dlsodpk
This is the 18 November 2003 version of
DLSODPK: Livermore Solver for Ordinary Differential equations,
with Preconditioned Krylov iteration methods for the
Newton correction linear systems.
This version is in double precision.
DLSODPK solves the initial value problem for stiff or nonstiff
systems of first order ODEs,
dy/dt = f(t,y) , or, in component form,
dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(NEQ)) (i = 1,...,NEQ).
Introduction
This is a modification of the DLSODE package which incorporates
various preconditioned Krylov subspace iteration methods for the
linear algebraic systems that arise in the case of stiff systems.
The linear systems that must be solved have the form
A * x = b , where A = identity - hl0 * (df/dy) .
Here hl0 is a scalar, and df/dy is the Jacobian matrix of partial
derivatives of f (NEQ by NEQ).
The particular Krylov method is chosen by setting the second digit,
MITER, in the method flag MF.
Currently, the values of MITER have the following meanings:
MITER = 1 means the preconditioned Scaled Incomplete
Orthogonalization Method (SPIOM).
2 means an incomplete version of the Preconditioned Scaled
Generalized Minimal Residual method (SPIGMR).
This is the best choice in general.
3 means the Preconditioned Conjugate Gradient method (PCG).
Recommended only when df/dy is symmetric or nearly so.
4 means the scaled Preconditioned Conjugate Gradient method
(PCGS). Recommended only when D-inverse * df/dy * D is
symmetric or nearly so, where D is the diagonal scaling
matrix with elements 1/EWT(i) (see RTOL/ATOL description).
9 means that only a user-supplied matrix P (approximating A)
will be used, with no Krylov iteration done. This option
allows the user to provide the complete linear system
solution algorithm, if desired.
The user can apply preconditioning to the linear system A*x = b,
by means of arbitrary matrices (the preconditioners).
In the case of SPIOM and SPIGMR, one can apply left and right
preconditioners P1 and P2, and the basic iterative method is then
applied to the matrix (P1-inverse)*A*(P2-inverse) instead of to the
matrix A. The product P1*P2 should be an approximation to matrix A
such that linear systems with P1 or P2 are easier to solve than with
A. Preconditioning from the left only or right only means using
P2 = identity or P1 = identity, respectively.
In the case of the PCG and PCGS methods, there is only one
preconditioner matrix P (but it can be the product of more than one).
It should approximate the matrix A but allow for relatively
easy solution of linear systems with coefficient matrix P.
For PCG, P should be positive definite symmetric, or nearly so,
and for PCGS, the scaled preconditioner D-inverse * P * D
should be symmetric or nearly so.
If the Jacobian J = df/dy splits in a natural way into a sum
J = J1 + J2, then one possible choice of preconditioners is
P1 = identity - hl0 * J1 and P2 = identity - hl0 * J2
provided each of these is easy to solve (or approximately solve).
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References:
1. Peter N. Brown and Alan C. Hindmarsh, Reduced Storage Matrix
Methods in Stiff ODE Systems, J. Appl. Math. & Comp., 31 (1989),
pp. 40-91; also L.L.N.L. Report UCRL-95088, Rev. 1, June 1987.
2. Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
North-Holland, Amsterdam, 1983, pp. 55-64.
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Authors: Alan C. Hindmarsh and Peter N. Brown
Center for Applied Scientific Computing, L-561
Lawrence Livermore National Laboratory
Livermore, CA 94551
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Summary of Usage
Communication between the user and the DLSODPK package, for normal
situations, is summarized here. This summary describes only a subset
of the full set of options available. See the full description for
details, including optional communication, nonstandard options,
and instructions for special situations. See also the demonstration
program distributed with this solver.
A. First provide a subroutine of the form:
SUBROUTINE F (NEQ, T, Y, YDOT)
DOUBLE PRECISION T, Y(*), YDOT(*)
which supplies the vector function f by loading YDOT(i) with f(i).
B. Next determine (or guess) whether or not the problem is stiff.
Stiffness occurs when the Jacobian matrix df/dy has an eigenvalue
whose real part is negative and large in magnitude, compared to the
reciprocal of the t span of interest. If the problem is nonstiff,
use a method flag MF = 10. If it is stiff, MF should be between 21
and 24, or possibly 29. MF = 22 is generally the best choice.
Use 23 or 24 only if symmetry is present. Use MF = 29 if the
complete linear system solution is to be provided by the user.
The following four parameters must also be set.
IWORK(1) = LWP = length of real array WP for preconditioning.
IWORK(2) = LIWP = length of integer array IWP for preconditioning.
IWORK(3) = JPRE = preconditioner type flag:
= 0 for no preconditioning (P1 = P2 = P = identity)
= 1 for left-only preconditioning (P2 = identity)
= 2 for right-only preconditioning (P1 = identity)
= 3 for two-sided preconditioning (and PCG or PCGS)
IWORK(4) = JACFLG = flag for whether JAC is called.
= 0 if JAC is not to be called,
= 1 if JAC is to be called.
Use JACFLG = 1 if JAC computes any nonconstant data for use in
preconditioning, such as Jacobian elements.
