 QQ1, QQ2, QQ3, QQ4, RKIN1, RKIN2, S, VERTD1, VERTD2, ZDN, ZUP
C
C Set diurnal rate coefficients.
C S = SIN(OM*T)
C IF (S .GT. 0.0E0) THEN
C Q3 = EXP(-A3/S)
C Q4 = EXP(-A4/S)
C ELSE
C Q3 = 0.0E0
C Q4 = 0.0E0
C ENDIF
C Loop over all grid points.
C DO 20 JZ = 1,MZ
C ZDN = 30.0E0 + (JZ - 1.5E0)*DZ
C ZUP = ZDN + DZ
C CZDN = VDCO*EXP(0.2E0*ZDN)
C CZUP = VDCO*EXP(0.2E0*ZUP)
C IBLOK0 = (JZ-1)*MX
C IDN = -MX
C IF (JZ .EQ. 1) IDN = MX
C IUP = MX
C IF (JZ .EQ. MZ) IUP = -MX
C DO 10 JX = 1,MX
C IBLOK = IBLOK0 + JX
C C1 = Y(1,IBLOK)
C C2 = Y(2,IBLOK)
C Set kinetic rate terms.
C QQ1 = Q1*C1*C3
C QQ2 = Q2*C1*C2
C QQ3 = Q3*C3
C QQ4 = Q4*C2
C RKIN1 = -QQ1 - QQ2 + 2.0E0*QQ3 + QQ4
C RKIN2 = QQ1 - QQ2 - QQ4
C Set vertical diffusion terms.
C C1DN = Y(1,IBLOK+IDN)
C C2DN = Y(2,IBLOK+IDN)
C C1UP = Y(1,IBLOK+IUP)
C C2UP = Y(2,IBLOK+IUP)
C VERTD1 = CZUP*(C1UP - C1) - CZDN*(C1 - C1DN)
C VERTD2 = CZUP*(C2UP - C2) - CZDN*(C2 - C2DN)
C Set horizontal diffusion and advection terms.
C ILEFT = -1
C IF (JX .EQ. 1) ILEFT = 1
C IRIGHT = 1
C IF (JX .EQ. MX) IRIGHT = -1
C C1LT = Y(1,IBLOK+ILEFT)
C C2LT = Y(2,IBLOK+ILEFT)
C C1RT = Y(1,IBLOK+IRIGHT)
C C2RT = Y(2,IBLOK+IRIGHT)
C HORD1 = HDCO*(C1RT - 2.0E0*C1 + C1LT)
C HORD2 = HDCO*(C2RT - 2.0E0*C2 + C2LT)
C HORAD1 = HACO*(C1RT - C1LT)
C HORAD2 = HACO*(C2RT - C2LT)
C Load all terms into YDOT.
C YDOT(1,IBLOK) = VERTD1 + HORD1 + HORAD1 + RKIN1
C YDOT(2,IBLOK) = VERTD2 + HORD2 + HORAD2 + RKIN2
C 10 CONTINUE
C 20 CONTINUE
C RETURN
C END
C
C SUBROUTINE JACBD (F, NEQ, T, Y, YSV, REWT, F0, F1, HRL1,
C 1 BD, IPBD, IER, RPAR, IPAR)
C EXTERNAL F
C REAL T, Y, YSV, REWT, F0, F1, HRL1, BD, RPAR
C DIMENSION Y(2, *), YSV(*), REWT(*), F0(*), F1(*), BD(2, 2, *),
C 1 IPBD(2, *)
C REAL Q1,Q2,Q3,Q4, A3,A4, OM, C3, DZ, HDCO,VDCO,HACO
C COMMON /PCOM/ Q1,Q2,Q3,Q4,A3,A4,OM,C3,DZ,HDCO,VDCO,HACO,MX,MZ,MM
C REAL C1, C2, CZDN, CZUP, DIAG, ZDN, ZUP
C
C Compute diagonal Jacobian blocks, multiplied by -HRL1
C (using q3 and q4 values computed on last F call).
C DO 20 JZ = 1,MZ
C ZDN = 30.0E0 + (JZ - 1.5E0)*DZ
C ZUP = ZDN + DZ
C CZDN = VDCO*EXP(0.2E0*ZDN)
C CZUP = VDCO*EXP(0.2E0*ZUP)
C DIAG = -(CZDN + CZUP + 2.0E0*HDCO)
C IBLOK0 = (JZ-1)*MX
C DO 10 JX = 1,MX
C IBLOK = IBLOK0 + JX
C C1 = Y(1,IBLOK)
C C2 = Y(2,IBLOK)
C BD(1,1,IBLOK) = -HRL1*( (-Q1*C3 - Q2*C2) + DIAG )
C BD(1,2,IBLOK) = -HRL1*( -Q2*C1 + Q4 )
C BD(2,1,IBLOK) = -HRL1*( Q1*C3 - Q2*C2 )
C BD(2,2,IBLOK) = -HRL1*( (-Q2*C1 - Q4) + DIAG )
C 10 CONTINUE
C 20 CONTINUE
C Add identity matrix and do LU decompositions on blocks.
C DO 40 IBLOK = 1,MM
C BD(1,1,IBLOK) = BD(1,1,IBLOK) + 1.0E0
C BD(2,2,IBLOK) = BD(2,2,IBLOK) + 1.0E0
C CALL SGEFA (BD(1,1,IBLOK), 2, 2, IPBD(1,IBLOK), IER)
C IF (IER .NE. 0) RETURN
C 40 CONTINUE
C RETURN
C END
C
C SUBROUTINE SOLBD (NEQ, T, Y, F0, WK, HRL1, BD, IPBD, V, LR, IER,
C 1 RPAR, IPAR)
C REAL T, Y, F0, WK, HRL1, BD, V, RPAR
C DIMENSION BD(2,2,*), IPBD(2,*), V(2,*)
C REAL Q1,Q2,Q3,Q4, A3,A4, OM, C3, DZ, HDCO,VDCO,HACO
C COMMON /PCOM/ Q1,Q2,Q3,Q4,A3,A4,OM,C3,DZ,HDCO,VDCO,HACO,MX,MZ,MM
C
C Solve the block-diagonal system Px = v using LU factors stored in BD
C and pivot data in IPBD, and return the solution in V.
C IER = 0
C DO 10 I = 1,MM
C CALL SGESL (BD(1,1,I), 2, 2, IPBD(1,I), V(1,I), 0)
C 10 CONTINUE
C RETURN
C END
C
C The output of this program, on a Cray-1 in single precision,
C is as follows:
C
C t = 7.20e+03 no. steps = 194 order = 5 stepsize = 1.17e+02
C c1 (bot.left/middle/top rt.) = 1.047e+04 2.964e+04 1.119e+04
C c2 (bot.left/middle/top rt.) = 2.527e+11 7.154e+11 2.700e+11
C
C t = 1.44e+04 no. steps = 227 order = 5 stepsize = 2.73e+02
C c1 (bot.left/middle/top rt.) = 6.659e+06 5.316e+06 7.301e+06
C c2 (bot.left/middle/top rt.) = 2.582e+11 2.057e+11 2.833e+11
C
C t = 2.16e+04 no. steps = 252 order = 5 stepsize = 4.21e+02
C c1 (bot.left/middle/top rt.) = 2.665e+07 1.036e+07 2.931e+07
C c2 (bot.left/middle/top rt.) = 2.993e+11 1.028e+11 3.313e+11
C
C t = 2.88e+04 no. steps = 291 order = 4 stepsize = 2.13e+02
C c1 (bot.left/middle/top rt.) = 8.702e+06 1.292e+07 9.650e+06
C c2 (bot.left/middle/top rt.) = 3.380e+11 5.029e+11 3.751e+11
C
C t = 3.60e+04 no. steps = 321 order = 5 stepsize = 9.90e+01
C c1 (bot.left/middle/top rt.) = 1.404e+04 2.029e+04 1.561e+04
C c2 (bot.left/middle/top rt.) = 3.387e+11 4.894e+11 3.765e+11
C
C t = 4.32e+04 no. steps = 374 order = 4 stepsize = 4.44e+02
C c1 (bot.left/middle/top rt.) = -5.457e-09 -4.365e-09 -6.182e-09
C c2 (bot.left/middle/top rt.) = 3.382e+11 1.355e+11 3.804e+11
C
C t = 5.04e+04 no. steps = 393 order = 5 stepsize = 5.22e+02
C c1 (bot.left/middle/top rt.) = 3.396e-12 2.798e-12 3.789e-12
C c2 (bot.left/middle/top rt.) = 3.358e+11 4.930e+11 3.864e+11
C
C t = 5.76e+04 no. steps = 407 order = 5 stepsize = 3.54e+02
C c1 (bot.left/middle/top rt.) = 7.738e-12 6.455e-12 8.598e-12
C c2 (bot.left/middle/top rt.) = 3.320e+11 9.650e+11 3.909e+11
C
C t = 6.48e+04 no. steps = 419 order = 5 stepsize = 5.90e+02
C c1 (bot.left/middle/top rt.) = -2.018e-11 -1.680e-11 -2.243e-11
C c2 (bot.left/middle/top rt.) = 3.313e+11 8.922e+11 3.963e+11
C
C t = 7.20e+04 no. steps = 432 order = 5 stepsize = 5.90e+02
C c1 (bot.left/middle/top rt.) = -2.837e-11 -2.345e-11 -3.166e-11
C c2 (bot.left/middle/top rt.) = 3.330e+11 6.186e+11 4.039e+11
C
C t = 7.92e+04 no. steps = 444 order = 5 stepsize = 5.90e+02
C c1 (bot.left/middle/top rt.) = -4.861e-14 -4.433e-14 -5.162e-14
C c2 (bot.left/middle/top rt.) = 3.334e+11 6.669e+11 4.120e+11
C
C t = 8.64e+04 no. steps = 456 order = 5 stepsize = 5.90e+02
C c1 (bot.left/middle/top rt.) = 2.511e-15 2.071e-15 2.802e-15
C c2 (bot.left/middle/top rt.) = 3.352e+11 9.107e+11 4.163e+11
C
C
C Final statistics:
C RWORK size = 3861 IWORK size = 230
C Number of steps = 456 Number of f evals. = 1317
C Number of prec. evals. = 82 Number of prec. solves = 1226
C Number of nonl. iters. = 571 Number of lin. iters. = 743
C Average Krylov subspace dimension (NLI/NNI) = 1.3012
C Number of conv. failures: nonlinear = 0 linear = 0
C-----------------------------------------------------------------------
C Full description of user interface to SVODPK.
C
C The user interface to SVODPK consists of the following parts.
C
C i. The call sequence to subroutine SVODPK, which is a driver
C routine for the solver. This includes descriptions of both
C the call sequence arguments and of user-supplied routines.
C Following these descriptions are
C * a description of optional input available through the
C call sequence,
C * a description of optional output (in the work arrays), and
C * instructions for interrupting and restarting a solution.
C
C ii. Descriptions of other routines in the SVODPK package that may be
C (optionally) called by the user. These provide the ability to
C alter error message handling, save and restore the internal
C COMMON, and obtain specified derivatives of the solution y(t).
C
C iii. Descriptions of COMMON blocks to be declared in overlay
C or similar environments.
C
C iv. Description of two routines in the SVODPK package, either of
C which the user may replace with the user's own version, if
C desired. These relate to the measurement of errors.
C
C-----------------------------------------------------------------------
C Part i. Call Sequence.
C
C The call sequence parameters used for input only are
C F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW,
C JAC, PSOL, MF,
C and those used for both input and output are
C Y, T, ISTATE.
C The work arrays RWORK and IWORK are also used for conditional and
C optional input and optional output. (The term output here refers
C to the return from subroutine SVODPK to the user's calling program.)
C
C The legality of input parameters will be thoroughly checked on the
C initial call for the problem, but not checked thereafter unless a
C change in input parameters is flagged by ISTATE = 3 in the input.
C
C The descriptions of the call arguments are as follows.
C
C F = The name of the user-supplied subroutine defining the
C ODE system. The system must be put in the first-order
C form dy/dt = f(t,y), where f is a vector-valued function
C of the scalar t and the vector y. Subroutine F is to
C compute the function f. It is to have the form
C SUBROUTINE F (NEQ, T, Y, YDOT, RPAR, IPAR)
C DIMENSION Y(NEQ), YDOT(NEQ), RPAR(*), IPAR(*)
C where NEQ, T, and Y are input, and the array YDOT = f(t,y)
C is output. Y and YDOT are arrays of length NEQ.
C (In the DIMENSION statement above, NEQ can be replaced by
C * to make Y and YDOT assumed size arrays.)
C Subroutine F should not alter Y or T.
C F must be declared EXTERNAL in the calling program.
C
C Subroutine F may access user-defined real and integer
C work arrays RPAR and IPAR, which are to be dimensioned
C in the user's calling (main) program.
C
C If quantities computed in the F routine are needed
C externally to SVODPK, an extra call to F should be made
C for this purpose, for consistent and accurate results.
C If only the derivative dy/dt is needed, use SVINDY instead.
C
C NEQ = The size of the ODE system (number of first order
C ordinary differential equations). Used only for input.
C NEQ may not be increased during the problem, but
C can be decreased (with ISTATE = 3 in the input).
C
C Y = A real array for the vector of dependent variables, of
C length NEQ or more. Used for both input and output on the
C first call (ISTATE = 1), and only for output on other calls.
C On the first call, Y must contain the vector of initial
C values. In the output, Y contains the computed solution
C evaluated at T. If desired, the Y array may be used
C for other purposes between calls to the solver.
C
C This array is passed as the Y argument in all calls to
C F, JAC, and PSOL.
C
C T = The independent variable. In the input, T is used only on
C the first call, as the initial point of the integration.
C In the output, after each call, T is the value at which a
C computed solution Y is evaluated (usually the same as TOUT).
C On an error return, T is the farthest point reached.
C
C TOUT = The next value of t at which a computed solution is desired.
C Used only for input.
C
C When starting the problem (ISTATE = 1), TOUT may be equal
C to T for one call, then should .ne. T for the next call.
C For the initial T, an input value of TOUT .ne. T is used
C in order to determine the direction of the integration
C (i.e. the algebraic sign of the step sizes) and the rough
C scale of the problem. Integration in either direction
C (forward or backward in t) is permitted.
C
C If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
C the first call (i.e. the first call with TOUT .ne. T).
C Otherwise, TOUT is required on every call.
C
C If ITASK = 1, 3, or 4, the values of TOUT need not be
C monotone, but a value of TOUT which backs up is limited
C to the current internal t interval, whose endpoints are
C TCUR - HU and TCUR. (See optional output, below, for
C TCUR and HU.)
C
C ITOL = An indicator for the type of error control. See
C description below under ATOL. Used only for input.
C
C RTOL = A relative error tolerance parameter, either a scalar or
C an array of length NEQ. See description below under ATOL.
C Input only.
C
C ATOL = An absolute error tolerance parameter, either a scalar or
C an array of length NEQ. Input only.
C
C The input parameters ITOL, RTOL, and ATOL determine
C the error control performed by the solver. The solver will
C control the vector e = (e(i)) of estimated local errors
C in Y, according to an inequality of the form
C rms-norm of ( e(i)/EWT(i) ) .le. 1,
C where EWT(i) = RTOL(i)*abs(Y(i)) + ATOL(i),
C and the rms-norm (root-mean-square norm) here is
C rms-norm(v) = sqrt(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
C is a vector of weights which must always be positive, and
C the values of RTOL and ATOL should all be non-negative.
C The following table gives the types (scalar/array) of
C RTOL and ATOL, and the corresponding form of EWT(i).
C
C ITOL RTOL ATOL EWT(i)
C 1 scalar scalar RTOL*ABS(Y(i)) + ATOL
C 2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
C 3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
C 4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
C
C When either of these parameters is a scalar, it need not
C be dimensioned in the user's calling program.
C
C If none of the above choices (with ITOL, RTOL, and ATOL
C fixed throughout the problem) is suitable, more general
C error controls can be obtained by substituting
C user-supplied routines for the setting of EWT and/or for
C the norm calculation. See Part iv below.
C
C If global errors are to be estimated by making a repeated
C run on the same problem with smaller tolerances, then all
C components of RTOL and ATOL (i.e. of EWT) should be scaled
C down uniformly.
C
C ITASK = An index specifying the task to be performed.
C Input only. ITASK has the following values and meanings.
C 1 means normal computation of output values of y(t) at
C t = TOUT (by overshooting and interpolating).
C 2 means take one step only and return.
C 3 means stop at the first internal mesh point at or
C beyond t = TOUT and return.
C 4 means normal computation of output values of y(t) at
C t = TOUT but without overshooting t = TCRIT.
C TCRIT must be input as RWORK(1). TCRIT may be equal to
C or beyond TOUT, but not behind it in the direction of
C integration. This option is useful if the problem
C has a singularity at or beyond t = TCRIT.
C 5 means take one step, without passing TCRIT, and return.
C TCRIT must be input as RWORK(1).
C
C Note: If ITASK = 4 or 5 and the solver reaches TCRIT
C (within roundoff), it will return T = TCRIT (exactly) to
C indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
C in which case answers at T = TOUT are returned first).
C
C ISTATE = an index used for input and output to specify the
C the state of the calculation.
C
C In the input, the values of ISTATE are as follows.
C 1 means this is the first call for the problem
C (initializations will be done). See note below.
C 2 means this is not the first call, and the calculation
C is to continue normally, with no change in any input
C parameters except possibly TOUT and ITASK.
C (If ITOL, RTOL, and/or ATOL are changed between calls
C with ISTATE = 2, the new values will be used but not
C tested for legality.)
C 3 means this is not the first call, and the
C calculation is to continue normally, but with
C a change in input parameters other than
C TOUT and ITASK. Changes are allowed in
C NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
C and any of the optional input except H0.
C
C Note: A preliminary call with TOUT = T is not counted
C as a first call here, as no initialization or checking of
C input is done. (Such a call is sometimes useful to include
C the initial conditions in the output.)
C Thus the first call for which TOUT .ne. T requires
C ISTATE = 1 in the input.
C
C In the output, ISTATE has the following values and meanings.
C 1 means nothing was done, as TOUT was equal to T with
C ISTATE = 1 in the input.
C 2 means the integration was performed successfully.
C -1 means an excessive amount of work (more than MXSTEP
C steps) was done on this call, before completing the
C requested task, but the integration was otherwise
C successful as far as T. (MXSTEP is an optional input
C and is normally 500.) To continue, the user may
C simply reset ISTATE to a value .gt. 1 and call again.
C (The excess work step counter will be reset to 0.)
C In addition, the user may increase MXSTEP to avoid
C this error return. (See optional input below.)
C -2 means too much accuracy was requested for the precision
C of the machine being used. This was detected before
C completing the requested task, but the integration
C was successful as far as T. To continue, the tolerance
C parameters must be reset, and ISTATE must be set
C to 3. The optional output TOLSF may be used for this
C purpose. (Note: If this condition is detected before
C taking any steps, then an illegal input return
C (ISTATE = -3) occurs instead.)
C -3 means illegal input was detected, before taking any
C integration steps. See written message for details.
C Note: If the solver detects an infinite loop of calls
C to the solver with illegal input, it will cause
C the run to stop.
C -4 means there were repeated error test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C The problem may have a singularity, or the input
C may be inappropriate.
C -5 means there were repeated convergence test failures on
C one attempted step, before completing the requested
C task, but the integration was successful as far as T.
C This may be caused by a poor preconditioner matrix.
C -6 means EWT(i) became zero for some i during the
C integration. Pure relative error control (ATOL(i)=0.0)
C was requested on a variable which has now vanished.
C The integration was successful as far as T.
C -7 means an unrecoverable error occurred in JAC or PSOL.
C Either JAC returned IER .ne. 0, or PSOL returned
C IER .lt. 0.
C
C Note: Since the normal output value of ISTATE is 2,
C it does not need to be reset for normal continuation.
C Also, since a negative input value of ISTATE will be
C regarded as illegal, a negative output value requires the
C user to change it, and possibly other input, before
C calling the solver again.
C
C IOPT = An integer flag to specify whether or not any optional
C input is being used on this call. Input only.
C The optional input is listed separately below.
C IOPT = 0 means no optional input is being used.
C Default values will be used in all cases.
C IOPT = 1 means optional input is being used.
C
C RWORK = A real working array (single precision).
C The length of RWORK must be at least
C 20 + NYH*(MAXORD + 1) + 3*NEQ + LENK + LWP where
C NYH = the initial value of NEQ,
C MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
C smaller value is given as an optional input),
C LENK = length of work space for Krylov-related data:
C LENK = 0 if MITER = 0,
C LENK = NEQ*(MAXL+3+MIN(1,MAXL-KMP))
C + (MAXL+3)*MAXL + 1 if MITER = 1,
C LENK = 3*NEQ if MITER = 9.
C LWP = length of real user work space for preconditioning.
C (See JAC/PSOL.)
C (See the MF description for METH and MITER.)
C Thus if MAXORD etc. have default values and NEQ is constant,
C this length is:
C 20 + 16*NEQ for MF = 10,
C 61 + 24*NEQ + LWP for MF = 11,
C 20 + 19*NEQ + LWP for MF = 19,
C 20 + 9*NEQ for MF = 20,
C 61 + 17*NEQ + LWP for MF = 21,
C 20 + 12*NEQ + LWP for MF = 29
C The first 20 words of RWORK are reserved for conditional
C and optional input and optional output.
C
C The following word in RWORK is a conditional input:
C RWORK(1) = TCRIT = critical value of t which the solver
C is not to overshoot. Required if ITASK is
C 4 or 5, and ignored otherwise. (See ITASK.)
C
C LRW = The length of the array RWORK, as declared by the user.
C (This will be checked by the solver.)
C
C IWORK = An integer work array. The length of IWORK must be at least
C 30 if MITER = 0 (MF = 10, 20), or
C 30 + LIWP otherwise (MF = 11, 21, 19, 29).
C LIWP = length of integer user work space for preconditioning.
C (See conditional input list following).
C
C The first 30 words of IWORK are reserved for conditional and
C optional input and optional output.
C
C The following 4 words in IWORK are conditional input,
C required if MITER .ge. 1:
C
C IWORK(1) = LWP = length of real array WP for use in
C preconditioning (part of RWORK array).
C IWORK(2) = LIWP = length of integer array IWP for use in
C preconditioning (part of IWORK array).
C The arrays WP and IWP are work arrays under the
C user's control, for use in the routines that
C perform preconditioning operations (JAC and PSOL).
C IWORK(3) = JPRE = preconditioner type flag:
C = 0 for no preconditioning (P1 = P2 = I
C = 1 for left-only preconditioning (P2 = I)
C = 2 for right-only preconditioning (P1 = I)
C = 3 for two-sided preconditioning
C IWORK(4) = JACFLG = flag for whether JAC is called.
C = 0 if JAC is not to be called,
C = 1 if JAC is to be called.
C Use JACFLG = 1 if JAC computes any nonconstant
C data needed in preconditioning operations,
C such as some of the Jacobian elements.
C
C
C LIW = the length of the array IWORK, as declared by the user.
C (This will be checked by the solver.)
C
C Note: The work arrays must not be altered between calls to SVODPK
C for the same problem, except possibly for the conditional and
C optional input, and except for the last 3*NEQ words of RWORK.
C The latter space is used for internal scratch space, and so is
C available for use by the user outside SVODPK between calls, if
C desired (but not for use by F or JAC).
C
C JAC = The name of the user-supplied routine (MITER = 1 or 9) to
C compute the Jacobian matrix, df/dy, as a function of
C the scalar t and the vector y. It is to have the form
C
C SUBROUTINE JAC (F, NEQ, T, Y, YSV, REWT, FTY, V, HRL1,
C 1 WP, IWP, IER, RPAR, IPAR)
C EXTERNAL F
C DIMENSION Y(NEQ), YSV(NEQ), REWT(NEQ), FTY(NEQ),
C 1 V(NEQ), WP(*), IWP(*), RPAR(*), IPAR(*)
C
C This routine must evaluate and preprocess any parts of the
C Jacobian matrix df/dy used in the preconditioners P1, P2 .
C The Y and FTY arrays contain the current values of y and
C f(t,y), respectively, and YSV also contains the current
C value of y. The array V is work space of length
C NEQ for use by JAC. REWT is the array of reciprocal error
C weights (1/ewt). JAC must multiply all computed Jacobian
C elements by the scalar -hrl1, add the identity matrix I and
C do any factorization operations called for, in preparation
C for solving linear systems with a coefficient matrix of
C P1 or P2. The matrix P1*P2 should be an approximation to
C I - hrl1 * (df/dy), where hrl1 is stored in HRL1. JAC should
C return IER = 0 if successful, and IER .ne. 0 if not.
C (If IER .ne. 0, a smaller time step will be tried.)
C The arrays WP (of length LWP) and IWP (of length LIWP)
C are for use by JAC and PSOL for work space and for storage
C of data needed for the solution of the preconditioner
C linear systems. Their lengths and contents are under the
C user's control.
C The JAC routine may save relevant Jacobian elements (or
C approximations) used in the preconditioners, along with the
C value of hrl1, and use these to reconstruct preconditioner
C matrices later without reevaluationg those elements.
C This may be cost-effective if JAC is called with hrl1
C considerably different from its earlier value, indicating
C that a corrector convergence failure has occurred because
C of the change in hrl1, not because of changes in the
C value of the Jacobian. In doing this, use the saved and
C current values of hrl1 to decide whether to use saved
C or reevaluated elements.
C JAC may alter V, but not Y, YSV, REWT, FTY, or HRL1.
C JAC must be declared external in the calling program.
C
C PSOL = the name of the user-supplied routine for the
C solution of preconditioner linear systems.
C It is to have the form
C SUBROUTINE PSOL (NEQ, T, Y, FTY, WK, HRL1, WP, IWP, B, LR,
C 1 IER, RPAR, IPAR)
C DIMENSION Y(NEQ), FTY(NEQ), WK(NEQ), WP(*), IWP(*),
C 1 B(NEQ), RPAR(*), IPAR(*)
C This routine must solve a linear system with b (stored in B)
C as right-hand side and one of the preconditioning matrices,
C P1 or P2, as coefficient matrix, and return the solution
C vector in B. LR is a flag concerning left vs. right
C preconditioning, input to PSOL. PSOL is to use P1 if LR = 1
C and P2 if LR = 2. In the case MITER = 9 (no Krylov
C iteration), LR will be 1 and then 2, according to JPRE, and
C PSOL is to return in B the desired approximate solution to
C A * x = b, where A = I - hrl1 * (df/dy). (hrl1 is stored in
C HRL1.) PSOL can use data generated in the JAC routine and
C stored in WP and IWP. The Y and FTY arrays contain the
C current values of y and f(t,y), respectively.
C The array WK is work space of length NEQ for use by PSOL.
C The argument HRL1 is the current value of the scalar appear-
C ing in the linear system. If the old value, as of the last
C JAC call, is needed, it must have been saved by JAC in WP.
C On return, PSOL should set the error flag IER as follows:
C IER = 0 if PSOL was successful,
C IER .gt. 0 on a recoverable error, meaning that the
C time step will be retried,
C IER .lt. 0 on an unrecoverable error, meaning that the
C solver is to stop immediately.
C PSOL may not alter Y, FTY, or HRL1.
C PSOL must be declared external in the calling program.
C
C MF = The method flag. Used only for input. The legal values of
C MF are 10, 11, 19, 20, 21, 29 .
C MF is a two-digit integer, MF = 10*METH + MITER .
C METH indicates the basic linear multistep method:
C METH = 1 means the implicit Adams method.
C METH = 2 means the method based on backward
C differentiation formulas (BDF-s).
C MITER indicates the corrector iteration method. Currently,
C the values of MITER have the following meanings:
C
C 0 means functional iteration is used (no Jacobian matrix
C is involved).
C
C 1 means SPIGMR, a scaled, preconditioned, incomplete version
C of GMRES, a generalized minimum residual method, is used.
C This is the best choice in general.
C
C 9 means that only a user-supplied matrix P (approximating A)
C will be used, with no Krylov iteration done internally to
C SVODPK. This option allows the user to provide the
C complete linear system solution algorithm, if desired.
C
C The user can apply preconditioning to the linear system A*x = b,
C by means of arbitrary matrices (the preconditioners).
C
C RPAR User-specified array used to communicate real parameters
C to user-supplied subroutines. If RPAR is a vector, then
C it must be dimensioned in the user's main program. If it
C is unused or a scalar, then it need not be dimensioned.
C
C IPAR User-specified array used to communicate integer parameter
C to user-supplied subroutines. The comments on dimensioning
C RPAR apply to IPAR.
C-----------------------------------------------------------------------
C Optional Inputs.
C
C The following is a list of the optional input provided for in the
C call sequence. (See also Part ii.) For each such input variable,
C this table lists its name as used in this documentation, its
C location in the call sequence, its meaning, and the default value.
C The use of any of this input requires IOPT = 1, and in that
C case all of this input is examined. A value of zero for any
C of these optional input variables will cause the default value to be
C used. Thus to use a subset of the optional input, simply preload
C locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
C then set those of interest to nonzero values.
C
C NAME LOCATION MEANING AND DEFAULT VALUE
C
C H0 RWORK(5) The step size to be attempted on the first step.
C The default value is determined by the solver.
C
C HMAX RWORK(6) The maximum absolute step size allowed.
C The default value is infinite.
C
C HMIN RWORK(7) The minimum absolute step size allowed.
C The default value is 0. (This lower bound is not
C enforced on the final step before reaching TCRIT
C when ITASK = 4 or 5.)
C
C DELT RWORK(8) Convergence test constant used in Krylov iteration
C algorithm. The default value is 0.05.
C
C MAXORD IWORK(5) The maximum order to be allowed. The default
C value is 12 if METH = 1, and 5 if METH = 2.
C If MAXORD exceeds the default value, it will
C be reduced to the default value.
C If MAXORD is changed during the problem, it may
C cause the current order to be reduced.
C
C MXSTEP IWORK(6) Maximum number of (internally defined) steps
C allowed during one call to the solver.
C The default value is 500.
C
C MXHNIL IWORK(7) Maximum number of messages printed (per problem)
C warning that T + H = T on a step (H = step size).
C This must be positive to result in a non-default
C value. The default value is 10.
C
C MAXL IWORK(8) maximum number of iterations in the SPIGMR
C algorithm (.le. NEQ). The default is
C MAXL = min(5, NEQ).
C
C KMP IWORK(9) number of vectors on which orthogonalization
C is done in the SPIGMR algorithm (.le. MAXL).
C The default is KMP = MAXL (complete GMRES method).
C See Ref. 2 for details on incomplete GMRES.
C Note: When KMP .lt. MAXL and MITER = 1, the length
C of RWORK must be set accordingly. See RWORK above.
C-----------------------------------------------------------------------
C Optional Outputs.
C
C As optional additional output from SVODPK, the variables listed
C below are quantities related to the performance of SVODPK
C which are available to the user. These are communicated by way of
C the work arrays, but also have internal mnemonic names as shown.
C Except where stated otherwise, all of this output is defined
C on any successful return from SVODPK, and on any return with
C ISTATE = -1, -2, -4, -5, -6, or -7. On an illegal input return
C (ISTATE = -3), they will be unchanged from their existing values
C (if any), except possibly for TOLSF, LENRW, and LENIW.
C On any error return, output relevant to the error will be defined,
C as noted below.
C
C NAME LOCATION MEANING
C
C HU RWORK(11) The step size in t last used (successfully).
C
C HCUR RWORK(12) The step size to be attempted on the next step.
C
C TCUR RWORK(13) The current value of the independent variable
C which the solver has actually reached, i.e. the
C current internal mesh point in t. In the output,
C TCUR will always be at least as far from the
C initial value of t as the current argument T,
C but may be farther (if interpolation was done).
C
C TOLSF RWORK(14) A tolerance scale factor, greater than 1.0,
C computed when a request for too much accuracy was
C detected (ISTATE = -3 if detected at the start of
C the problem, ISTATE = -2 otherwise). If ITOL is
C left unaltered but RTOL and ATOL are uniformly
C scaled up by a factor of TOLSF for the next call,
C then the solver is deemed likely to succeed.
C (The user may also ignore TOLSF and alter the
C tolerance parameters in any other way appropriate.)
C
C NST IWORK(11) The number of steps taken for the problem so far.
C
C NFE IWORK(12) The number of f evaluations for the problem so far.
C
C NPE IWORK(13) The number of preconditioner evaluations (JAC calls)
C so far.
C
C NQU IWORK(14) The method order last used (successfully).
C
C NQCUR IWORK(15) The order to be attempted on the next step.
C
C IMXER IWORK(16) The index of the component of largest magnitude in
C the weighted local error vector ( e(i)/EWT(i) ),
C on an error return with ISTATE = -4 or -5.
C
C LENRW IWORK(17) The length of RWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C LENIW IWORK(18) The length of IWORK actually required.
C This is defined on normal returns and on an illegal
C input return for insufficient storage.
C
C NNI IWORK(20) The number of nonlinear iterations so far (each of
C which calls the Krylov iterative linear solver).
C
C NCFN IWORK(21) The number of convergence failures of the nonlinear
C (Newton) iteration so far.
C Note: A measure of success is the overall rate of
C nonlinear convergence failures, NCFN/NST.
C
C NETF IWORK(22) The number of error test failures of the integrator
C so far.
C
C NLI IWORK(23) The number of linear iterations so far.
C Note: a measure of the success of SPIGMR algorithm
C is the average number of linear iterations per
C nonlinear iteration, given by NLI/NNI.
C If this is close to MAXL, MAXL may be too small.
C
C NPS IWORK(24) The number of preconditioning solve operations
C (PSOL calls) so far.
C
C NCFL IWORK(25) The number of convergence failures of the linear
C iteration so far.
C Note: A measure of success is the overall rate of
C linear convergence failures, NCFL/NNI.
C
C The following two arrays are segments of the RWORK array which
C may also be of interest to the user as optional output.
C For each array, the table below gives its internal name,
C its base address in RWORK, and its description.
C
C NAME BASE ADDRESS DESCRIPTION
C
C YH 21 The Nordsieck history array, of size NYH by
C (NQCUR + 1), where NYH is the initial value
C of NEQ. For j = 0,1,...,NQCUR, column j+1
C of YH contains HCUR**j/factorial(j) times
C the j-th derivative of the interpolating
C polynomial currently representing the
C solution, evaluated at t = TCUR.
C
C ACOR LENRW-NEQ+1 Array of size NEQ used for the accumulated
C corrections on each step, scaled in the output
C to represent the estimated local error in Y on
C the last step. This is the vector e in the
C description of the error control. Defined
C only on a successful return from SVODPK.
C
C-----------------------------------------------------------------------
C Interrupting and Restarting
C
C If the integration of a given problem by SVODPK is to be
C interrrupted and then later continued, such as when restarting
C an interrupted run or alternating between two or more ODE problems,
C the user should save, following the return from the last SVODPK call
C prior to the interruption, the contents of the call sequence
C variables and internal COMMON blocks, and later restore these
C values before the next SVODPK call for that problem. To save
C and restore the COMMON blocks, use subroutine SVKSRC, as
C described below in Part ii.
C
C In addition, if non-default values for either LUN or MFLAG are
C desired, an extra call to XSETUN and/or XSETF should be made just
C before continuing the integration. See Part ii below for details.
C
C-----------------------------------------------------------------------
C Part ii. Other Routines Callable.
C
C The following are optional calls which the user may make to
C gain additional capabilities in conjunction with SVODPK.
C (The routines XSETUN and XSETF are designed to conform to the
C SLATEC error handling package.)
C
C FORM OF CALL FUNCTION
C
C CALL XSETUN(LUN) Set the logical unit number, LUN, for
C output of messages from SVODPK, if
C the default is not desired.
C The default value of LUN is 6.
C
C CALL XSETF(MFLAG) Set a flag to control the printing of
C messages by SVODPK.
C MFLAG = 0 means do not print. (Danger:
C This risks losing valuable information.)
C MFLAG = 1 means print (the default).
C
C Either of the above calls may be made at
C any time and will take effect immediately.
C
C CALL SVKSRC(RSAV,ISAV,JOB) Saves and restores the contents of
C the internal COMMON blocks used by
C SVODPK. (See Part iii below.)
C RSAV must be a real array of length 52
C or more, and ISAV must be an integer
C array of length 52 or more.
C JOB=1 means save COMMON into RSAV/ISAV.
C JOB=2 means restore COMMON from RSAV/ISAV.
C
C SVKSRC is useful if one is
C interrupting a run and restarting
C later, or alternating between two or
C more problems solved with SVODPK.
C
C CALL SVINDY(,,,,,) Provide derivatives of y, of various
C (See below.) orders, at a specified point T, if
C desired. It may be called only after
C a successful return from SVODPK.
C
C The detailed instructions for using SVINDY are as follows.
C The form of the call is:
C
C CALL SVINDY (T, K, RWORK(21), NYH, DKY, IFLAG)
C
C The input parameters are:
C
C T = Value of independent variable where answers are desired
C (normally the same as the T last returned by SVODPK).
C For valid results, T must lie between TCUR - HU and TCUR.
C (See optional output for TCUR and HU.)
C K = Integer order of the derivative desired. K must satisfy
C 0 .le. K .le. NQCUR, where NQCUR is the current order
C (see optional output). The capability corresponding
C to K = 0, i.e. computing y(T), is already provided
C by SVODPK directly. Since NQCUR .ge. 1, the first
C derivative dy/dt is always available with SVINDY.
C RWORK(21) = The base address of the history array YH.
C NYH = Column length of YH, equal to the initial value of NEQ.
C
C The output parameters are:
C
C DKY = A real array of length NEQ containing the computed value
C of the K-th derivative of y(t).
C IFLAG = Integer flag, returned as 0 if K and T were legal,
C -1 if K was illegal, and -2 if T was illegal.
C On an error return, a message is also written.
C-----------------------------------------------------------------------
C Part iii. COMMON Blocks.