The arrays WP and IWP are work arrays under the user's control,
for use in the routines that perform preconditioning operations.
C. If the problem is stiff, you must supply two routines that deal
with the preconditioning of the linear systems to be solved.
These are as follows:
SUBROUTINE JAC (F, NEQ, T, Y, YSV, REWT, FTY, V, HL0, WP,IWP, IER)
DOUBLE PRECISION T, Y(*),YSV(*), REWT(*), FTY(*), V(*), HL0, WP(*)
INTEGER IWP(*)
This routine must evaluate and preprocess any parts of the
Jacobian matrix df/dy involved in the preconditioners P1, P2, P.
The Y and FTY arrays contain the current values of y and f(t,y),
respectively, and YSV also contains the current value of y.
The array V is work space of length NEQ.
JAC must multiply all computed Jacobian elements by the scalar
-HL0, add the identity matrix, and do any factorization
operations called for, in preparation for solving linear systems
with a coefficient matrix of P1, P2, or P. The matrix P1*P2 or P
should be an approximation to identity - HL0 * (df/dy).
JAC should return IER = 0 if successful, and IER .ne. 0 if not.
(If IER .ne. 0, a smaller time step will be tried.)
SUBROUTINE PSOL (NEQ, T, Y, FTY, WK, HL0, WP, IWP, B, LR, IER)
DOUBLE PRECISION T, Y(*), FTY(*), WK(*), HL0, WP(*), B(*)
INTEGER IWP(*)
This routine must solve a linear system with B as right-hand
side and one of the preconditioning matrices, P1, P2, or P, as
coefficient matrix, and return the solution vector in B.
LR is a flag concerning left vs right preconditioning, input
to PSOL. PSOL is to use P1 if LR = 1 and P2 if LR = 2.
In the case of the PCG or PCGS method, LR will be 3, and PSOL
should solve the system P*x = B with the preconditioner matrix P.
In the case MF = 29 (no Krylov iteration), LR will be 0,
and PSOL is to return in B the desired approximate solution
to A * x = B, where A = identity - HL0 * (df/dy).
PSOL can use data generated in the JAC routine and stored in
WP and IWP. WK is a work array of length NEQ.
The argument HL0 is the current value of the scalar appearing
in the linear system. If the old value, at the time of the last
JAC call, is needed, it must have been saved by JAC in WP.
On return, PSOL should set the error flag IER as follows:
IER = 0 if PSOL was successful,
IER .gt. 0 if a recoverable error occurred, meaning that the
time step will be retried,
IER .lt. 0 if an unrecoverable error occurred, meaning that the
solver is to stop immediately.
D. Write a main program which calls Subroutine DLSODPK once for
each point at which answers are desired. This should also provide
for possible use of logical unit 6 for output of error messages by
DLSODPK. On the first call to DLSODPK, supply arguments as follows:
F = name of subroutine for right-hand side vector f.
This name must be declared External in calling program.
NEQ = number of first order ODEs.
Y = array of initial values, of length NEQ.
T = the initial value of the independent variable.
TOUT = first point where output is desired (.ne. T).
ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
RTOL = relative tolerance parameter (scalar).
ATOL = absolute tolerance parameter (scalar or array).
the estimated local error in y(i) will be controlled so as
to be roughly less (in magnitude) than
EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
Thus the local error test passes if, in each component,
either the absolute error is less than ATOL (or ATOL(i)),
or the relative error is less than RTOL.
Use RTOL = 0.0 for pure absolute error control, and
use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
control. Caution: Actual (global) errors may exceed these
local tolerances, so choose them conservatively.
ITASK = 1 for normal computation of output values of y at t = TOUT.
ISTATE = integer flag (input and output). Set ISTATE = 1.
IOPT = 0 to indicate no optional inputs used.
RWORK = real work array of length at least:
20 + 16*NEQ for MF = 10,
45 + 17*NEQ + LWP for MF = 21,
61 + 17*NEQ + LWP for MF = 22,
20 + 15*NEQ + LWP for MF = 23 or 24,
20 + 12*NEQ + LWP for MF = 29.
LRW = declared length of RWORK (in user's dimension).
IWORK = integer work array of length at least:
30 for MF = 10,
35 + LIWP for MF = 21,
30 + LIWP for MF = 22, 23, 24, or 29.
LIW = declared length of IWORK (in user's dimension).
JAC,PSOL = names of subroutines for preconditioning.
These names must be declared External in the calling program.
MF = method flag. Standard values are:
10 for nonstiff (Adams) method.
21 for stiff (BDF) method, with preconditioned SIOM.
22 for stiff method, with preconditioned GMRES method.
23 for stiff method, with preconditioned CG method.
24 for stiff method, with scaled preconditioned CG method.
29 for stiff method, with user's PSOL routine only.
Note that the main program must declare arrays Y, RWORK, IWORK,
and possibly ATOL.
E. The output from the first call (or any call) is:
Y = array of computed values of y(t) vector.
T = corresponding value of independent variable (normally TOUT).
ISTATE = 2 if DLSODPK was successful, negative otherwise.
-1 means excess work done on this call (perhaps wrong MF).
-2 means excess accuracy requested (tolerances too small).
-3 means illegal input detected (see printed message).
-4 means repeated error test failures (check all inputs).
-5 means repeated convergence failures (perhaps bad JAC
or PSOL routine supplied or wrong choice of MF or
tolerances, or this solver is inappropriate).
-6 means error weight became zero during problem. (Solution
component i vanished, and ATOL or ATOL(i) = 0.)
-7 means an unrecoverable error occurred in PSOL.
F. To continue the integration after a successful return, simply
reset TOUT and call DLSODPK again. No other parameters need be reset.
Full Description of User Interface to DLSODPK
The user interface to DLSODPK consists of the following parts.
1. The call sequence to Subroutine DLSODPK, which is a driver
routine for the solver. This includes descriptions of both
the call sequence arguments and of user-supplied routines.
Following these descriptions is a description of
optional inputs available through the call sequence, and then
a description of optional outputs (in the work arrays).
2. Descriptions of other routines in the DLSODPK package that may be
(optionally) called by the user. These provide the ability to
alter error message handling, save and restore the internal
Common, and obtain specified derivatives of the solution y(t).
3. Descriptions of Common blocks to be declared in overlay
or similar environments, or to be saved when doing an interrupt
of the problem and continued solution later.
4. Description of two routines in the DLSODPK package, either of
which the user may replace with his/her own version, if desired.
These relate to the measurement of errors.
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Part 1. Call Sequence.
The call sequence parameters used for input only are
F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, PSOL, MF,
and those used for both input and output are
Y, T, ISTATE.
The work arrays RWORK and IWORK are also used for conditional and
optional inputs and optional outputs. (The term output here refers
to the return from Subroutine DLSODPK to the user's calling program.)
The legality of input parameters will be thoroughly checked on the
initial call for the problem, but not checked thereafter unless a
change in input parameters is flagged by ISTATE = 3 on input.
The descriptions of the call arguments are as follows.
F = the name of the user-supplied subroutine defining the
ODE system. The system must be put in the first-order
form dy/dt = f(t,y), where f is a vector-valued function
of the scalar t and the vector y. Subroutine F is to
compute the function f. It is to have the form
SUBROUTINE F (NEQ, T, Y, YDOT)
DOUBLE PRECISION T, Y(*), YDOT(*)
where NEQ, T, and Y are input, and the array YDOT = f(t,y)
is output. Y and YDOT are arrays of length NEQ.
Subroutine F should not alter Y(1),...,Y(NEQ).
F must be declared External in the calling program.
Subroutine F may access user-defined quantities in
NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
(dimensioned in F) and/or Y has length exceeding NEQ(1).
See the descriptions of NEQ and Y below.
If quantities computed in the F routine are needed
externally to DLSODPK, an extra call to F should be made
for this purpose, for consistent and accurate results.
If only the derivative dy/dt is needed, use DINTDY instead.
NEQ = the size of the ODE system (number of first order
ordinary differential equations). Used only for input.
NEQ may be decreased, but not increased, during the problem.
If NEQ is decreased (with ISTATE = 3 on input), the
remaining components of Y should be left undisturbed, if
these are to be accessed in the user-supplied subroutines.
Normally, NEQ is a scalar, and it is generally referred to
as a scalar in this user interface description. However,
NEQ may be an array, with NEQ(1) set to the system size.
(The DLSODPK package accesses only NEQ(1).) In either case,
this parameter is passed as the NEQ argument in all calls
to F, JAC, and PSOL. Hence, if it is an array, locations
NEQ(2),... may be used to store other integer data and pass
it to the user-supplied subroutines. Each such routine must
include NEQ in a Dimension statement in that case.
Y = a real array for the vector of dependent variables, of
length NEQ or more. Used for both input and output on the
first call (ISTATE = 1), and only for output on other calls.
On the first call, Y must contain the vector of initial
values. On output, Y contains the computed solution vector,
evaluated at T. If desired, the Y array may be used
for other purposes between calls to the solver.
This array is passed as the Y argument in all calls to F,
JAC, and PSOL. Hence its length may exceed NEQ, and locations
Y(NEQ+1),... may be used to store other real data and
pass it to the user-supplied subroutines. (The DLSODPK
package accesses only Y(1),...,Y(NEQ).)
T = the independent variable. On input, T is used only on the
first call, as the initial point of the integration.
On output, after each call, T is the value at which a
computed solution y is evaluated (usually the same as TOUT).
On an error return, T is the farthest point reached.
TOUT = the next value of t at which a computed solution is desired.
Used only for input.
When starting the problem (ISTATE = 1), TOUT may be equal
to T for one call, then should .ne. T for the next call.
For the initial T, an input value of TOUT .ne. T is used
in order to determine the direction of the integration
(i.e. the algebraic sign of the step sizes) and the rough
scale of the problem. Integration in either direction
(forward or backward in t) is permitted.
If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
the first call (i.e. the first call with TOUT .ne. T).
Otherwise, TOUT is required on every call.
If ITASK = 1, 3, or 4, the values of TOUT need not be
monotone, but a value of TOUT which backs up is limited
to the current internal T interval, whose endpoints are
TCUR - HU and TCUR (see optional outputs, below, for
TCUR and HU).