C If SVODPK is to be used in an overlay situation, the user
C must declare, in the primary overlay, the variables in:
C (1) the call sequence to SVODPK,
C (2) the three internal COMMON blocks
C /SVOD01/ of length 81 (48 single precision words
C followed by 33 integer words),
C /SVOD02/ of length 9 (1 single precision word
C followed by 8 integer words),
C /SVPK01/ of length 14 (3 single precision words
C followed by 11 integer words)
C
C If SVODPK is used on a system in which the contents of internal
C COMMON blocks are not preserved between calls, the user should
C declare the above three COMMON blocks in the calling (main) program
C to insure that their contents are preserved.
C
C-----------------------------------------------------------------------
C Part iv. Optionally Replaceable Solver Routines.
C
C Below are descriptions of two routines in the SVODPK package which
C relate to the measurement of errors. Either routine can be
C replaced by a user-supplied version, if desired. However, since such
C a replacement may have a major impact on performance, it should be
C done only when absolutely necessary, and only with great caution.
C (Note: The means by which the package version of a routine is
C superseded by the user's version may be system-dependent.)
C
C (a) SEWSET.
C The following subroutine is called just before each internal
C integration step, and sets the array of error weights, EWT, as
C described under ITOL/RTOL/ATOL above:
C SUBROUTINE SEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
C where NEQ, ITOL, RTOL, and ATOL are as in the SVODPK call sequence,
C YCUR contains the current dependent variable vector, and
C EWT is the array of weights set by SEWSET.
C
C If the user supplies this subroutine, it must return in EWT(i)
C (i = 1,...,NEQ) a positive quantity suitable for comparison with
C errors in Y(i). The EWT array returned by SEWSET is passed to the
C SVNORM routine (see below), and also used by SVODPK in the computation
C of the optional output IMXER, the diagonal Jacobian approximation,
C and the increments for difference quotient Jacobians.
C
C In the user-supplied version of SEWSET, it may be desirable to use
C the current values of derivatives of y. Derivatives up to order NQ
C are available from the history array YH, described above under
C Optional Output. In SEWSET, YH is identical to the YCUR array,
C extended to NQ + 1 columns with a column length of NYH and scale
C factors of h**j/factorial(j). On the first call for the problem,
C given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
C NYH is the initial value of NEQ. The quantities NQ, H, and NST
C can be obtained by including in SEWSET the statements
C COMMON /SVOD01/ RVOD(48), IVOD(33)
C COMMON /SVOD02/ HU, NCFN, NETF, NFE, NPE, NLU, NNI, NQU, NST
C NQ = IVOD(28)
C H = RVOD(21)
C Thus, for example, the current value of dy/dt can be obtained as
C YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
C unnecessary when NST = 0).
C
C (b) SVNORM.
C The following is a real function routine which computes the weighted
C root-mean-square norm of a vector v:
C D = SVNORM (N, V, W)
C where:
C N = the length of the vector,
C V = real array of length N containing the vector,
C W = real array of length N containing weights,
C D = sqrt( (1/N) * sum(V(i)*W(i))**2 ).
C SVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
C EWT is as set by subroutine SEWSET.
C
C If the user supplies this function, it should return a non-negative
C value of SVNORM suitable for use in the error control in SVODPK.
C None of the arguments should be altered by SVNORM.
C For example, a user-supplied SVNORM routine might:
C -substitute a max-norm of (V(i)*W(i)) for the RMS-norm, or
C -ignore some components of V in the norm, with the effect of
C suppressing the error control on those components of Y.
C-----------------------------------------------------------------------
C
C Revision History (YYYYMMDD)
C 19910315 DATE WRITTEN
C 19910415 Minor revisions to VODPK prologue.
C 19921106 In VSTEP, added ETAQ and ETAQM1 to SAVE statement.
C 19930701 IN VNLSK, moved line setting HRL1 below statement 220.
C 19940502 Minor revisions to VODPK prologue and internal comments.
C In VODPK, set JACFLG = 0 if 0 < MITER < 9 and JPRE = 0.
C In VNLSK, add conditions on rescaling of correction vector.
C 19940504 In demo programs, fixed logic in SOLSBG involving LR.
C 19970515 Minor revisions to VODPK prologue and internal comments.
C In VHIN, attached sign to H in second derivative estimation.
C 19981111 In VODPK, at end of Block B, when ISTATE = 3, jump to 200.
C 20020423 Major upgrade: Added *DECK lines. Renamed all routines and
C Common blocks for uniqueness across single/double prec.
C versions and for sharing of routines with VODE and ODEPACK.
C Changed name R1MACH to RUMACH.
C Converted intrinsic names to generic form.
C Numerous revisions to main prologue.
C Revisions to demo program - formats, intrinsics, comments.
C
C-----------------------------------------------------------------------
C Other Routines in the SVODPK Package.
C
C In addition to subroutine SVODPK, the SVODPK package includes the
C following subroutines and function routines:
C SVHIN computes an approximate step size for the initial step.
C SVINDY computes an interpolated value of the y vector at t = TOUT.
C SVSTEP is the core integrator, which does one step of the
C integration and the associated error control.
C SVSET sets all method coefficients and test constants.
C SVJUST adjusts the history array on a change of order.
C SVNLSK solves the underlying nonlinear system -- the corrector.
C SVSLPK manages solution of linear system in chord iteration.
C SVSPIG performs the SPIGMR algorithm.
C SVATV computes a scaled, preconditioned product (I-hrl1*J)*v.
C SORTHOG orthogonalizes a vector against previous basis vectors.
C SHEQR generates a QR factorization of a Hessenberg matrix.
C SHELS finds the least squares solution of a Hessenberg system.
C SVUSOL interfaces to the user's PSOL routine (MITER = 9).
C SEWSET sets the error weight vector EWT before each step.
C SVNORM computes the weighted r.m.s. norm of a vector.
C SVKSRC is a user-callable routine to save and restore
C the contents of the internal COMMON blocks.
C SAXPY, SCOPY, SDOT, SNRM2, and SSCAL are basic linear
C algebra modules (BLAS) used by this package.
C RUMACH computes the unit roundoff in a machine-independent manner.
C XERRWV, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
C error messages and warnings. XERRWV is machine-dependent.
C Note: SVNORM, SDOT, SNRM2, RUMACH, IXSAV, and IUMACH are function
C routines. All the others are subroutines.
C
C-----------------------------------------------------------------------
C
C Declarations for external routines and function subroutines called ---
EXTERNAL SVNLSK
REAL RUMACH, SVNORM
C
C Declarations for local variables -------------------------------------
C
LOGICAL IHIT, LAVD, LCFN, LCFL, LWARN
REAL ATOLI, AVDIM, BIG, EWTI, FOUR, H0, HMAX, HMX,
1 HUN, ONE, PT05, PT2, PT9, RCFL, RCFN, RH, RTOLI, SIZE,
2 TCRIT, TNEXT, TOLSF, TP, TWO, ZERO
INTEGER I, IER, IFLAG, IMXER, KGO, LENIW, LENIWK, LENRW, LENWK,
1 LENWM, LF0, LIWP, LWP, MORD, MXHNL0, MXSTP0, NCFL0, NCFN0,
2 NITER, NLI0, NNI0, NNID, NSTD, NSLAST, NWARN
CHARACTER*80 MSG
DIMENSION MORD(2)
C-----------------------------------------------------------------------
C The following Fortran-77 declarations are to cause the values of the
C listed (local) variables to be saved between calls to SVODPK.
C-----------------------------------------------------------------------
SAVE MORD, MXHNL0, MXSTP0
SAVE ZERO, ONE, TWO, FOUR, HUN, PT05, PT2, PT9
C
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL ACNRM, CCMXJ, CONP, CRATE, DRC, EL,
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU, TQ, TN, UROUND
INTEGER ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
1 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
2 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
3 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
4 NSLP, NYH
C
C Type declarations for labeled COMMON block SVOD02 --------------------
C
REAL HU
INTEGER NCFN, NETF, NFE, NPE, NLU, NNI, NQU, NST
C
C Type declarations for labeled COMMON block SVPK01 --------------------
C
REAL DELT, SQRTN, RSQRTN
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LVSAV, KMP, MAXL, MNEWT,
1 NLI, NPS, NCFL
C
C-----------------------------------------------------------------------
C The following internal COMMON blocks contain variables which are
C communicated between subroutines in the SVODPK package, or which are
C to be saved between calls to SVODPK.
C In each block, real variables precede integers.
C The block /SVOD01/ appears in subroutines SVODPK, SVINDY, SVSTEP,
C SVSET, SVJUST, SVNLSK, SVSLPK, SVATV, and SVKSRC.
C The block /SVOD02/ appears in subroutines SVODPK, SVINDY, SVSTEP,
C SVNLSK, SVSLPK, SVATV, and SVKSRC.
C The block /SVPK01/ appears in subroutines SVODPK, SVNLSK, SVSLPK,
C and SVKSRC.
C
C The variables stored in the internal COMMON blocks are as follows:
C
C ACNRM = Weighted r.m.s. norm of accumulated correction vectors.
C CCMXJ = Threshhold on DRC for updating the Jacobian. (See DRC.)
C CONP = The saved value of TQ(5).
C CRATE = Estimated corrector convergence rate constant.
C DRC = Relative change in H*RL1 since last VJAC call.
C EL = Real array of integration coefficients. See SVSET.
C ETA = Saved tentative ratio of new to old H.
C ETAMAX = Saved maximum value of ETA to be allowed.
C H = The step size.
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 HNEW = The step size to be attempted on the next step.
C HSCAL = Stepsize in scaling of YH array.
C PRL1 = The saved value of RL1.
C RC = Ratio of current H*RL1 to value on last VJAC call.
C RL1 = The reciprocal of the coefficient EL(1).
C TAU = Real vector of past NQ step sizes, length 13.
C TQ = A real vector of length 5 in which SVSET stores constants
C used for the convergence test, the error test, and the
C selection of H at a new order.
C TN = The independent variable, updated on each step taken.
C UROUND = The machine unit roundoff. The smallest positive real number
C such that 1.0 + UROUND .ne. 1.0
C ICF = Integer flag for convergence failure in SVNLSK:
C 0 means no failures.
C 1 means convergence failure with out of date Jacobian
C (recoverable error).
C 2 means convergence failure with current Jacobian or
C singular matrix (unrecoverable error).
C INIT = Saved integer flag indicating whether initialization of the
C problem has been done (INIT = 1) or not.
C IPUP = Saved flag to signal updating of Newton matrix.
C JCUR = Output flag from VJAC showing Jacobian status:
C JCUR = 0 means J is not current.
C JCUR = 1 means J is current.
C JSTART = Integer flag used as input to SVSTEP:
C 0 means perform the first step.
C 1 means take a new step continuing from the last.
C -1 means take the next step with a new value of MAXORD,
C HMIN, HMXI, N, METH, MITER, and/or matrix parameters.
C On return, SVSTEP sets JSTART = 1.
C JSV = Integer flag for Jacobian saving, = sign(MF).
C KFLAG = A completion code from SVSTEP 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, -4 fatal error in VNLS.
C KUTH = Input flag to SVSTEP showing whether H was reduced by the
C driver. KUTH = 1 if H was reduced, = 0 otherwise.
C L = Integer variable, NQ + 1, current order plus one.
C LMAX = MAXORD + 1 (used for dimensioning).
C LOCJS = A pointer to the saved Jacobian, whose storage starts at
C WM(LOCJS), if JSV = 1.
C LYH, LEWT, LACOR, LSAVF, LWM, LIWM = Saved integer pointers
C to segments of RWORK and IWORK.
C MAXORD = The maximum order of integration method to be allowed.
C METH/MITER = The method flags. See MF.
C MSBJ = The maximum number of steps between J evaluations, = 50.
C MXHNIL = Saved value of optional input MXHNIL.
C MXSTEP = Saved value of optional input MXSTEP.
C N = The number of first-order ODEs, = NEQ.
C NEWH = Saved integer to flag change of H.
C NEWQ = The method order to be used on the next step.
C NHNIL = Saved counter for occurrences of T + H = T.
C NQ = Integer variable, the current integration method order.
C NQNYH = Saved value of NQ*NYH.
C NQWAIT = A counter controlling the frequency of order changes.
C An order change is about to be considered if NQWAIT = 1.
C NSLJ = The number of steps taken as of the last Jacobian update.
C NSLP = Saved value of NST as of last Newton matrix update.
C NYH = Saved value of the initial value of NEQ.
C
C HU = The step size in t last used.
C NCFN = Number of nonlinear convergence failures so far.
C NETF = The number of error test failures of the integrator so far.
C NFE = The number of f evaluations for the problem so far.
C NPE = The number of preconditioner evaluations (JAC calls) so far.
C NLU = The number of matrix LU decompositions so far.
C NNI = Number of nonlinear iterations so far.
C NQU = The method order last used.
C NST = The number of steps taken for the problem so far.
C
C DELT = Convergence test constant in Krylov iterations.
C SQRTN = SQRT(NEQ), for use in weights in Krylov convergence tests.
C RSQRTN = 1.0/SQRTN, also for use in convergence weights.
C JPRE = Preconditioner type flag.
C JACFLG = Indicator for presence of user-supplied JAC routine.
C LOCWP = Location of start of user's WP array in WM work array.
C LOCIWP = Location of start of user's IWP array in IWM work array.
C LVSAV = Saved pointer to VSAV array in RWORK.
C KMP = Number of vectors on which orthogonalization is done in
C Krylov iteration.
C MAXL = Maximum dimension of Krylov subspace used.
C MNEWT = Newton iteration index.
C NLI = Number of linear (Krylov) iterations done.
C NPS = Number of preconditioner solvers (PSOL calls) done.
C NCFL = Number of convergence failures in Krylov iteration.
C-----------------------------------------------------------------------
COMMON /SVOD01/ ACNRM, CCMXJ, CONP, CRATE, DRC, EL(13),
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU(13), TQ(5), TN, UROUND,
3 ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
4 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
5 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
6 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
7 NSLP, NYH
COMMON /SVOD02/ HU, NCFN, NETF, NFE, NPE, NLU, NNI, NQU, NST
COMMON /SVPK01/ DELT, SQRTN, RSQRTN, JPRE, JACFLG, LOCIWP,
1 LOCWP, LVSAV, KMP, MAXL, MNEWT, NLI, NPS, NCFL
C
DATA MORD(1),MORD(2)/12,5/, MXSTP0/500/, MXHNL0/10/
DATA ZERO/0.0E0/, ONE/1.0E0/, TWO/2.0E0/, FOUR/4.0E0/,
1 PT05/0.05E0/, PT2/0.2E0/, PT9/0.9E0/, HUN/100.0E0/
C-----------------------------------------------------------------------
C Block A.
C This code block is executed on every call.
C It tests ISTATE and ITASK for legality and branches appropriately.
C If ISTATE .gt. 1 but the flag INIT shows that initialization has
C not yet been done, an error return occurs.
C If ISTATE = 1 and TOUT = T, jump to Block G and return immediately.
C-----------------------------------------------------------------------
IF (ISTATE .LT. 1 .OR. ISTATE .GT. 3) GO TO 601
IF (ITASK .LT. 1 .OR. ITASK .GT. 5) GO TO 602
IF (ISTATE .EQ. 1) GO TO 10
IF (INIT .EQ. 0) GO TO 603
IF (ISTATE .EQ. 2) GO TO 200
GO TO 20
10 INIT = 0
IF (TOUT .EQ. T) RETURN
C-----------------------------------------------------------------------
C Block B.
C The next code block is executed for the initial call (ISTATE = 1),
C or for a continuation call with parameter changes (ISTATE = 3).
C It contains checking of all inputs and various initializations.
C
C First check legality of the non-optional inputs NEQ, ITOL, IOPT, MF.
C-----------------------------------------------------------------------
20 IF (NEQ .LE. 0) GO TO 604
IF (ISTATE .EQ. 1) GO TO 25
IF (NEQ .GT. N) GO TO 605
25 N = NEQ
IF (ITOL .LT. 1 .OR. ITOL .GT. 4) GO TO 606
IF (IOPT .LT. 0 .OR. IOPT .GT. 1) GO TO 607
JSV = 0
METH = MF/10
MITER = MF - 10*METH
IF (METH .LT. 1 .OR. METH .GT. 2) GO TO 608
IF (MITER .LT. 0) GO TO 608
IF (MITER .GT. 1 .AND. MITER .LT. 9) GO TO 608
IF (MITER .GE. 1) JPRE = IWORK(3)
JACFLG = 0
IF (MITER .GE. 1) JACFLG = IWORK(4)
IF (MITER .GE. 1 .AND. MITER .NE. 9 .AND. JPRE .EQ. 0) JACFLG = 0
C Next process and check the optional inputs. --------------------------
IF (IOPT .EQ. 1) GO TO 40
MAXORD = MORD(METH)
MXSTEP = MXSTP0
MXHNIL = MXHNL0
IF (ISTATE .EQ. 1) H0 = ZERO
HMXI = ZERO
HMIN = ZERO
MAXL = MIN(5,N)
KMP = MAXL
DELT = PT05
GO TO 60
40 MAXORD = IWORK(5)
IF (MAXORD .LT. 0) GO TO 611
IF (MAXORD .EQ. 0) MAXORD = 100
MAXORD = MIN(MAXORD,MORD(METH))
MXSTEP = IWORK(6)
IF (MXSTEP .LT. 0) GO TO 612
IF (MXSTEP .EQ. 0) MXSTEP = MXSTP0
MXHNIL = IWORK(7)
IF (MXHNIL .LT. 0) GO TO 613
IF (MXHNIL .EQ. 0) MXHNIL = MXHNL0
IF (ISTATE .NE. 1) GO TO 50
H0 = RWORK(5)
IF ((TOUT - T)*H0 .LT. ZERO) GO TO 614
50 HMAX = RWORK(6)
IF (HMAX .LT. ZERO) GO TO 615
HMXI = ZERO
IF (HMAX .GT. ZERO) HMXI = ONE/HMAX
HMIN = RWORK(7)
IF (HMIN .LT. ZERO) GO TO 616
MAXL = IWORK(8)
IF (MAXL .EQ. 0) MAXL = 5
MAXL = MIN(MAXL,N)
KMP = IWORK(9)
IF (KMP .EQ. 0 .OR. KMP .GT. MAXL) KMP = MAXL
DELT = RWORK(8)
IF (DELT .EQ. 0.0E0) DELT = PT05
C-----------------------------------------------------------------------
C Set work array pointers and check lengths lrw and liw.
C Pointers to segments of RWORK and iwork are named by prefixing l to
C the name of the segment. e.g., the segment YH starts at RWORK(LYH).
C Segments of RWORK (in order) are YH, WM, EWT, SAVF, VSAV, ACOR.
C Within WM, LOCWP is the location of the WP work array,
C and within IWM, LOCIWP is the location of the IWP work array.
C-----------------------------------------------------------------------
60 LYH = 21
IF (ISTATE .EQ. 1) NYH = N
LWM = LYH + (MAXORD + 1)*NYH
IF (MITER .EQ. 0) LENWK = 0
IF (MITER .EQ. 1)
1 LENWK = N*(MAXL+2+MIN(1,MAXL-KMP)) + (MAXL+3)*MAXL + 1
IF (MITER .EQ. 9) LENWK = 2*N
LWP = 0
IF (MITER .GE. 1) LWP = IWORK(1)
LENWM = LENWK + LWP
LOCWP = LENWK + 1
LEWT = LWM + LENWM
LSAVF = LEWT + N
LVSAV = LSAVF + N
LACOR = LVSAV + N
IF (MITER .EQ. 0) LACOR = LVSAV
LENRW = LACOR + N - 1
IWORK(17) = LENRW
LIWM = 31
LENIWK = 0
LIWP = 0
IF (MITER .GE. 1) LIWP = IWORK(2)
LENIW = 30 + LENIWK + LIWP
LOCIWP = LENIWK + 1
IWORK(18) = LENIW
IF (LENRW .GT. LRW) GO TO 617
IF (LENIW .GT. LIW) GO TO 618
C Check RTOL and ATOL for legality. ------------------------------------
RTOLI = RTOL(1)
ATOLI = ATOL(1)
DO 70 I = 1, N
IF (ITOL .GE. 3) RTOLI = RTOL(I)
IF (ITOL .EQ. 2 .OR. ITOL .EQ. 4) ATOLI = ATOL(I)
IF (RTOLI .LT. ZERO) GO TO 619
IF (ATOLI .LT. ZERO) GO TO 620
70 CONTINUE
C Load SQRT(N) and its reciprocal in common. ---------------------------
SQRTN = SQRT(REAL(N))
RSQRTN = ONE/SQRTN
IF (ISTATE .EQ. 1) GO TO 100
C If ISTATE = 3, set flag to signal parameter changes to SVSTEP. -------
JSTART = -1
IF (NQ .LE. MAXORD) GO TO 200
C MAXORD was reduced below NQ. Copy YH(*,MAXORD+2) into SAVF. ---------
CALL SCOPY (N, RWORK(LWM), 1, RWORK(LSAVF), 1)
GO TO 200
C-----------------------------------------------------------------------
C Block C.
C The next block is for the initial call only (ISTATE = 1).
C It contains all remaining initializations, the initial call to F,
C and the calculation of the initial step size.
C The error weights in EWT are inverted after being loaded.
C-----------------------------------------------------------------------
100 UROUND = RUMACH()
TN = T
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 110
TCRIT = RWORK(1)
IF ((TCRIT - TOUT)*(TOUT - T) .LT. ZERO) GO TO 625
IF (H0 .NE. ZERO .AND. (T + H0 - TCRIT)*H0 .GT. ZERO)
1 H0 = TCRIT - T
110 JSTART = 0
CCMXJ = PT2
MSBJ = 50
NHNIL = 0
NST = 0
NSLAST = 0
HU = ZERO
NQU = 0
NPE = 0
NLI0 = 0
NNI0 = 0
NCFN0 = 0
NCFL0 = 0
NWARN = 0
NNI = 0
NLI = 0
NPS = 0
NETF = 0
NCFN = 0
NCFL = 0
C Initial call to F. (LF0 points to YH(*,2).) -------------------------
LF0 = LYH + NYH
CALL F (N, T, Y, RWORK(LF0), RPAR, IPAR)
NFE = 1
C Load the initial value vector in YH. ---------------------------------
CALL SCOPY (N, Y, 1, RWORK(LYH), 1)
C Load and invert the EWT array. (H is temporarily set to 1.0.) -------
NQ = 1
H = ONE
CALL SEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
DO 120 I = 1, N
IF (RWORK(I+LEWT-1) .LE. ZERO) GO TO 621
120 RWORK(I+LEWT-1) = ONE/RWORK(I+LEWT-1)
IF (H0 .NE. ZERO) GO TO 180
C Call SVHIN to set initial step size H0 to be attempted. --------------
CALL SVHIN (N, T, RWORK(LYH), RWORK(LF0), F, RPAR, IPAR, TOUT,
1 UROUND, RWORK(LEWT), ITOL, ATOL, Y, RWORK(LACOR), H0,
2 NITER, IER)
NFE = NFE + NITER
IF (IER .NE. 0) GO TO 622
C Adjust H0 if necessary to meet HMAX bound. ---------------------------
180 RH = ABS(H0)*HMXI
IF (RH .GT. ONE) H0 = H0/RH
C Load H with H0 and scale YH(*,2) by H0. ------------------------------
H = H0
CALL SSCAL (N, H0, RWORK(LF0), 1)
GO TO 270
C-----------------------------------------------------------------------
C Block D.
C The next code block is for continuation calls only (ISTATE = 2 or 3)
C and is to check stop conditions before taking a step.
C-----------------------------------------------------------------------
200 NSLAST = NST
KUTH = 0
NLI0 = NLI
NNI0 = NNI
NCFN0 = NCFN
NCFL0 = NCFL
NWARN = 0
GO TO (210, 250, 220, 230, 240), ITASK
210 IF ((TN - TOUT)*H .LT. ZERO) GO TO 250
CALL SVINDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
220 TP = TN - HU*(ONE + HUN*UROUND)
IF ((TP - TOUT)*H .GT. ZERO) GO TO 623
IF ((TN - TOUT)*H .LT. ZERO) GO TO 250
GO TO 400
230 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. ZERO) GO TO 624
IF ((TCRIT - TOUT)*H .LT. ZERO) GO TO 625
IF ((TN - TOUT)*H .LT. ZERO) GO TO 245
CALL SVINDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
IF (IFLAG .NE. 0) GO TO 627
T = TOUT
GO TO 420
240 TCRIT = RWORK(1)
IF ((TN - TCRIT)*H .GT. ZERO) GO TO 624
245 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. HUN*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + HNEW*(ONE + FOUR*UROUND)
IF ((TNEXT - TCRIT)*H .LE. ZERO) GO TO 250
H = (TCRIT - TN)*(ONE - FOUR*UROUND)
KUTH = 1
C-----------------------------------------------------------------------
C Block E.
C The next block is normally executed for all calls and contains
C the call to the one-step core integrator SVSTEP.
C
C This is a looping point for the integration steps.
C
C First check for too many steps being taken,
C check for poor Newton/Krylov performance, update EWT (if not at
C start of problem), check for too much accuracy being requested, and
C check for H below the roundoff level in T.
C-----------------------------------------------------------------------
250 CONTINUE
IF ((NST-NSLAST) .GE. MXSTEP) GO TO 500
CALL SEWSET (N, ITOL, RTOL, ATOL, RWORK(LYH), RWORK(LEWT))
NSTD = NST - NSLAST
NNID = NNI - NNI0
IF (NSTD .LT. 10 .OR. NNID .EQ. 0) GO TO 255
AVDIM = REAL(NLI - NLI0)/REAL(NNID)
RCFN = REAL(NCFN - NCFN0)/REAL(NSTD)
RCFL = REAL(NCFL - NCFL0)/REAL(NNID)
LAVD = AVDIM .GT. (MAXL - PT05)
LCFN = RCFN .GT. PT9
LCFL = RCFL .GT. PT9
LWARN = LAVD .OR. LCFN .OR. LCFL
IF (.NOT.LWARN) GO TO 255
NWARN = NWARN + 1
IF (NWARN .GT. 10) GO TO 255
IF (LAVD) THEN
MSG = 'SVODPK- Warning. Poor iterative algorithm performance '
CALL XERRWV (MSG, 56, 111, 0, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' at T = R1. Average no. of linear iterations = R2 '
CALL XERRWV (MSG, 56, 111, 0, 0, 0, 0, 2, TN, AVDIM)
ENDIF
IF (LCFN) THEN
MSG = 'SVODPK- Warning. Poor iterative algorithm performance '
CALL XERRWV (MSG, 56, 112, 0, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' at T = R1. Nonlinear convergence failure rate = R2'
CALL XERRWV (MSG, 56, 112, 0, 0, 0, 0, 2, TN, RCFN)
ENDIF
IF (LCFL) THEN
MSG = 'SVODPK- Warning. Poor iterative algorithm performance '
CALL XERRWV (MSG, 56, 113, 0, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' at T = R1. Linear convergence failure rate = R2 '
CALL XERRWV (MSG, 56, 113, 0, 0, 0, 0, 2, TN, RCFL)
ENDIF
255 CONTINUE
DO 260 I = 1, N
IF (RWORK(I+LEWT-1) .LE. ZERO) GO TO 510
260 RWORK(I+LEWT-1) = ONE/RWORK(I+LEWT-1)
270 TOLSF = UROUND*SVNORM (N, RWORK(LYH), RWORK(LEWT))
IF (TOLSF .LE. ONE) GO TO 280
TOLSF = TOLSF*TWO
IF (NST .EQ. 0) GO TO 626
GO TO 520
280 IF ((TN + H) .NE. TN) GO TO 290
NHNIL = NHNIL + 1
IF (NHNIL .GT. MXHNIL) GO TO 290
MSG = 'SVODPK- Warning: internal T (=R1) and H (=R2) are'
CALL XERRWV (MSG, 50, 101, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG=' such that in the machine, T + H = T on the next step '
CALL XERRWV (MSG, 60, 101, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' (H = step size). solver will continue anyway'
CALL XERRWV (MSG, 50, 101, 1, 0, 0, 0, 2, TN, H)
IF (NHNIL .LT. MXHNIL) GO TO 290
MSG = 'SVODPK- Above warning has been issued I1 times. '
CALL XERRWV (MSG, 50, 102, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' it will not be issued again for this problem'
CALL XERRWV (MSG, 50, 102, 1, 1, MXHNIL, 0, 0, ZERO, ZERO)
290 CONTINUE
C-----------------------------------------------------------------------
C CALL SVSTEP (Y, YH, NYH, YH, EWT, SAVF, VSAV, ACOR, WM, IWM,
C F, JAC, PSOL, SVNLSK, RPAR, IPAR)
C-----------------------------------------------------------------------
CALL SVSTEP (Y, RWORK(LYH), NYH, RWORK(LYH), RWORK(LEWT),
1 RWORK(LSAVF), RWORK(LVSAV), RWORK(LACOR), RWORK(LWM),
2 IWORK(LIWM), F, JAC, PSOL, SVNLSK, RPAR, IPAR)
KGO = 1 - KFLAG
GO TO (300, 530, 540, 550, 555), KGO
C-----------------------------------------------------------------------
C Block F.
C The following block handles the case of a successful return from the
C core integrator (KFLAG = 0). Test for stop conditions.
C-----------------------------------------------------------------------
300 INIT = 1
KUTH = 0
GO TO (310, 400, 330, 340, 350), ITASK
C ITASK = 1. if TOUT has been reached, interpolate. -------------------
310 IF ((TN - TOUT)*H .LT. ZERO) GO TO 250
CALL SVINDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
C ITASK = 3. Jump to exit if TOUT was reached. ------------------------
330 IF ((TN - TOUT)*H .GE. ZERO) GO TO 400
GO TO 250
C ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary.
340 IF ((TN - TOUT)*H .LT. ZERO) GO TO 345
CALL SVINDY (TOUT, 0, RWORK(LYH), NYH, Y, IFLAG)
T = TOUT
GO TO 420
345 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. HUN*UROUND*HMX
IF (IHIT) GO TO 400
TNEXT = TN + H*(ONE + FOUR*UROUND)
IF ((TNEXT - TCRIT)*H .LE. ZERO) GO TO 250
H = (TCRIT - TN)*(ONE - FOUR*UROUND)
KUTH = 1
GO TO 250
C ITASK = 5. See if TCRIT was reached and jump to exit. ---------------
350 HMX = ABS(TN) + ABS(H)
IHIT = ABS(TN - TCRIT) .LE. HUN*UROUND*HMX
C-----------------------------------------------------------------------
C Block G.
C The following block handles all successful returns from SVODPK.
C If ITASK .ne. 1, Y is loaded from YH and T is set accordingly.
C ISTATE is set to 2, and the optional outputs are loaded into the
C work arrays before returning.
C-----------------------------------------------------------------------
400 CONTINUE
CALL SCOPY (N, RWORK(LYH), 1, Y, 1)
T = TN
IF (ITASK .NE. 4 .AND. ITASK .NE. 5) GO TO 420
IF (IHIT) T = TCRIT
420 ISTATE = 2
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NPE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(20) = NNI
IWORK(21) = NCFN
IWORK(22) = NETF
IWORK(23) = NLI
IWORK(24) = NPS
IWORK(25) = NCFL
RETURN
C-----------------------------------------------------------------------
C Block H.
C The following block handles all unsuccessful returns other than
C those for illegal input. First the error message routine is called.
C if there was an error test or convergence test failure, IMXER is set.
C Then Y is loaded from YH, and T is set to TN. The optional outputs
C are loaded into the work arrays before returning.
C-----------------------------------------------------------------------
C The maximum number of steps was taken before reaching TOUT. ----------
500 MSG = 'SVODPK- At current T (=R1), MXSTEP (=I1) steps '
CALL XERRWV (MSG, 50, 201, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' taken on this call before reaching TOUT '
CALL XERRWV (MSG, 50, 201, 1, 1, MXSTEP, 0, 1, TN, ZERO)
ISTATE = -1
GO TO 580
C EWT(i) .le. 0.0 for some i (not at start of problem). ----------------
510 EWTI = RWORK(LEWT+I-1)
MSG = 'SVODPK- At T (=R1), EWT(I1) has become R2 .le. 0.'
CALL XERRWV (MSG, 50, 202, 1, 1, I, 0, 2, TN, EWTI)
ISTATE = -6
GO TO 580
C Too much accuracy requested for machine precision. -------------------
520 MSG = 'SVODPK- At T (=R1), too much accuracy requested '
CALL XERRWV (MSG, 50, 203, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' for precision of machine: See TOLSF (=R2) '
CALL XERRWV (MSG, 50, 203, 1, 0, 0, 0, 2, TN, TOLSF)
RWORK(14) = TOLSF
ISTATE = -2
GO TO 580
C KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. -----
530 MSG = 'SVODPK- At T(=R1) and step size H(=R2), the error'
CALL XERRWV (MSG, 50, 204, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' test failed repeatedly or with abs(H) = HMIN'
CALL XERRWV (MSG, 50, 204, 1, 0, 0, 0, 2, TN, H)
ISTATE = -4
GO TO 560
C KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ----
540 MSG = 'SVODPK- At T (=R1) and step size H (=R2), the '
CALL XERRWV (MSG, 50, 205, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' corrector convergence failed repeatedly '
CALL XERRWV (MSG, 50, 205, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' or with abs(H) = HMIN '
CALL XERRWV (MSG, 30, 205, 1, 0, 0, 0, 2, TN, H)
ISTATE = -5
GO TO 560
C KFLAG = -3. Unrecoverable error from JAC. ---------------------------
550 MSG = 'SVODPK- at T (=R1) an unrecoverable error return '
CALL XERRWV(MSG, 50, 206, 0, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' was made from subroutine JAC '
CALL XERRWV(MSG, 40, 206, 0, 0, 0, 0, 1, TN, ZERO)
ISTATE = -7
GO TO 580
C KFLAG = -4. Unrecoverable error from PSOL. --------------------------
555 MSG = 'SVODPK- at T (=R1) an unrecoverable error return '
CALL XERRWV(MSG, 50, 207, 0, 0, 0, 0, 0, ZERO, ZERO)
MSG = ' was made from subroutine PSOL '
CALL XERRWV(MSG, 40, 207, 0, 0, 0, 0, 1, TN, ZERO)
ISTATE = -7
GO TO 580
C Compute IMXER if relevant. -------------------------------------------
560 BIG = ZERO
IMXER = 1
DO 570 I = 1, N
SIZE = ABS(RWORK(I+LACOR-1)*RWORK(I+LEWT-1))
IF (BIG .GE. SIZE) GO TO 570
BIG = SIZE
IMXER = I
570 CONTINUE
IWORK(16) = IMXER
C Set Y vector, T, and optional outputs. -------------------------------
580 CONTINUE
CALL SCOPY (N, RWORK(LYH), 1, Y, 1)
T = TN
RWORK(11) = HU
RWORK(12) = H
RWORK(13) = TN
IWORK(11) = NST
IWORK(12) = NFE
IWORK(13) = NPE
IWORK(14) = NQU
IWORK(15) = NQ
IWORK(20) = NNI
IWORK(21) = NCFN
IWORK(22) = NETF
IWORK(23) = NLI
IWORK(24) = NPS
IWORK(25) = NCFL
RETURN
C-----------------------------------------------------------------------
C Block I.
C The following block handles all error returns due to illegal input
C (ISTATE = -3), as detected before calling the core integrator.
C Call the error message routine and then return.