ITOL = an indicator for the type of error control. See
description below under ATOL. Used only for input.
RTOL = a relative error tolerance parameter, either a scalar or
an array of length NEQ. See description below under ATOL.
Input only.
ATOL = an absolute error tolerance parameter, either a scalar or
an array of length NEQ. Input only.
The input parameters ITOL, RTOL, and ATOL determine
the error control performed by the solver. The solver will
control the vector E = (E(i)) of estimated local errors
in y, according to an inequality of the form
RMS-norm of ( E(i)/EWT(i) ) .le. 1,
where EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
and the RMS-norm (root-mean-square norm) here is
RMS-norm(v) = SQRT(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
is a vector of weights which must always be positive, and
the values of RTOL and ATOL should all be non-negative.
the following table gives the types (scalar/array) of
RTOL and ATOL, and the corresponding form of EWT(i).
ITOL RTOL ATOL EWT(i)
1 scalar scalar RTOL*ABS(Y(i)) + ATOL
2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
When either of these parameters is a scalar, it need not
be dimensioned in the user's calling program.
If none of the above choices (with ITOL, RTOL, and ATOL
fixed throughout the problem) is suitable, more general
error controls can be obtained by substituting
user-supplied routines for the setting of EWT and/or for
the norm calculation. See Part 4 below.
If global errors are to be estimated by making a repeated
run on the same problem with smaller tolerances, then all
components of RTOL and ATOL (i.e. of EWT) should be scaled
down uniformly.
ITASK = an index specifying the task to be performed.
Input only. ITASK has the following values and meanings.
1 means normal computation of output values of y(t) at
t = TOUT (by overshooting and interpolating).
2 means take one step only and return.
3 means stop at the first internal mesh point at or
beyond t = TOUT and return.
4 means normal computation of output values of y(t) at
t = TOUT but without overshooting t = TCRIT.
TCRIT must be input as RWORK(1). TCRIT may be equal to
or beyond TOUT, but not behind it in the direction of
integration. This option is useful if the problem
has a singularity at or beyond t = TCRIT.
5 means take one step, without passing TCRIT, and return.
TCRIT must be input as RWORK(1).
Note: If ITASK = 4 or 5 and the solver reaches TCRIT
(within roundoff), it will return T = TCRIT (exactly) to
indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
in which case answers at t = TOUT are returned first).
ISTATE = an index used for input and output to specify the
the state of the calculation.
On input, the values of ISTATE are as follows.
1 means this is the first call for the problem
(initializations will be done). See note below.
2 means this is not the first call, and the calculation
is to continue normally, with no change in any input
parameters except possibly TOUT and ITASK.
(If ITOL, RTOL, and/or ATOL are changed between calls
with ISTATE = 2, the new values will be used but not
tested for legality.)
3 means this is not the first call, and the
calculation is to continue normally, but with
a change in input parameters other than
TOUT and ITASK. Changes are allowed in
NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
and any of the optional inputs except H0.
Note: A preliminary call with TOUT = T is not counted
as a first call here, as no initialization or checking of
input is done. (Such a call is sometimes useful for the
purpose of outputting the initial conditions.)
Thus the first call for which TOUT .ne. T requires
ISTATE = 1 on input.
On output, ISTATE has the following values and meanings.
1 means nothing was done; TOUT = T and ISTATE = 1 on input.
2 means the integration was performed successfully.
-1 means an excessive amount of work (more than MXSTEP
steps) was done on this call, before completing the
requested task, but the integration was otherwise
successful as far as T. (MXSTEP is an optional input
and is normally 500.) To continue, the user may
simply reset ISTATE to a value .gt. 1 and call again
(the excess work step counter will be reset to 0).
In addition, the user may increase MXSTEP to avoid
this error return (see below on optional inputs).
-2 means too much accuracy was requested for the precision
of the machine being used. This was detected before
completing the requested task, but the integration
was successful as far as T. To continue, the tolerance
parameters must be reset, and ISTATE must be set
to 3. The optional output TOLSF may be used for this
purpose. (Note: If this condition is detected before
taking any steps, then an illegal input return
(ISTATE = -3) occurs instead.)
-3 means illegal input was detected, before taking any
integration steps. See written message for details.
Note: If the solver detects an infinite loop of calls
to the solver with illegal input, it will cause
the run to stop.
-4 means there were repeated error test failures on
one attempted step, before completing the requested
task, but the integration was successful as far as T.
The problem may have a singularity, or the input
may be inappropriate.
-5 means there were repeated convergence test failures on
one attempted step, before completing the requested
task, but the integration was successful as far as T.
-6 means EWT(i) became zero for some i during the
integration. Pure relative error control (ATOL(i)=0.0)
was requested on a variable which has now vanished.
The integration was successful as far as T.
-7 means the PSOL routine returned an unrecoverable error
flag (IER .lt. 0). The integration was successful as
far as T.
Note: since the normal output value of ISTATE is 2,
it does not need to be reset for normal continuation.