C-----------------------------------------------------------------------
601 MSG = 'SVODPK- ISTATE (=I1) illegal '
CALL XERRWV (MSG, 30, 1, 1, 1, ISTATE, 0, 0, ZERO, ZERO)
IF (ISTATE .LT. 0) GO TO 800
GO TO 700
602 MSG = 'SVODPK- ITASK (=I1) illegal '
CALL XERRWV (MSG, 30, 2, 1, 1, ITASK, 0, 0, ZERO, ZERO)
GO TO 700
603 MSG='SVODPK- ISTATE (=I1) .gt. 1 but SVODPK not initialized '
CALL XERRWV (MSG, 60, 3, 1, 1, ISTATE, 0, 0, ZERO, ZERO)
GO TO 700
604 MSG = 'SVODPK- NEQ (=I1) .lt. 1 '
CALL XERRWV (MSG, 30, 4, 1, 1, NEQ, 0, 0, ZERO, ZERO)
GO TO 700
605 MSG = 'SVODPK- ISTATE = 3 and NEQ increased (I1 to I2) '
CALL XERRWV (MSG, 50, 5, 1, 2, N, NEQ, 0, ZERO, ZERO)
GO TO 700
606 MSG = 'SVODPK- ITOL (=I1) illegal '
CALL XERRWV (MSG, 30, 6, 1, 1, ITOL, 0, 0, ZERO, ZERO)
GO TO 700
607 MSG = 'SVODPK- IOPT (=I1) illegal '
CALL XERRWV (MSG, 30, 7, 1, 1, IOPT, 0, 0, ZERO, ZERO)
GO TO 700
608 MSG = 'SVODPK- MF (=I1) illegal '
CALL XERRWV (MSG, 30, 8, 1, 1, MF, 0, 0, ZERO, ZERO)
GO TO 700
611 MSG = 'SVODPK- MAXORD (=I1) .lt. 0 '
CALL XERRWV (MSG, 30, 11, 1, 1, MAXORD, 0, 0, ZERO, ZERO)
GO TO 700
612 MSG = 'SVODPK- MXSTEP (=I1) .lt. 0 '
CALL XERRWV (MSG, 30, 12, 1, 1, MXSTEP, 0, 0, ZERO, ZERO)
GO TO 700
613 MSG = 'SVODPK- MXHNIL (=I1) .lt. 0 '
CALL XERRWV (MSG, 30, 13, 1, 1, MXHNIL, 0, 0, ZERO, ZERO)
GO TO 700
614 MSG = 'SVODPK- TOUT (=R1) behind T (=R2) '
CALL XERRWV (MSG, 40, 14, 1, 0, 0, 0, 2, TOUT, T)
MSG = ' integration direction is given by H0 (=R1) '
CALL XERRWV (MSG, 50, 14, 1, 0, 0, 0, 1, H0, ZERO)
GO TO 700
615 MSG = 'SVODPK- HMAX (=R1) .lt. 0.0 '
CALL XERRWV (MSG, 30, 15, 1, 0, 0, 0, 1, HMAX, ZERO)
GO TO 700
616 MSG = 'SVODPK- HMIN (=R1) .lt. 0.0 '
CALL XERRWV (MSG, 30, 16, 1, 0, 0, 0, 1, HMIN, ZERO)
GO TO 700
617 CONTINUE
MSG='SVODPK- RWORK length needed, LENRW (=I1), exceeds LRW (=I2)'
CALL XERRWV (MSG, 60, 17, 1, 2, LENRW, LRW, 0, ZERO, ZERO)
GO TO 700
618 CONTINUE
MSG='SVODPK- IWORK length needed, LENIW (=I1), exceeds LIW (=I2)'
CALL XERRWV (MSG, 60, 18, 1, 2, LENIW, LIW, 0, ZERO, ZERO)
GO TO 700
619 MSG = 'SVODPK- RTOL(I1) is R1 .lt. 0.0 '
CALL XERRWV (MSG, 40, 19, 1, 1, I, 0, 1, RTOLI, ZERO)
GO TO 700
620 MSG = 'SVODPK- ATOL(I1) is R1 .lt. 0.0 '
CALL XERRWV (MSG, 40, 20, 1, 1, I, 0, 1, ATOLI, ZERO)
GO TO 700
621 EWTI = RWORK(LEWT+I-1)
MSG = 'SVODPK- EWT(I1) is R1 .le. 0.0 '
CALL XERRWV (MSG, 40, 21, 1, 1, I, 0, 1, EWTI, ZERO)
GO TO 700
622 CONTINUE
MSG='SVODPK- TOUT (=R1) too close to T(=R2) to start integration'
CALL XERRWV (MSG, 60, 22, 1, 0, 0, 0, 2, TOUT, T)
GO TO 700
623 CONTINUE
MSG='SVODPK- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) '
CALL XERRWV (MSG, 60, 23, 1, 1, ITASK, 0, 2, TOUT, TP)
GO TO 700
624 CONTINUE
MSG='SVODPK- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) '
CALL XERRWV (MSG, 60, 24, 1, 0, 0, 0, 2, TCRIT, TN)
GO TO 700
625 CONTINUE
MSG='SVODPK- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) '
CALL XERRWV (MSG, 60, 25, 1, 0, 0, 0, 2, TCRIT, TOUT)
GO TO 700
626 MSG = 'SVODPK- At start of problem, too much accuracy '
CALL XERRWV (MSG, 50, 26, 1, 0, 0, 0, 0, ZERO, ZERO)
MSG=' requested for precision of machine: See TOLSF (=R1) '
CALL XERRWV (MSG, 60, 26, 1, 0, 0, 0, 1, TOLSF, ZERO)
RWORK(14) = TOLSF
GO TO 700
627 MSG='SVODPK- Trouble from SVINDY. ITASK = I1, TOUT = R1 '
CALL XERRWV (MSG, 60, 27, 1, 1, ITASK, 0, 1, TOUT, ZERO)
C
700 CONTINUE
ISTATE = -3
RETURN
C
800 MSG = 'SVODPK- Run aborted: apparent infinite loop '
CALL XERRWV (MSG, 50, 303, 2, 0, 0, 0, 0, ZERO, ZERO)
RETURN
C----------------------- End of Subroutine SVODPK ----------------------
END
*DECK SVNLSK
SUBROUTINE SVNLSK (Y, YH, LDYH, VSAV, SAVF, EWT, ACOR, IWM, WM,
1 F, JAC, PSOL, NFLAG, RPAR, IPAR)
C
EXTERNAL F, JAC, PSOL
REAL Y, YH, VSAV, SAVF, EWT, ACOR, WM, RPAR
INTEGER IWM, LDYH, NFLAG, IPAR
DIMENSION Y(*), YH(LDYH, *), SAVF(*), VSAV(*), EWT(*), ACOR(*),
1 IWM(*), WM(*), RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C Call sequence input -- YH, LDYH, F, JAC, EWT, ACOR, IWM, WM,
C NFLAG, RPAR, IPAR
C Call sequence output -- Y, YH, VSAV, SAVF, ACOR, IWM, WM, NFLAG
C COMMON block variables accessed:
C /SVOD01/ ACNRM, CRATE, DRC, H, ICF, IPUP, JCUR, JSTART,
C METH, MITER, N, NSLP, RC, RL1, TN, TQ
C /SVOD02/ NFE, NNI, NPE, NST
C /SVPK01/ JACFLG, LOCIWP, LOCWP, MNEWT
C Subroutines called: F, JAC, PSOL, SAXPY, SCOPY, SSCAL, SVSLPK
C Function subroutines called: SVNORM
C-----------------------------------------------------------------------
C Subroutine SVNLSK is a nonlinear system solver, which uses either
C functional iteration (MITER = 0), or a combination of an inexact
C Newton method and preconditioned Krylov iteration (MITER .gt. 0)
C to solve the implicit system for the corrector y vector.
C It calls Subroutine JAC (user-supplied) for preprocessing the
C preconditioner, and Subroutine SVSLPK for the Krylov iteration.
C
C In addition to variables described elsewhere, communication with
C SVNLSK uses the following variables:
C
C Y = The dependent variable, a vector of length N, input.
C YH = The Nordsieck (Taylor) array, LDYH by LMAX, input
C and output. On input, it contains predicted values.
C LDYH = A constant .ge. N, the first dimension of YH, input.
C VSAV = A work array of length N.
C SAVF = A work array of length N.
C EWT = An error weight vector of length N, input.
C ACOR = A work array of length N, used for the accumulated
C corrections to the predicted y vector.
C WM,IWM = Real and integer work arrays associated with matrix
C operations in Newton iteration (MITER .ne. 0).
C F = Dummy name for user-supplied routine for f.
C JAC = Dummy name for user-supplied routine for Jacobian data
C and associated preconditioner matrix.
C PSOL = Dummy name for user-supplied subroutine to solve
C preconditioner linear system.
C NFLAG = Input/output flag, with values and meanings as follows:
C INPUT
C 0 first call for this time step.
C -1 convergence failure in previous call to SVNLSK.
C -2 error test failure in SVSTEP.
C OUTPUT
C 0 successful completion of nonlinear solver.
C -1 convergence failure or failure in JAC.
C -2 unrecoverable error in matrix preprocessing
C (cannot occur here).
C -3 unrecoverable error in PSOL.
C RPAR, IPAR = Dummy names for user's real and integer work arrays.
C
C IPUP = Own variable flag with values and meanings as follows:
C 0, do not update preconditioner.
C MITER .ne. 0, update the preconditioner, because it is
C the initial step, user input changed,
C there was an error test failure, or an
C update is indicated by a change in the
C scalar RC or step counter NST.
C
C For more details, see comments in driver subroutine.
C-----------------------------------------------------------------------
C
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL ACNRM, CCMXJ, CONP, CRATE, DRC, EL,
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU, TQ, TN, UROUND
INTEGER ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
1 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
2 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
3 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
4 NSLP, NYH
C
C Type declarations for labeled COMMON block SVOD02 --------------------
C
REAL HU
INTEGER NCFN, NETF, NFE, NPE, NLU, NNI, NQU, NST
C
C Type declarations for labeled COMMON block SVPK01 --------------------
C
REAL DELT, SQRTN, RSQRTN
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LVSAV, KMP, MAXL, MNEWT,
1 NLI, NPS, NCFL
C
COMMON /SVOD01/ ACNRM, CCMXJ, CONP, CRATE, DRC, EL(13),
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU(13), TQ(5), TN, UROUND,
3 ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
4 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
5 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
6 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
7 NSLP, NYH
COMMON /SVOD02/ HU, NCFN, NETF, NFE, NPE, NLU, NNI, NQU, NST
COMMON /SVPK01/ DELT, SQRTN, RSQRTN, JPRE, JACFLG, LOCIWP,
1 LOCWP, LVSAV, KMP, MAXL, MNEWT, NLI, NPS, NCFL
C
C Type declarations for local variables --------------------------------
C
REAL CCMAX, CRDOWN, CSCALE, DEL, DCON, DELP, HRL1,RDIV
REAL ONE, TWO, ZERO
INTEGER I, IERPJ, IERSL, M, MAXCOR, MSBP
C
C Type declaration for function subroutines called ---------------------
C
REAL SVNORM
C-----------------------------------------------------------------------
C The following Fortran-77 declarations are to cause the values of the
C listed (local) variables to be saved between calls to SVODPK.
SAVE CCMAX, CRDOWN, MAXCOR, MSBP, RDIV
SAVE ONE, TWO, ZERO
C-----------------------------------------------------------------------
DATA CCMAX /0.3E0/, CRDOWN /0.3E0/, MAXCOR /3/, MSBP /20/,
1 RDIV /2.0E0/
DATA ONE /1.0E0/, TWO /2.0E0/, ZERO /0.0E0/
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(*). The YH array is not altered in the corrector loop.
C-----------------------------------------------------------------------
IF (JSTART .EQ. 0) NSLP = 0
IF (NFLAG .EQ. 0) ICF = 0
IF (NFLAG .EQ. -2) IPUP = MITER
IF ( (JSTART .EQ. 0) .OR. (JSTART .EQ. -1) ) IPUP = MITER
IF (JACFLG .EQ. 0) THEN
IPUP = 0
CRATE = ONE
GO TO 220
ENDIF
DRC = ABS(RC-ONE)
IF (DRC .GT. CCMAX .OR. NST .GE. NSLP+MSBP) IPUP = MITER
220 M = 0
HRL1 = H*RL1
DELP = ZERO
MNEWT = 0
CALL SCOPY (N, YH(1,1), 1, Y, 1 )
CALL F (N, TN, Y, SAVF, RPAR, IPAR)
NFE = NFE + 1
IF (IPUP .LE. 0) GO TO 250
C-----------------------------------------------------------------------
C If indicated, the preconditioner matrix 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-----------------------------------------------------------------------
JCUR = 1
IERPJ = 0
CALL JAC (F, N, TN, Y, YH, EWT, SAVF, ACOR, HRL1,
1 WM(LOCWP), IWM(LOCIWP), IERPJ, RPAR, IPAR)
NPE = NPE + 1
IPUP = 0
RC = ONE
DRC = ZERO
CRATE = ONE
NSLP = NST
IF (IERPJ .NE. 0) GO TO 420
250 DO 260 I = 1, N
260 ACOR(I) = ZERO
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) = RL1*(H*SAVF(I) - YH(I,2))
290 Y(I) = SAVF(I) - ACOR(I)
DEL = SVNORM (N, Y, EWT)
DO 300 I = 1, N
300 Y(I) = YH(I,1) + SAVF(I)
CALL SCOPY (N, SAVF, 1, ACOR, 1)
GO TO 400
C-----------------------------------------------------------------------
C In the case of the Newton method, compute the corrector error,
C and solve the linear system with that as right-hand side and
C A as coefficient matrix. In the case of Modified Newton iteration
C with BDF, the correction is scaled by the factor 2/(1+RC) to
C account for changes in H*RL1 since the last JAC call.
C-----------------------------------------------------------------------
350 DO 360 I = 1, N
360 VSAV(I) = HRL1*SAVF(I) - (RL1*YH(I,2) + ACOR(I))
CALL SVSLPK (Y, SAVF, VSAV, EWT, WM, IWM, F, PSOL, IERSL,
1 RPAR, IPAR)
NNI = NNI + 1
IF (METH .EQ. 2 .AND. JACFLG .EQ. 1
1 .AND. MITER .EQ. 9 .AND. RC .NE. ONE) THEN
CSCALE = TWO/(ONE + RC)
CALL SSCAL (N, CSCALE, VSAV, 1)
ENDIF
IF (IERSL .LT. 0) GO TO 440
IF (IERSL .GT. 0) GO TO 410
DEL = SVNORM (N, VSAV, EWT)
CALL SAXPY (N, ONE, VSAV, 1, ACOR, 1)
DO 380 I = 1, N
380 Y(I) = YH(I,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(CRDOWN*CRATE,DEL/DELP)
DCON = DEL*MIN(ONE,CRATE)/TQ(4)
IF (DCON .LE. ONE) GO TO 450
M = M + 1
IF (M .EQ. MAXCOR) GO TO 410
IF (M .GE. 2 .AND. DEL .GT. RDIV*DELP) GO TO 410
MNEWT = M
DELP = DEL
CALL F (N, TN, Y, SAVF, RPAR, IPAR)
NFE = NFE + 1
GO TO 270
C
410 IF (MITER .EQ. 0 .OR. JCUR .EQ. 1 .OR. JACFLG .EQ. 0) GO TO 420
ICF = 1
IPUP = MITER
GO TO 220
C
420 CONTINUE
ICF = 2
NFLAG = -1
RETURN
440 CONTINUE
NFLAG = -3
RETURN
C Return for successful step. ------------------------------------------
450 NFLAG = 0
JCUR = 0
ICF = 0
IF (M .EQ. 0) ACNRM = DEL
IF (M .GT. 0) ACNRM = SVNORM (N, ACOR, EWT)
RETURN
C----------------------- End of Subroutine SVNLSK ----------------------
END
*DECK SVSLPK
SUBROUTINE SVSLPK (Y, SAVF, X, EWT, WM, IWM, F, PSOL, IERSL,
1 RPAR, IPAR)
EXTERNAL F, PSOL
REAL Y, SAVF, X, EWT, WM, RPAR
INTEGER IWM, IERSL, IPAR
DIMENSION Y(*), SAVF(*), X(*), EWT(*), WM(*), IWM(*),
1 RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C Call sequence input -- Y, SAVF, X, EWT, F, PSOL, RPAR, IPAR
C Call sequence output -- Y, SAVF, X, WM, IWM, IERSL
C COMMON block variables accessed:
C /SVOD01/ H, RL1, TQ, TN, MITER, N
C /SVPK01/ DELT, SQRTN, RSQRTN, JPRE, LOCIWP, LOCWP,
C KMP, MAXL, MNEWT, NLI, NPS, NCFL
C Subroutines called: F, PSOL, SCOPY, SSCAL, SVSPIG, SVUSOL
C-----------------------------------------------------------------------
C This routine interfaces with SVSPIG or SVUSOL for the solution of
C the linear system arising from a Newton iteration (MITER .ne. 0).
C
C In addition to variables described elsewhere, communication with
C SVSLPK 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. Otherwise,
C the stepsize is reduced (forcing a new evalua-
C tion of the preconditioner) and the step is
C repeated.
C IERSL = -1 means there was a nonrecoverable error in the
C iterative solver. The stepsize is reduced in
C SVSTEP and the step is repeated.
C-----------------------------------------------------------------------
C
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL ACNRM, CCMXJ, CONP, CRATE, DRC, EL,
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU, TQ, TN, UROUND
INTEGER ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
1 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
2 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
3 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
4 NSLP, NYH
C
C Type declarations for labeled COMMON block SVPK01 --------------------
C
REAL DELT, SQRTN, RSQRTN
INTEGER JPRE, JACFLG, LOCWP, LOCIWP, LVSAV, KMP, MAXL, MNEWT,
1 NLI, NPS, NCFL
C
C Type declarations for local variables --------------------------------
C
REAL DELTA, HRL1
INTEGER IFLAG, LB, LDL, LGMR, LHES, LQ, LV, LWK, MAXLP1, NPSL
C-----------------------------------------------------------------------
COMMON /SVOD01/ ACNRM, CCMXJ, CONP, CRATE, DRC, EL(13),
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU(13), TQ(5), TN, UROUND,
3 ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
4 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
5 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
6 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
7 NSLP, NYH
COMMON /SVPK01/ DELT, SQRTN, RSQRTN, JPRE, JACFLG, LOCIWP,
1 LOCWP, LVSAV, KMP, MAXL, MNEWT, NLI, NPS, NCFL
C-----------------------------------------------------------------------
C
IERSL = 0
HRL1 = H*RL1
DELTA = DELT*TQ(4)
IF (MITER .EQ. 1) THEN
C-----------------------------------------------------------------------
C Use the SPIGMR algorithm to solve the linear system A*x = -f.
C-----------------------------------------------------------------------
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 SVSPIG (TN, Y, SAVF, WM(LB), EWT, N, MAXL, MAXLP1, KMP,
1 DELTA, HRL1, JPRE, MNEWT, F, PSOL, NPSL, X, WM(LV), WM(LHES),
2 WM(LQ), LGMR, WM(LOCWP), IWM(LOCIWP), WM(LWK), WM(LDL),
3 RPAR, IPAR, IFLAG)
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
ELSE IF (MITER .EQ. 9) THEN
C-----------------------------------------------------------------------
C Use SVUSOL, which interfaces to PSOL, to solve the linear system
C (No Krylov iteration).
C-----------------------------------------------------------------------
LB = 1
LWK = LB + N
CALL SCOPY (N, X, 1, WM(LB), 1)
CALL SVUSOL (N, TN, Y, SAVF, WM(LB), EWT, DELTA, HRL1, JPRE,
1 MNEWT, PSOL, NPSL, X, WM(LOCWP), IWM(LOCIWP), WM(LWK),
2 RPAR, IPAR, IFLAG)
NPS = NPS + NPSL
IF (IFLAG .NE. 0) NCFL = NCFL + 1
IF (IFLAG .EQ. 3) IERSL = 1
IF (IFLAG .LT. 0) IERSL = -1
RETURN
ENDIF
C----------------------- End of Subroutine SVSLPK ----------------------
END
*DECK SVSPIG
SUBROUTINE SVSPIG (TN, Y, SAVF, B, WGHT, N, MAXL, MAXLP1,
1 KMP, DELTA, HB0, JPRE, MNEWT, F, PSOL, NPSL, X, V, HES, Q,
2 LGMR, WP, IWP, WK, DL, RPAR, IPAR, IFLAG)
EXTERNAL F, PSOL
REAL TN, Y, SAVF, B, WGHT, DELTA, HB0, X, V, HES,
1 Q, WP, WK, DL, RPAR
INTEGER N, MAXL, MAXLP1, KMP, JPRE, MNEWT, NPSL, LGMR, IWP,
1 IFLAG, IPAR
DIMENSION Y(*), SAVF(*), B(*), WGHT(*), X(*), V(N,*),
1 HES(MAXLP1,*), Q(*), WP(*), IWP(*), WK(*), DL(*),
2 RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C Call sequence input -- TN, Y, SAVF, B, WGHT, N, MAXL, MAXLP1, DELTA,
C HB0, JPRE, MNEWT, F, PSOL, RPAR, IPAR
C Call sequence output -- B, KMP, DELTA, NPSL, X, V, HES, Q, LGMR, WP,
C IWP, WK, DL, RPAR, IPAR, IFLAG
C COMMON block variables accessed: None
C Subroutines called: F, SORTHOG, PSOL, SAXPY, SCOPY, SHELS, SHEQR,
C SSCAL, SVATV
C Function subroutines called: SNRM2
C-----------------------------------------------------------------------
C This routine solves the linear system A * x = b using SPIGMR,
C a scaled preconditioned incomplete version of the generalized
C minimum residual method GMRES.
C An initial guess of x = 0 is assumed.
C-----------------------------------------------------------------------
C
C On entry
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 SVSPIG.)
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 HB0 = current value of (step size h) * (coefficient beta0).
C
C JPRE = preconditioner type flag.
C
C MNEWT = Newton iteration counter (.ge. 0).
C
C WK = real work array used by routine SVATV 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
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 the current
C order of the upper Hessenberg 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-----------------------------------------------------------------------
C
C Type declarations for local variables --------------------------------
C
REAL BNRM, BNRM0, C, DLNRM, PROD, RHO, S, SNORMW, TEM
INTEGER I, IER, INFO, IP1, I2, J, K, LL, LLP1
C
C Type declaration for function subroutines called ---------------------
C
REAL SNRM2
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 (N, TN, Y, SAVF, WK, HB0, WP, IWP, B, 1,
1 IER, RPAR, IPAR)
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 SVATV 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 SVATV (Y, SAVF, V(1,LL), WGHT, X, F, PSOL, RPAR, IPAR,
1 V(1,LL+1), WK, WP, IWP, HB0, 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 (N, TN, Y, SAVF, WK, HB0, WP, IWP, X, 2,
1 IER, RPAR, IPAR)
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 SVSPIG ----------------------
END
*DECK SVATV
SUBROUTINE SVATV (Y, SAVF, V, WGHT, FTEM, F, PSOL, RPAR, IPAR,
1 Z, VTEM, WP, IWP, HB0, JPRE, IER, NPSL)
EXTERNAL F, PSOL
REAL Y, SAVF, V, WGHT, FTEM, RPAR, Z, VTEM, WP, HB0
INTEGER IPAR, IWP, JPRE, IER, NPSL
DIMENSION Y(*), SAVF(*), V(*), WGHT(*), FTEM(*), Z(*),
1 VTEM(*), WP(*), IWP(*), RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C Call sequence input -- Y, SAVF, V, WGHT, F, PSOL, RPAR, IPAR,
C WP, IWP, HB0, NPSL
C Call sequence output --Z, IER, NPSL
C COMMON block variables accessed:
C /SVOD01/ TN, N
C /SVOD02/ NFE
C Subroutines called: F, PSOL, SCOPY
C Function subroutines called: SNRM2
C-----------------------------------------------------------------------
C This routine computes the product
C
C (D-inverse)*(P1-inverse)*(I - hb0*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 L2 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 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 HB0 = current value of (step size h) * (coefficient beta0).
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
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL ACNRM, CCMXJ, CONP, CRATE, DRC, EL,
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU, TQ, TN, UROUND
INTEGER ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
1 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
2 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
3 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
4 NSLP, NYH
C
C Type declarations for labeled COMMON block SVOD02 --------------------
C
REAL HU
INTEGER NCFN, NETF, NFE, NPE, NLU, NNI, NQU, NST
C
C Type declarations for local variables --------------------------------
C
REAL FAC, RNORM, TEMPN
INTEGER I
C
C Type declaration for function subroutines called ---------------------
C
REAL SNRM2
C
COMMON /SVOD01/ ACNRM, CCMXJ, CONP, CRATE, DRC, EL(13),
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU(13), TQ(5), TN, UROUND,
3 ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
4 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
5 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
6 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
7 NSLP, NYH
COMMON /SVOD02/ HU, NCFN, NETF, NFE, NPE, NLU, NNI, NQU, NST
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 = HB0
GO TO 60
C
C JPRE = 2 or 3. Apply inverse of right preconditioner to VTEM. -------
30 CONTINUE
CALL PSOL (N, TN, Y, SAVF, FTEM, HB0, WP, IWP, VTEM, 2,
1 IER, RPAR, IPAR)
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 = HB0*TEMPN
C
C For all JPRE, call F with incremented Y argument, and restore Y. -----
60 CONTINUE
CALL F (N, TN, Y, FTEM, RPAR, IPAR)
NFE = NFE + 1
CALL SCOPY (N, Z, 1, Y, 1)
C Set Z = (I - HB0*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 (N, TN, Y, SAVF, FTEM, HB0, WP, IWP, Z, 1,
1 IER, RPAR, IPAR)
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 SVATV -----------------------
END
*DECK SVUSOL
SUBROUTINE SVUSOL (N, TN, Y, SAVF, B, WGHT, DELTA, HB0, JPRE,
1 MNEWT, PSOL, NPSL, X, WP, IWP, WK, RPAR, IPAR, IFLAG)
EXTERNAL PSOL
REAL TN, Y, SAVF, B, WGHT, DELTA, HB0, X, WP, WK, RPAR
INTEGER N, JPRE, MNEWT, NPSL, IWP, IPAR, IFLAG
DIMENSION Y(*), SAVF(*), B(*), WGHT(*), X(*),
1 WP(*), IWP(*), WK(*), RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C This routine solves the linear system A * x = b using only
C calls 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 1 (if JPRE = 1 or 3),
C then 2 (if JPRE = 2 or 3).
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 HB0 = current value of (step size h) * (coefficient beta0).
C
C JPRE = preconditioner type flag.
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-----------------------------------------------------------------------
C
C Type declarations for local variables --------------------------------
C
REAL BNRM
INTEGER I, IER
C
C Type declaration for function subroutines called ---------------------
C
REAL 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 Apply inverse of left preconditioner to vector b. --------------------
30 IER = 0
IF (JPRE .EQ. 0 .OR. JPRE .EQ. 2) GO TO 40
CALL PSOL (N, TN, Y, SAVF, WK, HB0, WP, IWP, B, 1,
1 IER, RPAR,IPAR)
NPSL = 1
IF (IER .NE. 0) GO TO 100
C Apply inverse of right preconditioner to result, and copy to X. ------
40 IF (JPRE .LE. 1) GO TO 50
CALL PSOL (N, TN, Y, SAVF, WK, HB0, WP, IWP, B, 2,
1 IER, RPAR, IPAR)
NPSL = NPSL + 1
IF (IER .NE. 0) GO TO 100
50 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 SVUSOL ----------------------
END
*DECK SVKSRC
SUBROUTINE SVKSRC (RSAV, ISAV, JOB)
REAL RSAV
INTEGER ISAV, JOB
DIMENSION RSAV(*), ISAV(*)
C-----------------------------------------------------------------------
C Call sequence input -- RSAV, ISAV, JOB
C Call sequence output -- RSAV, ISAV
C COMMON block variables accessed: all of /SVOD01/, /SVOD02/, /SVPK01/
C
C Subroutines/functions called by SVKSRC: None
C-----------------------------------------------------------------------
C This routine saves or restores (depending on JOB) the contents of the
C COMMON blocks SVOD01, SVOD02, SVPK01, used internally by SVODPK.
C
C RSAV = real array of length 52 or more.
C ISAV = integer array of length 52 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
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL RVOD1
INTEGER IVOD1
C
C Type declarations for labeled COMMON block SVOD02 --------------------
C
REAL RVOD2
INTEGER IVOD2
C
C Type declarations for labeled COMMON block SVPK01 --------------------
C
REAL RVPK1
INTEGER IVPK1
C
C Type declarations for local variables --------------------------------
C
INTEGER I, IOFF, LENIV1, LENIV2, LENRV1, LENRV2, LRVK1, LIVK1
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 LENRV1, LENIV1, LENRV2, LENIV2, LRVK1, LIVK1
C-----------------------------------------------------------------------
COMMON /SVOD01/ RVOD1(48), IVOD1(33)
COMMON /SVOD02/ RVOD2(1), IVOD2(8)
COMMON /SVPK01/ RVPK1(3), IVPK1(11)
DATA LENRV1 /48/, LENIV1 /33/, LENRV2 /1/, LENIV2 /8/,
1 LRVK1 /3/, LIVK1 /11/
C
IF (JOB .EQ. 2) GO TO 100
DO 10 I = 1, LENRV1
10 RSAV(I) = RVOD1(I)
DO 12 I = 1, LENRV2
12 RSAV(LENRV1+I) = RVOD2(I)
IOFF = LENRV1 + LENRV2
DO 14 I = 1, LRVK1
14 RSAV(IOFF+I) = RVPK1(I)
C
DO 20 I = 1, LENIV1
20 ISAV(I) = IVOD1(I)
DO 22 I = 1, LENIV2
22 ISAV(LENIV1+I) = IVOD2(I)
IOFF = LENIV1 + LENIV2
DO 24 I = 1, LIVK1
24 ISAV(IOFF+I) = IVPK1(I)
C
RETURN
C
100 CONTINUE
DO 110 I = 1, LENRV1
110 RVOD1(I) = RSAV(I)
DO 112 I = 1, LENRV2
112 RVOD2(I) = RSAV(LENRV1+I)
IOFF = LENRV1 + LENRV2
DO 114 I = 1, LRVK1
114 RVPK1(I) = RSAV(IOFF+I)
C
DO 120 I = 1, LENIV1
120 IVOD1(I) = ISAV(I)
DO 122 I = 1, LENIV2
122 IVOD2(I) = ISAV(LENIV1+I)
IOFF = LENIV1 + LENIV2
DO 124 I = 1, LIVK1
124 IVPK1(I) = ISAV(IOFF+I)
C
RETURN
C----------------------- End of Subroutine SVKSRC ----------------------
END
*DECK SVHIN
SUBROUTINE SVHIN (N, T0, Y0, YDOT, F, RPAR, IPAR, TOUT, UROUND,
1 EWT, ITOL, ATOL, Y, TEMP, H0, NITER, IER)
EXTERNAL F
REAL T0, Y0, YDOT, RPAR, TOUT, UROUND, EWT, ATOL, Y,
1 TEMP, H0
INTEGER N, IPAR, ITOL, NITER, IER
DIMENSION Y0(*), YDOT(*), EWT(*), ATOL(*), Y(*),
1 TEMP(*), RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C Call sequence input -- N, T0, Y0, YDOT, F, RPAR, IPAR, 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 SVHIN: F
C Function routines called by SVHIN: 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 w.r.m.s.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 SVHIN is done with the following variables:
C
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 RPAR, IPAR = Dummy names for user's real and integer work arrays.
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, H, HALF, HG, HLB, HNEW, HRAT,
1 HUB, HUN, PT1, T1, TDIST, TROUND, TWO, YDDNRM
INTEGER I, ITER
C
C Type declaration for function subroutines called ---------------------
C
REAL SVNORM
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. --------
H = SIGN (HG, TOUT - T0)
T1 = T0 + H
DO 60 I = 1, N
60 Y(I) = Y0(I) + H*YDOT(I)
CALL F (N, T1, Y, TEMP, RPAR, IPAR)
DO 70 I = 1, N
70 TEMP(I) = (TEMP(I) - YDOT(I))/H
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. Then exit. ----
80 H0 = HNEW*HALF
IF (H0 .LT. HLB) H0 = HLB
IF (H0 .GT. HUB) H0 = HUB
90 H0 = SIGN(H0, TOUT - T0)
NITER = ITER
IER = 0
RETURN
C Error return for TOUT - T0 too small. --------------------------------
100 IER = -1
RETURN
C----------------------- End of Subroutine SVHIN -----------------------
END
*DECK SVINDY
SUBROUTINE SVINDY (T, K, YH, LDYH, DKY, IFLAG)
REAL T, YH, DKY
INTEGER K, LDYH, IFLAG
DIMENSION YH(LDYH,*), DKY(*)
C-----------------------------------------------------------------------
C Call sequence input -- T, K, YH, LDYH
C Call sequence output -- DKY, IFLAG
C COMMON block variables accessed:
C /SVOD01/ -- H, TN, UROUND, L, N, NQ
C /SVOD02/ -- HU
C
C Subroutines called by SVINDY.: SSCAL, XERRWV
C Function routines called by SVINDY: None
C-----------------------------------------------------------------------
C SVINDY 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, 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 Discussion above and comments in driver explain all variables.
C-----------------------------------------------------------------------
C
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL ACNRM, CCMXJ, CONP, CRATE, DRC, EL,
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU, TQ, TN, UROUND
INTEGER ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
1 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
2 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
3 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
4 NSLP, NYH
C
C Type declarations for labeled COMMON block SVOD02 --------------------
C
REAL HU
INTEGER NCFN, NETF, NFE, NJE, NLU, NNI, NQU, NST
C
C Type declarations for local variables --------------------------------
C
REAL C, HUN, R, S, TFUZZ, TN1, TP, ZERO
INTEGER I, IC, J, JB, JB2, JJ, JJ1, JP1
CHARACTER*80 MSG
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 HUN, ZERO
C
COMMON /SVOD01/ ACNRM, CCMXJ, CONP, CRATE, DRC, EL(13),
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU(13), TQ(5), TN, UROUND,
3 ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
4 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
5 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
6 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
7 NSLP, NYH
COMMON /SVOD02/ HU, NCFN, NETF, NFE, NJE, NLU, NNI, NQU, NST
C
DATA HUN /100.0E0/, ZERO /0.0E0/
C
IFLAG = 0
IF (K .LT. 0 .OR. K .GT. NQ) GO TO 80
TFUZZ = HUN*UROUND*SIGN(ABS(TN) + ABS(HU), HU)
TP = TN - HU - TFUZZ
TN1 = TN + TFUZZ
IF ((T-TP)*(T-TN1) .GT. ZERO) 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 = REAL(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 = REAL(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)
CALL SSCAL (N, R, DKY, 1)
RETURN
C
80 MSG = 'SVINDY-- K (=I1) illegal '
CALL XERRWV (MSG, 30, 51, 1, 1, K, 0, 0, ZERO, ZERO)
IFLAG = -1
RETURN
90 MSG = 'SVINDY-- T (=R1) illegal '
CALL XERRWV (MSG, 30, 52, 1, 0, 0, 0, 1, T, ZERO)
MSG=' T not in interval TCUR - HU (= R1) to TCUR (=R2) '
CALL XERRWV (MSG, 60, 52, 1, 0, 0, 0, 2, TP, TN)
IFLAG = -2
RETURN
C----------------------- End of Subroutine SVINDY ----------------------
END
*DECK SVSTEP
SUBROUTINE SVSTEP (Y, YH, LDYH, YH1, EWT, SAVF, VSAV, ACOR,
1 WM, IWM, F, JAC, PSOL, VNLS, RPAR, IPAR)
EXTERNAL F, JAC, PSOL, VNLS
REAL Y, YH, YH1, EWT, SAVF, VSAV, ACOR, WM, RPAR
INTEGER LDYH, IWM, IPAR
DIMENSION Y(*), YH(LDYH,*), YH1(*), EWT(*), SAVF(*), VSAV(*),
1 ACOR(*), WM(*), IWM(*), RPAR(*), IPAR(*)
C-----------------------------------------------------------------------
C Call sequence input -- Y, YH, LDYH, YH1, EWT, SAVF, VSAV,
C ACOR, WM, IWM, F, JAC, PSOL, VNLS, RPAR, IPAR
C Call sequence output -- YH, ACOR, WM, IWM
C COMMON block variables accessed:
C /SVOD01/ ACNRM, EL(13), H, HMIN, HMXI, HNEW, HSCAL, RC, TAU(13),
C TQ(5), TN, JCUR, JSTART, KFLAG, KUTH,
C L, LMAX, MAXORD, N, NEWQ, NQ, NQWAIT
C /SVOD02/ HU, NCFN, NETF, NFE, NQU, NST
C
C Subroutines called by SVSTEP: F, SAXPY, SCOPY, SSCAL,
C SVJUST, VNLS, SVSET
C Function routines called by SVSTEP: SVNORM
C-----------------------------------------------------------------------
C SVSTEP performs one step of the integration of an initial value
C problem for a system of ordinary differential equations.
C SVSTEP calls subroutine VNLS for the solution of the nonlinear system
C arising in the time step. Thus it is independent of the problem
C Jacobian structure and the type of nonlinear system solution method.
C SVSTEP returns a completion flag KFLAG (in COMMON).
C A return with KFLAG = -1 or -2 means either ABS(H) = HMIN or 10
C consecutive failures occurred. On a return with KFLAG negative,
C the values of TN and the YH array are as of the beginning of the last
C step, and H is the last step size attempted.
C
C Communication with SVSTEP is done with the following variables:
C
C Y = An array of length N used for the dependent variable vector.
C YH = An LDYH 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 LDYH = A constant integer .ge. N, the first dimension of YH.
C N is the number of ODEs in the system.
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 VSAV = A work array of length N passed to subroutine VNLS.
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 VNLS.
C F = Dummy name for the user supplied subroutine for f.
C JAC = Dummy name for the user supplied Jacobian subroutine.
C PSOL = Dummy name for the subroutine passed to VNLS, for
C possible use there.
C VNLS = Dummy name for the nonlinear system solving subroutine,
C whose real name is dependent on the method used.
C RPAR, IPAR = Dummy names for user's real and integer work arrays.
C-----------------------------------------------------------------------
C
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL ACNRM, CCMXJ, CONP, CRATE, DRC, EL,
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU, TQ, TN, UROUND
INTEGER ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
1 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
2 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
3 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
4 NSLP, NYH
C
C Type declarations for labeled COMMON block SVOD02 --------------------
C
REAL HU
INTEGER NCFN, NETF, NFE, NJE, NLU, NNI, NQU, NST
C
C Type declarations for local variables --------------------------------
C
REAL ADDON, BIAS1,BIAS2,BIAS3, CNQUOT, DDN, DSM, DUP,
1 ETACF, ETAMIN, ETAMX1, ETAMX2, ETAMX3, ETAMXF,
2 ETAQ, ETAQM1, ETAQP1, FLOTL, ONE, ONEPSM,
3 R, THRESH, TOLD, ZERO
INTEGER I, I1, I2, IBACK, J, JB, KFC, KFH, MXNCF, NCF, NFLAG
C
C Type declaration for function subroutines called ---------------------
C
REAL SVNORM
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 ADDON, BIAS1, BIAS2, BIAS3,
1 ETACF, ETAMIN, ETAMX1, ETAMX2, ETAMX3, ETAMXF, ETAQ, ETAQM1,
2 KFC, KFH, MXNCF, ONEPSM, THRESH, ONE, ZERO
C-----------------------------------------------------------------------
COMMON /SVOD01/ ACNRM, CCMXJ, CONP, CRATE, DRC, EL(13),
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU(13), TQ(5), TN, UROUND,
3 ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
4 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
5 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
6 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
7 NSLP, NYH
COMMON /SVOD02/ HU, NCFN, NETF, NFE, NJE, NLU, NNI, NQU, NST
C
DATA KFC/-3/, KFH/-7/, MXNCF/10/
DATA ADDON /1.0E-6/, BIAS1 /6.0E0/, BIAS2 /6.0E0/,
1 BIAS3 /10.0E0/, ETACF /0.25E0/, ETAMIN /0.1E0/,
2 ETAMXF /0.2E0/, ETAMX1 /1.0E4/, ETAMX2 /10.0E0/,
3 ETAMX3 /10.0E0/, ONEPSM /1.00001E0/, THRESH /1.5E0/
DATA ONE/1.0E0/, ZERO/0.0E0/
C
KFLAG = 0
TOLD = TN
NCF = 0
JCUR = 0
NFLAG = 0
IF (JSTART .GT. 0) GO TO 20
IF (JSTART .EQ. -1) GO TO 100
C-----------------------------------------------------------------------
C On the first call, the order is set to 1, and other variables are
C initialized. ETAMAX is the maximum ratio by which H can be increased
C in a single step. It is normally 10, but is larger during the
C first step to compensate for the small initial H. If a failure
C occurs (in corrector convergence or error test), ETAMAX is set to 1
C for the next increase.