Also, since a negative input value of ISTATE will be
regarded as illegal, a negative output value requires the
user to change it, and possibly other inputs, before
calling the solver again.
IOPT = an integer flag to specify whether or not any optional
inputs are being used on this call. Input only.
The optional inputs are listed separately below.
IOPT = 0 means no optional inputs are being used.
Default values will be used in all cases.
IOPT = 1 means one or more optional inputs are being used.
RWORK = a real working array (double precision).
The length of RWORK must be at least
20 + NYH*(MAXORD + 1) + 3*NEQ + LENLS + LWP where
NYH = the initial value of NEQ,
MAXORD = 12 (if METH = 1) or 5 (if METH = 2) (unless a
smaller value is given as an optional input),
LENLS = length of work space for linear system (Krylov)
method, excluding preconditioning:
LENLS = 0 if MITER = 0,
LENLS = NEQ*(MAXL+3) + MAXL**2 if MITER = 1,
LENLS = NEQ*(MAXL+3+MIN(1,MAXL-KMP))
+ (MAXL+3)*MAXL + 1 if MITER = 2,
LENLS = 6*NEQ if MITER = 3 or 4,
LENLS = 3*NEQ if MITER = 9.
(See the MF description for METH and MITER, and the
list of optional inputs for MAXL and KMP.)
LWP = length of real user work space for preconditioning
(see JAC/PSOL).
Thus if default values are used and NEQ is constant,
this length is:
20 + 16*NEQ for MF = 10,
45 + 24*NEQ + LWP FOR MF = 11,
61 + 24*NEQ + LWP FOR MF = 12,
20 + 22*NEQ + LWP FOR MF = 13 OR 14,
20 + 19*NEQ + LWP FOR MF = 19,
20 + 9*NEQ FOR MF = 20,
45 + 17*NEQ + LWP FOR MF = 21,
61 + 17*NEQ + LWP FOR MF = 22,
20 + 15*NEQ + LWP FOR MF = 23 OR 24,
20 + 12*NEQ + LWP for MF = 29.
The first 20 words of RWORK are reserved for conditional
and optional inputs and optional outputs.
The following word in RWORK is a conditional input:
RWORK(1) = TCRIT = critical value of t which the solver
is not to overshoot. Required if ITASK is
4 or 5, and ignored otherwise. (See ITASK.)
LRW = the length of the array RWORK, as declared by the user.
(This will be checked by the solver.)
IWORK = an integer work array. The length of IWORK must be at least
30 if MITER = 0 (MF = 10 or 20),
30 + MAXL + LIWP if MITER = 1 (MF = 11, 21),
30 + LIWP if MITER = 2, 3, 4, or 9.
MAXL = 5 unless a different optional input value is given.
LIWP = length of integer user work space for preconditioning
(see conditional input list following).
The first few words of IWORK are used for conditional and
optional inputs and optional outputs.
The following 4 words in IWORK are conditional inputs,
required if MITER .ge. 1:
IWORK(1) = LWP = length of real array WP for use in
preconditioning (part of RWORK array).
IWORK(2) = LIWP = length of integer array IWP for use in
preconditioning (part of IWORK array).
The arrays WP and IWP are work arrays under the
user's control, for use in the routines that
perform preconditioning operations (JAC and PSOL).
IWORK(3) = JPRE = preconditioner type flag:
= 0 for no preconditioning (P1 = P2 = P = identity)
= 1 for left-only preconditioning (P2 = identity)
= 2 for right-only preconditioning (P1 = identity)
= 3 for two-sided preconditioning (and PCG or PCGS)
IWORK(4) = JACFLG = flag for whether JAC is called.
= 0 if JAC is not to be called,
= 1 if JAC is to be called.
Use JACFLG = 1 if JAC computes any nonconstant
data needed in preconditioning operations,
such as some of the Jacobian elements.
LIW = the length of the array IWORK, as declared by the user.
(This will be checked by the solver.)
Note: The work arrays must not be altered between calls to DLSODPK
for the same problem, except possibly for the conditional and
optional inputs, and except for the last 3*NEQ words of RWORK.
The latter space is used for internal scratch space, and so is
available for use by the user outside DLSODPK between calls, if
desired (but not for use by any of the user-supplied subroutines).
JAC = the name of the user-supplied routine to compute any
Jacobian elements (or approximations) involved in the
matrix preconditioning operations (MITER .ge. 1).
It is to have the form
SUBROUTINE JAC (F, NEQ, T, Y, YSV, REWT, FTY, V,
1 HL0, WP, IWP, IER)
DOUBLE PRECISION T, Y(*),YSV(*), REWT(*), FTY(*), V(*),
1 HL0, WP(*)
INTEGER IWP(*)
This routine must evaluate and preprocess any parts of the
Jacobian matrix df/dy used in the preconditioners P1, P2, P.
the Y and FTY arrays contain the current values of y and
f(t,y), respectively, and YSV also contains the current
value of y. The array V is work space of length
NEQ for use by JAC. REWT is the array of reciprocal error
weights (1/EWT). JAC must multiply all computed Jacobian
elements by the scalar -HL0, add the identity matrix, and do
any factorization operations called for, in preparation
for solving linear systems with a coefficient matrix of
P1, P2, or P. The matrix P1*P2 or P should be an
approximation to identity - HL0 * (df/dy). JAC should
return IER = 0 if successful, and IER .ne. 0 if not.