C-----------------------------------------------------------------------
LMAX = MAXORD + 1
NQ = 1
L = 2
NQNYH = NQ*LDYH
TAU(1) = H
PRL1 = ONE
RC = ZERO
ETAMAX = ETAMX1
NQWAIT = 2
HSCAL = H
GO TO 200
C-----------------------------------------------------------------------
C Take preliminary actions on a normal continuation step (JSTART.GT.0).
C If the driver changed H, then ETA must be reset and NEWH set to 1.
C If a change of order was dictated on the previous step, then
C it is done here and appropriate adjustments in the history are made.
C On an order decrease, the history array is adjusted by SVJUST.
C On an order increase, the history array is augmented by a column.
C On a change of step size H, the history array YH is rescaled.
C-----------------------------------------------------------------------
20 CONTINUE
IF (KUTH .EQ. 1) THEN
ETA = MIN(ETA,H/HSCAL)
NEWH = 1
ENDIF
50 IF (NEWH .EQ. 0) GO TO 200
IF (NEWQ .EQ. NQ) GO TO 150
IF (NEWQ .LT. NQ) THEN
CALL SVJUST (YH, LDYH, -1)
NQ = NEWQ
L = NQ + 1
NQWAIT = L
GO TO 150
ENDIF
IF (NEWQ .GT. NQ) THEN
CALL SVJUST (YH, LDYH, 1)
NQ = NEWQ
L = NQ + 1
NQWAIT = L
GO TO 150
ENDIF
C-----------------------------------------------------------------------
C The following block handles preliminaries needed when JSTART = -1.
C If N was reduced, zero out part of YH to avoid undefined references.
C If MAXORD was reduced to a value less than the tentative order NEWQ,
C then NQ is set to MAXORD, and a new H ratio ETA is chosen.
C Otherwise, we take the same preliminary actions as for JSTART .gt. 0.
C In any case, NQWAIT is reset to L = NQ + 1 to prevent further
C changes in order for that many steps.
C The new H ratio ETA is limited by the input H if KUTH = 1,
C by HMIN if KUTH = 0, and by HMXI in any case.
C Finally, the history array YH is rescaled.
C-----------------------------------------------------------------------
100 CONTINUE
LMAX = MAXORD + 1
IF (N .EQ. LDYH) GO TO 120
I1 = 1 + (NEWQ + 1)*LDYH
I2 = (MAXORD + 1)*LDYH
IF (I1 .GT. I2) GO TO 120
DO 110 I = I1, I2
110 YH1(I) = ZERO
120 IF (NEWQ .LE. MAXORD) GO TO 140
FLOTL = REAL(LMAX)
IF (MAXORD .LT. NQ-1) THEN
DDN = SVNORM (N, SAVF, EWT)/TQ(1)
ETA = ONE/((BIAS1*DDN)**(ONE/FLOTL) + ADDON)
ENDIF
IF (MAXORD .EQ. NQ .AND. NEWQ .EQ. NQ+1) ETA = ETAQ
IF (MAXORD .EQ. NQ-1 .AND. NEWQ .EQ. NQ+1) THEN
ETA = ETAQM1
CALL SVJUST (YH, LDYH, -1)
ENDIF
IF (MAXORD .EQ. NQ-1 .AND. NEWQ .EQ. NQ) THEN
DDN = SVNORM (N, SAVF, EWT)/TQ(1)
ETA = ONE/((BIAS1*DDN)**(ONE/FLOTL) + ADDON)
CALL SVJUST (YH, LDYH, -1)
ENDIF
ETA = MIN(ETA,ONE)
NQ = MAXORD
L = LMAX
140 IF (KUTH .EQ. 1) ETA = MIN(ETA,ABS(H/HSCAL))
IF (KUTH .EQ. 0) ETA = MAX(ETA,HMIN/ABS(HSCAL))
ETA = ETA/MAX(ONE,ABS(HSCAL)*HMXI*ETA)
NEWH = 1
NQWAIT = L
IF (NEWQ .LE. MAXORD) GO TO 50
C Rescale the history array for a change in H by a factor of ETA. ------
150 R = ONE
DO 180 J = 2, L
R = R*ETA
CALL SSCAL (N, R, YH(1,J), 1 )
180 CONTINUE
H = HSCAL*ETA
HSCAL = H
RC = RC*ETA
NQNYH = NQ*LDYH
C-----------------------------------------------------------------------
C This section computes the predicted values by effectively
C multiplying the YH array by the Pascal triangle matrix.
C SVSET is called to calculate all integration coefficients.
C RC is the ratio of new to old values of the coefficient H/EL(2)=h/l1.
C-----------------------------------------------------------------------
200 TN = TN + H
I1 = NQNYH + 1
DO 220 JB = 1, NQ
I1 = I1 - LDYH
DO 210 I = I1, NQNYH
210 YH1(I) = YH1(I) + YH1(I+LDYH)
220 CONTINUE
CALL SVSET
RL1 = ONE/EL(2)
RC = RC*(RL1/PRL1)
PRL1 = RL1
C
C Call the nonlinear system solver. ------------------------------------
C
CALL VNLS (Y, YH, LDYH, VSAV, SAVF, EWT, ACOR, IWM, WM,
1 F, JAC, PSOL, NFLAG, RPAR, IPAR)
C
IF (NFLAG .EQ. 0) GO TO 450
C-----------------------------------------------------------------------
C The VNLS routine failed to achieve convergence (NFLAG .NE. 0).
C The YH array is retracted to its values before prediction.
C The step size H is reduced and the step is retried, if possible.
C Otherwise, an error exit is taken.
C-----------------------------------------------------------------------
NCF = NCF + 1
NCFN = NCFN + 1
ETAMAX = ONE
TN = TOLD
I1 = NQNYH + 1
DO 430 JB = 1, NQ
I1 = I1 - LDYH
DO 420 I = I1, NQNYH
420 YH1(I) = YH1(I) - YH1(I+LDYH)
430 CONTINUE
IF (NFLAG .LT. -1) GO TO 680
IF (ABS(H) .LE. HMIN*ONEPSM) GO TO 670
IF (NCF .EQ. MXNCF) GO TO 670
ETA = ETACF
ETA = MAX(ETA,HMIN/ABS(H))
NFLAG = -1
GO TO 150
C-----------------------------------------------------------------------
C The corrector has converged (NFLAG = 0). The local error test is
C made and control passes to statement 500 if it fails.
C-----------------------------------------------------------------------
450 CONTINUE
DSM = ACNRM/TQ(2)
IF (DSM .GT. ONE) GO TO 500
C-----------------------------------------------------------------------
C After a successful step, update the YH and TAU arrays and decrement
C NQWAIT. If NQWAIT is then 1 and NQ .lt. MAXORD, then ACOR is saved
C for use in a possible order increase on the next step.
C If ETAMAX = 1 (a failure occurred this step), keep NQWAIT .ge. 2.
C-----------------------------------------------------------------------
KFLAG = 0
NST = NST + 1
HU = H
NQU = NQ
DO 470 IBACK = 1, NQ
I = L - IBACK
470 TAU(I+1) = TAU(I)
TAU(1) = H
DO 480 J = 1, L
CALL SAXPY (N, EL(J), ACOR, 1, YH(1,J), 1 )
480 CONTINUE
NQWAIT = NQWAIT - 1
IF ((L .EQ. LMAX) .OR. (NQWAIT .NE. 1)) GO TO 490
CALL SCOPY (N, ACOR, 1, YH(1,LMAX), 1 )
CONP = TQ(5)
490 IF (ETAMAX .NE. ONE) GO TO 560
IF (NQWAIT .LT. 2) NQWAIT = 2
NEWQ = NQ
NEWH = 0
ETA = ONE
HNEW = H
GO TO 690
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 the
C same order. After repeated failures, H is forced to decrease
C more rapidly.
C-----------------------------------------------------------------------
500 KFLAG = KFLAG - 1
NETF = NETF + 1
NFLAG = -2
TN = TOLD
I1 = NQNYH + 1
DO 520 JB = 1, NQ
I1 = I1 - LDYH
DO 510 I = I1, NQNYH
510 YH1(I) = YH1(I) - YH1(I+LDYH)
520 CONTINUE
IF (ABS(H) .LE. HMIN*ONEPSM) GO TO 660
ETAMAX = ONE
IF (KFLAG .LE. KFC) GO TO 530
C Compute ratio of new H to current H at the current order. ------------
FLOTL = REAL(L)
ETA = ONE/((BIAS2*DSM)**(ONE/FLOTL) + ADDON)
ETA = MAX(ETA,HMIN/ABS(H),ETAMIN)
IF ((KFLAG .LE. -2) .AND. (ETA .GT. ETAMXF)) ETA = ETAMXF
GO TO 150
C-----------------------------------------------------------------------
C Control reaches this section if 3 or more consecutive failures
C have occurred. It is assumed that the elements of the YH array
C have accumulated errors of the wrong order. The order is reduced
C by one, if possible. Then H is reduced by a factor of 0.1 and
C the step is retried. After a total of 7 consecutive failures,
C an exit is taken with KFLAG = -1.
C-----------------------------------------------------------------------
530 IF (KFLAG .EQ. KFH) GO TO 660
IF (NQ .EQ. 1) GO TO 540
ETA = MAX(ETAMIN,HMIN/ABS(H))
CALL SVJUST (YH, LDYH, -1)
L = NQ
NQ = NQ - 1
NQWAIT = L
GO TO 150
540 ETA = MAX(ETAMIN,HMIN/ABS(H))
H = H*ETA
HSCAL = H
TAU(1) = H
CALL F (N, TN, Y, SAVF, RPAR, IPAR)
NFE = NFE + 1
DO 550 I = 1, N
550 YH(I,2) = H*SAVF(I)
NQWAIT = 10
GO TO 200
C-----------------------------------------------------------------------
C If NQWAIT = 0, an increase or decrease in order by one is considered.
C Factors ETAQ, ETAQM1, ETAQP1 are computed by which H could
C be multiplied at order q, q-1, or q+1, respectively.
C The largest of these is determined, and the new order and
C step size set accordingly.
C A change of H or NQ is made only if H increases by at least a
C factor of THRESH. If an order change is considered and rejected,
C then NQWAIT is set to 2 (reconsider it after 2 steps).
C-----------------------------------------------------------------------
C Compute ratio of new H to current H at the current order. ------------
560 FLOTL = REAL(L)
ETAQ = ONE/((BIAS2*DSM)**(ONE/FLOTL) + ADDON)
IF (NQWAIT .NE. 0) GO TO 600
NQWAIT = 2
ETAQM1 = ZERO
IF (NQ .EQ. 1) GO TO 570
C Compute ratio of new H to current H at the current order less one. ---
DDN = SVNORM (N, YH(1,L), EWT)/TQ(1)
ETAQM1 = ONE/((BIAS1*DDN)**(ONE/(FLOTL - ONE)) + ADDON)
570 ETAQP1 = ZERO
IF (L .EQ. LMAX) GO TO 580
C Compute ratio of new H to current H at current order plus one. -------
CNQUOT = (TQ(5)/CONP)*(H/TAU(2))**L
DO 575 I = 1, N
575 SAVF(I) = ACOR(I) - CNQUOT*YH(I,LMAX)
DUP = SVNORM (N, SAVF, EWT)/TQ(3)
ETAQP1 = ONE/((BIAS3*DUP)**(ONE/(FLOTL + ONE)) + ADDON)
580 IF (ETAQ .GE. ETAQP1) GO TO 590
IF (ETAQP1 .GT. ETAQM1) GO TO 620
GO TO 610
590 IF (ETAQ .LT. ETAQM1) GO TO 610
600 ETA = ETAQ
NEWQ = NQ
GO TO 630
610 ETA = ETAQM1
NEWQ = NQ - 1
GO TO 630
620 ETA = ETAQP1
NEWQ = NQ + 1
CALL SCOPY (N, ACOR, 1, YH(1,LMAX), 1)
C Test tentative new H against THRESH, ETAMAX, and HMXI, then exit. ----
630 IF (ETA .LT. THRESH .OR. ETAMAX .EQ. ONE) GO TO 640
ETA = MIN(ETA,ETAMAX)
ETA = ETA/MAX(ONE,ABS(H)*HMXI*ETA)
NEWH = 1
HNEW = H*ETA
GO TO 690
640 NEWQ = NQ
NEWH = 0
ETA = ONE
HNEW = H
GO TO 690
C-----------------------------------------------------------------------
C All returns are made through this section.
C On a successful return, ETAMAX is reset and ACOR is scaled.
C-----------------------------------------------------------------------
660 KFLAG = -1
GO TO 720
670 KFLAG = -2
GO TO 720
680 IF (NFLAG .EQ. -2) KFLAG = -3
IF (NFLAG .EQ. -3) KFLAG = -4
GO TO 720
690 ETAMAX = ETAMX3
IF (NST .LE. 10) ETAMAX = ETAMX2
700 R = ONE/TQ(2)
CALL SSCAL (N, R, ACOR, 1)
720 JSTART = 1
RETURN
C----------------------- End of Subroutine SVSTEP ----------------------
END
*DECK SVSET
SUBROUTINE SVSET
C-----------------------------------------------------------------------
C Call sequence communication: None
C COMMON block variables accessed:
C /SVOD01/ -- EL(13), H, TAU(13), TQ(5), L(= NQ + 1),
C METH, NQ, NQWAIT
C
C Subroutines called by SVSET: None
C Function routines called by SVSET: None
C-----------------------------------------------------------------------
C SVSET is called by SVSTEP and sets coefficients for use there.
C
C For each order NQ, the coefficients in EL are calculated by use of
C the generating polynomial lambda(x), with coefficients EL(i).
C lambda(x) = EL(1) + EL(2)*x + ... + EL(NQ+1)*(x**NQ).
C For the backward differentiation formulas,
C NQ-1
C lambda(x) = (1 + x/xi*(NQ)) * product (1 + x/xi(i) ) .
C i = 1
C For the Adams formulas,
C NQ-1
C (d/dx) lambda(x) = c * product (1 + x/xi(i) ) ,
C i = 1
C lambda(-1) = 0, lambda(0) = 1,
C where c is a normalization constant.
C In both cases, xi(i) is defined by
C H*xi(i) = t sub n - t sub (n-i)
C = H + TAU(1) + TAU(2) + ... TAU(i-1).
C
C
C In addition to variables described previously, communication
C with SVSET uses the following:
C TAU = A vector of length 13 containing the past NQ values
C of H.
C EL = A vector of length 13 in which vset stores the
C coefficients for the corrector formula.
C TQ = A vector of length 5 in which vset stores constants
C used for the convergence test, the error test, and the
C selection of H at a new order.
C METH = The basic method indicator.
C NQ = The current order.
C L = NQ + 1, the length of the vector stored in EL, and
C the number of columns of the YH array being used.
C NQWAIT = A counter controlling the frequency of order changes.
C An order change is about to be considered if NQWAIT = 1.
C-----------------------------------------------------------------------
C
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL ACNRM, CCMXJ, CONP, CRATE, DRC, EL,
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU, TQ, TN, UROUND
INTEGER ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
1 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
2 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
3 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
4 NSLP, NYH
C
C Type declarations for local variables --------------------------------
C
REAL AHATN0, ALPH0, CNQM1, CORTES, CSUM, ELP, EM,
1 EM0, FLOTI, FLOTL, FLOTNQ, HSUM, ONE, RXI, RXIS, S, SIX,
2 T1, T2, T3, T4, T5, T6, TWO, XI, ZERO
INTEGER I, IBACK, J, JP1, NQM1, NQM2
C
DIMENSION EM(13)
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 CORTES, ONE, SIX, TWO, ZERO
C
COMMON /SVOD01/ ACNRM, CCMXJ, CONP, CRATE, DRC, EL(13),
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU(13), TQ(5), TN, UROUND,
3 ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
4 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
5 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
6 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
7 NSLP, NYH
C
DATA CORTES /0.1E0/
DATA ONE /1.0E0/, SIX /6.0E0/, TWO /2.0E0/, ZERO /0.0E0/
C
FLOTL = REAL(L)
NQM1 = NQ - 1
NQM2 = NQ - 2
GO TO (100, 200), METH
C
C Set coefficients for Adams methods. ----------------------------------
100 IF (NQ .NE. 1) GO TO 110
EL(1) = ONE
EL(2) = ONE
TQ(1) = ONE
TQ(2) = TWO
TQ(3) = SIX*TQ(2)
TQ(5) = ONE
GO TO 300
110 HSUM = H
EM(1) = ONE
FLOTNQ = FLOTL - ONE
DO 115 I = 2, L
115 EM(I) = ZERO
DO 150 J = 1, NQM1
IF ((J .NE. NQM1) .OR. (NQWAIT .NE. 1)) GO TO 130
S = ONE
CSUM = ZERO
DO 120 I = 1, NQM1
CSUM = CSUM + S*EM(I)/REAL(I+1)
120 S = -S
TQ(1) = EM(NQM1)/(FLOTNQ*CSUM)
130 RXI = H/HSUM
DO 140 IBACK = 1, J
I = (J + 2) - IBACK
140 EM(I) = EM(I) + EM(I-1)*RXI
HSUM = HSUM + TAU(J)
150 CONTINUE
C Compute integral from -1 to 0 of polynomial and of x times it. -------
S = ONE
EM0 = ZERO
CSUM = ZERO
DO 160 I = 1, NQ
FLOTI = REAL(I)
EM0 = EM0 + S*EM(I)/FLOTI
CSUM = CSUM + S*EM(I)/(FLOTI+ONE)
160 S = -S
C In EL, form coefficients of normalized integrated polynomial. --------
S = ONE/EM0
EL(1) = ONE
DO 170 I = 1, NQ
170 EL(I+1) = S*EM(I)/REAL(I)
XI = HSUM/H
TQ(2) = XI*EM0/CSUM
TQ(5) = XI/EL(L)
IF (NQWAIT .NE. 1) GO TO 300
C For higher order control constant, multiply polynomial by 1+x/xi(q). -
RXI = ONE/XI
DO 180 IBACK = 1, NQ
I = (L + 1) - IBACK
180 EM(I) = EM(I) + EM(I-1)*RXI
C Compute integral of polynomial. --------------------------------------
S = ONE
CSUM = ZERO
DO 190 I = 1, L
CSUM = CSUM + S*EM(I)/REAL(I+1)
190 S = -S
TQ(3) = FLOTL*EM0/CSUM
GO TO 300
C
C Set coefficients for BDF methods. ------------------------------------
200 DO 210 I = 3, L
210 EL(I) = ZERO
EL(1) = ONE
EL(2) = ONE
ALPH0 = -ONE
AHATN0 = -ONE
HSUM = H
RXI = ONE
RXIS = ONE
IF (NQ .EQ. 1) GO TO 240
DO 230 J = 1, NQM2
C In EL, construct coefficients of (1+x/xi(1))*...*(1+x/xi(j+1)). ------
HSUM = HSUM + TAU(J)
RXI = H/HSUM
JP1 = J + 1
ALPH0 = ALPH0 - ONE/REAL(JP1)
DO 220 IBACK = 1, JP1
I = (J + 3) - IBACK
220 EL(I) = EL(I) + EL(I-1)*RXI
230 CONTINUE
ALPH0 = ALPH0 - ONE/REAL(NQ)
RXIS = -EL(2) - ALPH0
HSUM = HSUM + TAU(NQM1)
RXI = H/HSUM
AHATN0 = -EL(2) - RXI
DO 235 IBACK = 1, NQ
I = (NQ + 2) - IBACK
235 EL(I) = EL(I) + EL(I-1)*RXIS
240 T1 = ONE - AHATN0 + ALPH0
T2 = ONE + REAL(NQ)*T1
TQ(2) = ABS(ALPH0*T2/T1)
TQ(5) = ABS(T2/(EL(L)*RXI/RXIS))
IF (NQWAIT .NE. 1) GO TO 300
CNQM1 = RXIS/EL(L)
T3 = ALPH0 + ONE/REAL(NQ)
T4 = AHATN0 + RXI
ELP = T3/(ONE - T4 + T3)
TQ(1) = ABS(ELP/CNQM1)
HSUM = HSUM + TAU(NQ)
RXI = H/HSUM
T5 = ALPH0 - ONE/REAL(NQ+1)
T6 = AHATN0 - RXI
ELP = T2/(ONE - T6 + T5)
TQ(3) = ABS(ELP*RXI*(FLOTL + ONE)*T5)
300 TQ(4) = CORTES*TQ(2)
RETURN
C----------------------- End of Subroutine SVSET -----------------------
END
*DECK SVJUST
SUBROUTINE SVJUST (YH, LDYH, IORD)
REAL YH
INTEGER LDYH, IORD
DIMENSION YH(LDYH,*)
C-----------------------------------------------------------------------
C Call sequence input -- YH, LDYH, IORD
C Call sequence output -- YH
C COMMON block input -- NQ, METH, LMAX, HSCAL, TAU(13), N
C COMMON block variables accessed:
C /SVOD01/ -- HSCAL, TAU(13), LMAX, METH, N, NQ,
C
C Subroutines called by SVJUST: SAXPY
C Function routines called by SVJUST: None
C-----------------------------------------------------------------------
C This subroutine adjusts the YH array on reduction of order,
C and also when the order is increased for the stiff option (METH = 2).