(If IER .ne. 0, a smaller time step will be tried.)
The arrays WP (of length LWP) and IWP (of length LIWP)
are for use by JAC and PSOL for work space and for storage
of data needed for the solution of the preconditioner
linear systems. Their lengths and contents are under the
user's control.
The JAC routine may save relevant Jacobian elements (or
approximations) used in the preconditioners, along with the
value of HL0, and use these to reconstruct preconditioner
matrices later without reevaluationg those elements.
This may be cost-effective if JAC is called with HL0
considerably different from its earlier value, indicating
that a corrector convergence failure has occurred because
of the change in HL0, not because of changes in the
value of the Jacobian. In doing this, use the saved and
current values of HL0 to decide whether to use saved
or reevaluated elements.
JAC may alter V, but may not alter Y, YSV, REWT, FTY, or HL0.
JAC must be declared External in the calling program.
Subroutine JAC may access user-defined quantities in
NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
(dimensioned in JAC) and/or Y has length exceeding NEQ(1).
See the descriptions of NEQ and Y above.
PSOL = the name of the user-supplied routine for the
solution of preconditioner linear systems.
It is to have the form
SUBROUTINE PSOL (NEQ, T, Y, FTY, WK,HL0, WP,IWP, B, LR,IER)
DOUBLE PRECISION T, Y(*), FTY(*), WK(*), HL0, WP(*), B(*)
INTEGER IWP(*)
This routine must solve a linear system with B as right-hand
side and one of the preconditioning matrices, P1, P2, or P,
as coefficient matrix, and return the solution vector in B.
LR is a flag concerning left vs right preconditioning, input
to PSOL. PSOL is to use P1 if LR = 1 and P2 if LR = 2.
In the case of the PCG or PCGS method, LR will be 3, and PSOL
should solve the system P*x = B with the preconditioner P.
In the case MITER = 9 (no Krylov iteration), LR will be 0,
and PSOL is to return in B the desired approximate solution
to A * x = B, where A = identity - HL0 * (df/dy).
PSOL can use data generated in the JAC routine and stored in
WP and IWP.
The Y and FTY arrays contain the current values of y and
f(t,y), respectively. The array WK is work space of length
NEQ for use by PSOL.
The argument HL0 is the current value of the scalar appearing
in the linear system. If the old value, as of the last
JAC call, is needed, it must have been saved by JAC in WP.
On return, PSOL should set the error flag IER as follows:
IER = 0 if PSOL was successful,
IER .gt. 0 on a recoverable error, meaning that the
time step will be retried,
IER .lt. 0 on an unrecoverable error, meaning that the
solver is to stop immediately.
PSOL may not alter Y, FTY, or HL0.
PSOL must be declared External in the calling program.
Subroutine PSOL may access user-defined quantities in
NEQ(2),... and Y(NEQ(1)+1),... if NEQ is an array
(dimensioned in PSOL) and/or Y has length exceeding NEQ(1).
See the descriptions of NEQ and Y above.
MF = the method flag. Used only for input. The legal values of
MF are 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, and 29.
MF has decimal digits METH and MITER: MF = 10*METH + MITER.
METH indicates the basic linear multistep method:
METH = 1 means the implicit Adams method.
METH = 2 means the method based on Backward
Differentiation Formulas (BDFs).
MITER indicates the corrector iteration method:
MITER = 0 means functional iteration (no linear system
is involved).
MITER = 1 means Newton iteration with Scaled Preconditioned
Incomplete Orthogonalization Method (SPIOM)
for the linear systems.
MITER = 2 means Newton iteration with Scaled Preconditioned
Generalized Minimal Residual method (SPIGMR)
for the linear systems.
MITER = 3 means Newton iteration with Preconditioned
Conjugate Gradient method (PCG)
for the linear systems.
MITER = 4 means Newton iteration with scaled Preconditioned
Conjugate Gradient method (PCGS)
for the linear systems.
MITER = 9 means Newton iteration with only the
user-supplied PSOL routine called (no Krylov
iteration) for the linear systems.
JPRE is ignored, and PSOL is called with LR = 0.
See comments in the introduction about the choice of MITER.
If MITER .ge. 1, the user must supply routines JAC and PSOL
(the names are arbitrary) as described above.
For MITER = 0, dummy arguments can be used.
-----------------------------------------------------------------------
Optional Inputs.
The following is a list of the optional inputs provided for in the
call sequence. (See also Part 2.) For each such input variable,
this table lists its name as used in this documentation, its
location in the call sequence, its meaning, and the default value.
The use of any of these inputs requires IOPT = 1, and in that
case all of these inputs are examined. A value of zero for any
of these optional inputs will cause the default value to be used.
Thus to use a subset of the optional inputs, simply preload
locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and
then set those of interest to nonzero values.
Name Location Meaning and Default Value
H0 RWORK(5) the step size to be attempted on the first step.
The default value is determined by the solver.
HMAX RWORK(6) the maximum absolute step size allowed.
The default value is infinite.
HMIN RWORK(7) the minimum absolute step size allowed.
The default value is 0. (This lower bound is not
enforced on the final step before reaching TCRIT
when ITASK = 4 or 5.)
DELT RWORK(8) convergence test constant in Krylov iteration
algorithm. The default is .05.
MAXORD IWORK(5) the maximum order to be allowed. The default
value is 12 if METH = 1, and 5 if METH = 2.
If MAXORD exceeds the default value, it will
be reduced to the default value.
If MAXORD is changed during the problem, it may
cause the current order to be reduced.
MXSTEP IWORK(6) maximum number of (internally defined) steps
allowed during one call to the solver.
The default value is 500.
MXHNIL IWORK(7) maximum number of messages printed (per problem)
warning that T + H = T on a step (H = step size).
This must be positive to result in a non-default
value. The default value is 10.
MAXL IWORK(8) maximum number of iterations in the SPIOM, SPIGMR,
PCG, or PCGS algorithm (.le. NEQ).
The default is MAXL = MIN(5,NEQ).
KMP IWORK(9) number of vectors on which orthogonalization
is done in SPIOM or SPIGMR algorithm (.le. MAXL).
The default is KMP = MAXL.
Note: When KMP .lt. MAXL and MF = 22, the length
of RWORK must be defined accordingly. See
the definition of RWORK above.
-----------------------------------------------------------------------
Optional Outputs.
As optional additional output from DLSODPK, the variables listed
below are quantities related to the performance of DLSODPK
which are available to the user. These are communicated by way of
the work arrays, but also have internal mnemonic names as shown.
Except where stated otherwise, all of these outputs are defined
on any successful return from DLSODPK, and on any return with
ISTATE = -1, -2, -4, -5, -6, or -7. On an illegal input return
(ISTATE = -3), they will be unchanged from their existing values
(if any), except possibly for TOLSF, LENRW, and LENIW.
On any error return, outputs relevant to the error will be defined,
as noted below.
Name Location Meaning
HU RWORK(11) the step size in t last used (successfully).
HCUR RWORK(12) the step size to be attempted on the next step.
TCUR RWORK(13) the current value of the independent variable
which the solver has actually reached, i.e. the
current internal mesh point in t. On output, TCUR
will always be at least as far as the argument
T, but may be farther (if interpolation was done).
TOLSF RWORK(14) a tolerance scale factor, greater than 1.0,
computed when a request for too much accuracy was
detected (ISTATE = -3 if detected at the start of
the problem, ISTATE = -2 otherwise). If ITOL is
left unaltered but RTOL and ATOL are uniformly
scaled up by a factor of TOLSF for the next call,
then the solver is deemed likely to succeed.
(The user may also ignore TOLSF and alter the
tolerance parameters in any other way appropriate.)
NST IWORK(11) the number of steps taken for the problem so far.
NFE IWORK(12) the number of f evaluations for the problem so far.
NPE IWORK(13) the number of calls to JAC so far (for Jacobian
evaluation associated with preconditioning).
NQU IWORK(14) the method order last used (successfully).
NQCUR IWORK(15) the order to be attempted on the next step.
IMXER IWORK(16) the index of the component of largest magnitude in
the weighted local error vector ( E(i)/EWT(i) ),
on an error return with ISTATE = -4 or -5.
LENRW IWORK(17) the length of RWORK actually required.
This is defined on normal returns and on an illegal
input return for insufficient storage.
LENIW IWORK(18) the length of IWORK actually required.
This is defined on normal returns and on an illegal
input return for insufficient storage.
NNI IWORK(19) number of nonlinear iterations so far (each of
which calls an iterative linear solver).
NLI IWORK(20) number of linear iterations so far.
Note: A measure of the success of algorithm is
the average number of linear iterations per
nonlinear iteration, given by NLI/NNI.
If this is close to MAXL, MAXL may be too small.
NPS IWORK(21) number of preconditioning solve operations
(PSOL calls) so far.
NCFN IWORK(22) number of convergence failures of the nonlinear
(Newton) iteration so far.
Note: A measure of success is the overall
rate of nonlinear convergence failures, NCFN/NST.
NCFL IWORK(23) number of convergence failures of the linear
iteration so far.
Note: A measure of success is the overall
rate of linear convergence failures, NCFL/NNI.
The following two arrays are segments of the RWORK array which
may also be of interest to the user as optional outputs.
For each array, the table below gives its internal name,
its base address in RWORK, and its description.
Name Base Address Description
YH 21 the Nordsieck history array, of size NYH by
(NQCUR + 1), where NYH is the initial value
of NEQ. For j = 0,1,...,NQCUR, column j+1
of YH contains HCUR**j/factorial(j) times
the j-th derivative of the interpolating
polynomial currently representing the solution,
evaluated at t = TCUR.
ACOR LENRW-NEQ+1 array of size NEQ used for the accumulated
corrections on each step, scaled on output
to represent the estimated local error in y
on the last step. This is the vector E in
the description of the error control. It is
defined only on a successful return from
DLSODPK.