C Communication with SVJUST uses the following:
C IORD = An integer flag used when METH = 2 to indicate an order
C increase (IORD = +1) or an order decrease (IORD = -1).
C HSCAL = Step size H used in scaling of Nordsieck array YH.
C (If IORD = +1, SVJUST assumes that HSCAL = TAU(1).)
C See References 1 and 2 for details.
C-----------------------------------------------------------------------
C
C Type declarations for labeled COMMON block SVOD01 --------------------
C
REAL ACNRM, CCMXJ, CONP, CRATE, DRC, EL,
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU, TQ, TN, UROUND
INTEGER ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
1 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
2 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
3 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
4 NSLP, NYH
C
C Type declarations for local variables --------------------------------
C
REAL ALPH0, ALPH1, HSUM, ONE, PROD, T1, XI,XIOLD, ZERO
INTEGER I, IBACK, J, JP1, LP1, NQM1, NQM2, NQP1
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 ONE, ZERO
C
COMMON /SVOD01/ ACNRM, CCMXJ, CONP, CRATE, DRC, EL(13),
1 ETA, ETAMAX, H, HMIN, HMXI, HNEW, HSCAL, PRL1,
2 RC, RL1, TAU(13), TQ(5), TN, UROUND,
3 ICF, INIT, IPUP, JCUR, JSTART, JSV, KFLAG, KUTH,
4 L, LMAX, LYH, LEWT, LACOR, LSAVF, LWM, LIWM,
5 LOCJS, MAXORD, METH, MITER, MSBJ, MXHNIL, MXSTEP,
6 N, NEWH, NEWQ, NHNIL, NQ, NQNYH, NQWAIT, NSLJ,
7 NSLP, NYH
C
DATA ONE /1.0E0/, ZERO /0.0E0/
C
IF ((NQ .EQ. 2) .AND. (IORD .NE. 1)) RETURN
NQM1 = NQ - 1
NQM2 = NQ - 2
GO TO (100, 200), METH
C-----------------------------------------------------------------------
C Nonstiff option...
C Check to see if the order is being increased or decreased.
C-----------------------------------------------------------------------
100 CONTINUE
IF (IORD .EQ. 1) GO TO 180
C Order decrease. ------------------------------------------------------
DO 110 J = 1, LMAX
110 EL(J) = ZERO
EL(2) = ONE
HSUM = ZERO
DO 130 J = 1, NQM2
C Construct coefficients of x*(x+xi(1))*...*(x+xi(j)). -----------------
HSUM = HSUM + TAU(J)
XI = HSUM/HSCAL
JP1 = J + 1
DO 120 IBACK = 1, JP1
I = (J + 3) - IBACK
120 EL(I) = EL(I)*XI + EL(I-1)
130 CONTINUE
C Construct coefficients of integrated polynomial. ---------------------
DO 140 J = 2, NQM1
140 EL(J+1) = REAL(NQ)*EL(J)/REAL(J)
C Subtract correction terms from YH array. -----------------------------
DO 170 J = 3, NQ
DO 160 I = 1, N
160 YH(I,J) = YH(I,J) - YH(I,L)*EL(J)
170 CONTINUE
RETURN
C Order increase. ------------------------------------------------------
C Zero out next column in YH array. ------------------------------------
180 CONTINUE
LP1 = L + 1
DO 190 I = 1, N
190 YH(I,LP1) = ZERO
RETURN
C-----------------------------------------------------------------------
C Stiff option...
C Check to see if the order is being increased or decreased.
C-----------------------------------------------------------------------
200 CONTINUE
IF (IORD .EQ. 1) GO TO 300
C Order decrease. ------------------------------------------------------
DO 210 J = 1, LMAX
210 EL(J) = ZERO
EL(3) = ONE
HSUM = ZERO
DO 230 J = 1,NQM2
C Construct coefficients of x*x*(x+xi(1))*...*(x+xi(j)). ---------------
HSUM = HSUM + TAU(J)
XI = HSUM/HSCAL
JP1 = J + 1
DO 220 IBACK = 1, JP1
I = (J + 4) - IBACK
220 EL(I) = EL(I)*XI + EL(I-1)
230 CONTINUE
C Subtract correction terms from YH array. -----------------------------
DO 250 J = 3,NQ
DO 240 I = 1, N
240 YH(I,J) = YH(I,J) - YH(I,L)*EL(J)
250 CONTINUE
RETURN
C Order increase. ------------------------------------------------------
300 DO 310 J = 1, LMAX
310 EL(J) = ZERO
EL(3) = ONE
ALPH0 = -ONE
ALPH1 = ONE
PROD = ONE
XIOLD = ONE
HSUM = HSCAL
IF (NQ .EQ. 1) GO TO 340
DO 330 J = 1, NQM1
C Construct coefficients of x*x*(x+xi(1))*...*(x+xi(j)). ---------------
JP1 = J + 1
HSUM = HSUM + TAU(JP1)
XI = HSUM/HSCAL
PROD = PROD*XI
ALPH0 = ALPH0 - ONE/REAL(JP1)
ALPH1 = ALPH1 + ONE/XI
DO 320 IBACK = 1, JP1
I = (J + 4) - IBACK
320 EL(I) = EL(I)*XIOLD + EL(I-1)
XIOLD = XI
330 CONTINUE
340 CONTINUE
T1 = (-ALPH0 - ALPH1)/PROD
C Load column L + 1 in YH array. ---------------------------------------
LP1 = L + 1
DO 350 I = 1, N
350 YH(I,LP1) = T1*YH(I,LMAX)
C Add correction terms to YH array. ------------------------------------
NQP1 = NQ + 1
DO 370 J = 3, NQP1
CALL SAXPY (N, EL(J), YH(1,LP1), 1, YH(1,J), 1 )
370 CONTINUE
RETURN
C----------------------- End of Subroutine SVJUST ----------------------
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 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 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 XERRWV
SUBROUTINE XERRWV (MSG, NMES, NERR, LEVEL, NI, I1, I2, NR, R1, R2)
C***BEGIN PROLOGUE XERRWV
C***SUBSIDIARY
C***PURPOSE Write error message with values.
C***CATEGORY R3C
C***TYPE SINGLE PRECISION (XERRWV-S, XERRWD-D)
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C
C Subroutines XERRWV, XSETF, XSETUN, and the function routine IXSAV,
C as given here, constitute a simplified version of the SLATEC error
C handling package.
C
C All arguments are input arguments.
C
C MSG = The message (character array).
C NMES = The length of MSG (number of characters).
C NERR = The error number (not used).
C LEVEL = The error level..
C 0 or 1 means recoverable (control returns to caller).
C 2 means fatal (run is aborted--see note below).
C NI = Number of integers (0, 1, or 2) to be printed with message.
C I1,I2 = Integers to be printed, depending on NI.
C NR = Number of reals (0, 1, or 2) to be printed with message.
C R1,R2 = Reals to be printed, depending on NR.
C
C Note.. this routine is machine-dependent and specialized for use
C in limited context, in the following ways..
C 1. The argument MSG is assumed to be of type CHARACTER, and
C the message is printed with a format of (1X,A).
C 2. The message is assumed to take only one line.
C Multi-line messages are generated by repeated calls.
C 3. If LEVEL = 2, control passes to the statement STOP
C to abort the run. This statement may be machine-dependent.
C 4. R1 and R2 are assumed to be in single precision and are printed
C in E21.13 format.
C
C***ROUTINES CALLED IXSAV
C***REVISION HISTORY (YYMMDD)
C 791129 DATE WRITTEN
C 890413 Heavily revised, with Common eliminated. (ACH, PNB)
C 921118 Replaced MFLGSV/LUNSAV by IXSAV. (ACH)
C 930329 Modified prologue to SLATEC format. (FNF)
C 930407 Changed MSG from CHARACTER*1 array to variable. (FNF)
C 930922 Minor cosmetic change. (FNF)
C***END PROLOGUE XERRWV
C
C*Internal Notes:
C
C For a different default logical unit number, IXSAV (or a subsidiary
C routine that it calls) will need to be modified.
C For a different run-abort command, change the statement following
C statement 100 at the end.
C-----------------------------------------------------------------------
C Subroutines called by XERRWV.. None
C Function routine called by XERRWV.. IXSAV
C-----------------------------------------------------------------------
C**End
C
C Declare arguments.
C
REAL R1, R2
INTEGER NMES, NERR, LEVEL, NI, I1, I2, NR
CHARACTER*(*) MSG
C
C Declare local variables.
C
INTEGER LUNIT, IXSAV, MESFLG
C
C Get logical unit number and message print flag.
C
C***FIRST EXECUTABLE STATEMENT XERRWV
LUNIT = IXSAV (1, 0, .FALSE.)
MESFLG = IXSAV (2, 0, .FALSE.)
IF (MESFLG .EQ. 0) GO TO 100
C
C Write the message.
C
WRITE (LUNIT,10) MSG
10 FORMAT(1X,A)
IF (NI .EQ. 1) WRITE (LUNIT, 20) I1
20 FORMAT(6X,'In above message, I1 =',I10)
IF (NI .EQ. 2) WRITE (LUNIT, 30) I1,I2
30 FORMAT(6X,'In above message, I1 =',I10,3X,'I2 =',I10)
IF (NR .EQ. 1) WRITE (LUNIT, 40) R1
40 FORMAT(6X,'In above message, R1 =',E21.13)
IF (NR .EQ. 2) WRITE (LUNIT, 50) R1,R2
50 FORMAT(6X,'In above, R1 =',E21.13,3X,'R2 =',E21.13)
C
C Abort the run if LEVEL = 2.
C
100 IF (LEVEL .NE. 2) RETURN
STOP
C----------------------- End of Subroutine XERRWV ----------------------
END
*DECK XSETF
SUBROUTINE XSETF (MFLAG)
C***BEGIN PROLOGUE XSETF
C***PURPOSE Reset the error print control flag.
C***CATEGORY R3A
C***TYPE ALL (XSETF-A)
C***KEYWORDS ERROR CONTROL
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C
C XSETF sets the error print control flag to MFLAG:
C MFLAG=1 means print all messages (the default).
C MFLAG=0 means no printing.
C
C***SEE ALSO XERRWD, XERRWV
C***REFERENCES (NONE)
C***ROUTINES CALLED IXSAV
C***REVISION HISTORY (YYMMDD)
C 921118 DATE WRITTEN
C 930329 Added SLATEC format prologue. (FNF)
C 930407 Corrected SEE ALSO section. (FNF)
C 930922 Made user-callable, and other cosmetic changes. (FNF)
C***END PROLOGUE XSETF
C
C Subroutines called by XSETF.. None
C Function routine called by XSETF.. IXSAV
C-----------------------------------------------------------------------
C**End
INTEGER MFLAG, JUNK, IXSAV
C
C***FIRST EXECUTABLE STATEMENT XSETF
IF (MFLAG .EQ. 0 .OR. MFLAG .EQ. 1) JUNK = IXSAV (2,MFLAG,.TRUE.)
RETURN
C----------------------- End of Subroutine XSETF -----------------------
END
*DECK XSETUN
SUBROUTINE XSETUN (LUN)
C***BEGIN PROLOGUE XSETUN
C***PURPOSE Reset the logical unit number for error messages.
C***CATEGORY R3B
C***TYPE ALL (XSETUN-A)
C***KEYWORDS ERROR CONTROL
C***DESCRIPTION
C
C XSETUN sets the logical unit number for error messages to LUN.
C
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***SEE ALSO XERRWD, XERRWV
C***REFERENCES (NONE)
C***ROUTINES CALLED IXSAV
C***REVISION HISTORY (YYMMDD)
C 921118 DATE WRITTEN
C 930329 Added SLATEC format prologue. (FNF)
C 930407 Corrected SEE ALSO section. (FNF)
C 930922 Made user-callable, and other cosmetic changes. (FNF)
C***END PROLOGUE XSETUN
C
C Subroutines called by XSETUN.. None
C Function routine called by XSETUN.. IXSAV
C-----------------------------------------------------------------------
C**End
INTEGER LUN, JUNK, IXSAV
C
C***FIRST EXECUTABLE STATEMENT XSETUN
IF (LUN .GT. 0) JUNK = IXSAV (1,LUN,.TRUE.)
RETURN
C----------------------- End of Subroutine XSETUN ----------------------
END
*DECK IXSAV
INTEGER FUNCTION IXSAV (IPAR, IVALUE, ISET)
C***BEGIN PROLOGUE IXSAV
C***SUBSIDIARY
C***PURPOSE Save and recall error message control parameters.
C***CATEGORY R3C
C***TYPE ALL (IXSAV-A)
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C
C IXSAV saves and recalls one of two error message parameters:
C LUNIT, the logical unit number to which messages are printed, and
C MESFLG, the message print flag.
C This is a modification of the SLATEC library routine J4SAVE.
C
C Saved local variables..
C LUNIT = Logical unit number for messages. The default is obtained
C by a call to IUMACH (may be machine-dependent).
C MESFLG = Print control flag..
C 1 means print all messages (the default).
C 0 means no printing.
C
C On input..
C IPAR = Parameter indicator (1 for LUNIT, 2 for MESFLG).
C IVALUE = The value to be set for the parameter, if ISET = .TRUE.
C ISET = Logical flag to indicate whether to read or write.
C If ISET = .TRUE., the parameter will be given
C the value IVALUE. If ISET = .FALSE., the parameter
C will be unchanged, and IVALUE is a dummy argument.
C
C On return..
C IXSAV = The (old) value of the parameter.
C
C***SEE ALSO XERRWD, XERRWV
C***ROUTINES CALLED IUMACH
C***REVISION HISTORY (YYMMDD)
C 921118 DATE WRITTEN
C 930329 Modified prologue to SLATEC format. (FNF)
C 930915 Added IUMACH call to get default output unit. (ACH)
C 930922 Minor cosmetic changes. (FNF)
C 010425 Type declaration for IUMACH added. (ACH)
C***END PROLOGUE IXSAV
C
C Subroutines called by IXSAV.. None
C Function routine called by IXSAV.. IUMACH
C-----------------------------------------------------------------------
C**End
LOGICAL ISET
INTEGER IPAR, IVALUE
C-----------------------------------------------------------------------
INTEGER IUMACH, LUNIT, MESFLG
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 routine.
C-----------------------------------------------------------------------
SAVE LUNIT, MESFLG
DATA LUNIT/-1/, MESFLG/1/
C
C***FIRST EXECUTABLE STATEMENT IXSAV
IF (IPAR .EQ. 1) THEN
IF (LUNIT .EQ. -1) LUNIT = IUMACH()
IXSAV = LUNIT
IF (ISET) LUNIT = IVALUE
ENDIF
C
IF (IPAR .EQ. 2) THEN
IXSAV = MESFLG
IF (ISET) MESFLG = IVALUE
ENDIF
C
RETURN
C----------------------- End of Function IXSAV -------------------------
END
*DECK IUMACH
INTEGER FUNCTION IUMACH()
C***BEGIN PROLOGUE IUMACH
C***PURPOSE Provide standard output unit number.
C***CATEGORY R1
C***TYPE INTEGER (IUMACH-I)
C***KEYWORDS MACHINE CONSTANTS
C***AUTHOR Hindmarsh, Alan C., (LLNL)
C***DESCRIPTION
C *Usage:
C INTEGER LOUT, IUMACH
C LOUT = IUMACH()
C
C *Function Return Values:
C LOUT : the standard logical unit for Fortran output.
C
C***REFERENCES (NONE)
C***ROUTINES CALLED (NONE)
C***REVISION HISTORY (YYMMDD)
C 930915 DATE WRITTEN
C 930922 Made user-callable, and other cosmetic changes. (FNF)
C***END PROLOGUE IUMACH
C
C*Internal Notes:
C The built-in value of 6 is standard on a wide range of Fortran
C systems. This may be machine-dependent.
C**End
C***FIRST EXECUTABLE STATEMENT IUMACH
IUMACH = 6
C
RETURN
C----------------------- End of Function IUMACH ------------------------
END
*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 RUMSUM
C***REVISION HISTORY (YYYYMMDD)
C 19930216 DATE WRITTEN
C 19930818 Added SLATEC-format prologue. (FNF)
C 20030707 Added RUMSUM to force normal storage of COMP. (ACH)
C***END PROLOGUE RUMACH
C
REAL U, COMP
C***FIRST EXECUTABLE STATEMENT RUMACH
U = 1.0E0
10 U = U*0.5E0
CALL RUMSUM(1.0E0, U, COMP)
IF (COMP .NE. 1.0E0) GO TO 10
RUMACH = U*2.0E0
RETURN
C----------------------- End of Function RUMACH ------------------------
END
SUBROUTINE RUMSUM(A,B,C)
C Routine to force normal storing of A + B, for RUMACH.
REAL A, B, C
C = A + B
RETURN
END