-----------------------------------------------------------------------
Part 2. Other Routines Callable.
The following are optional calls which the user may make to
gain additional capabilities in conjunction with DLSODPK.
(The routines XSETUN and XSETF are designed to conform to the
SLATEC error handling package.)
Form of Call Function
CALL XSETUN(LUN) Set the logical unit number, LUN, for
output of messages from DLSODPK, if
the default is not desired.
The default value of lun is 6.
CALL XSETF(MFLAG) Set a flag to control the printing of
messages by DLSODPK.
MFLAG = 0 means do not print. (Danger:
This risks losing valuable information.)
MFLAG = 1 means print (the default).
Either of the above calls may be made at
any time and will take effect immediately.
CALL DSRCPK(RSAV,ISAV,JOB) saves and restores the contents of
the internal Common blocks used by
DLSODPK (see Part 3 below).
RSAV must be a real array of length 222
or more, and ISAV must be an integer
array of length 50 or more.
JOB=1 means save Common into RSAV/ISAV.
JOB=2 means restore Common from RSAV/ISAV.
DSRCPK is useful if one is
interrupting a run and restarting
later, or alternating between two or
more problems solved with DLSODPK.
CALL DINTDY(,,,,,) Provide derivatives of y, of various
(See below) orders, at a specified point t, if
desired. It may be called only after
a successful return from DLSODPK.
The detailed instructions for using DINTDY are as follows.
The form of the call is:
CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
The input parameters are:
T = value of independent variable where answers are desired
(normally the same as the T last returned by DLSODPK).
for valid results, T must lie between TCUR - HU and TCUR.
(See optional outputs for TCUR and HU.)
K = integer order of the derivative desired. K must satisfy
0 .le. K .le. NQCUR, where NQCUR is the current order
(see optional outputs). The capability corresponding
to K = 0, i.e. computing y(T), is already provided
by DLSODPK directly. Since NQCUR .ge. 1, the first
derivative dy/dt is always available with DINTDY.
RWORK(21) = the base address of the history array YH.
NYH = column length of YH, equal to the initial value of NEQ.
The output parameters are:
DKY = a real array of length NEQ containing the computed value
of the K-th derivative of y(t).
IFLAG = integer flag, returned as 0 if K and T were legal,
-1 if K was illegal, and -2 if T was illegal.
On an error return, a message is also written.
-----------------------------------------------------------------------
Part 3. Common Blocks.
If DLSODPK is to be used in an overlay situation, the user
must declare, in the primary overlay, the variables in:
(1) the call sequence to DLSODPK, and
(2) the two internal Common blocks
/DLS001/ of length 255 (218 double precision words
followed by 37 integer words),
/DLPK01/ of length 17 (4 double precision words
followed by 13 integer words).
If DLSODPK is used on a system in which the contents of internal
Common blocks are not preserved between calls, the user should
declare the above Common blocks in the calling program to insure
that their contents are preserved.
If the solution of a given problem by DLSODPK is to be interrupted
and then later continued, such as when restarting an interrupted run
or alternating between two or more problems, the user should save,
following the return from the last DLSODPK call prior to the
interruption, the contents of the call sequence variables and the
internal Common blocks, and later restore these values before the
next DLSODPK call for that problem. To save and restore the Common
blocks, use Subroutine DSRCPK (see Part 2 above).
-----------------------------------------------------------------------
Part 4. Optionally Replaceable Solver Routines.
below are descriptions of two routines in the DLSODPK package which
relate to the measurement of errors. Either routine can be
replaced by a user-supplied version, if desired. However, since such
a replacement may have a major impact on performance, it should be
done only when absolutely necessary, and only with great caution.
(Note: The means by which the package version of a routine is
superseded by the user's version may be system-dependent.)
(a) DEWSET.
The following subroutine is called just before each internal
integration step, and sets the array of error weights, EWT, as
described under ITOL/RTOL/ATOL above:
SUBROUTINE DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
where NEQ, ITOL, RTOL, and ATOL are as in the DLSODPK call sequence,
YCUR contains the current dependent variable vector, and
EWT is the array of weights set by DEWSET.
If the user supplies this subroutine, it must return in EWT(i)
(i = 1,...,NEQ) a positive quantity suitable for comparing errors
in y(i) to. The EWT array returned by DEWSET is passed to the DVNORM
routine (see below), and also used by DLSODPK in the computation
of the optional output IMXER, the diagonal Jacobian approximation,
and the increments for difference quotient Jacobians.
In the user-supplied version of DEWSET, it may be desirable to use
the current values of derivatives of y. Derivatives up to order NQ
are available from the history array YH, described above under
optional outputs. In DEWSET, YH is identical to the YCUR array,
extended to NQ + 1 columns with a column length of NYH and scale
factors of H**j/factorial(j). On the first call for the problem,
given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
NYH is the initial value of NEQ. The quantities NQ, H, and NST
can be obtained by including in DEWSET the statements:
DOUBLE PRECISION RLS
COMMON /DLS001/ RLS(218),ILS(37)
NQ = ILS(33)
NST = ILS(34)
H = RLS(212)
Thus, for example, the current value of dy/dt can be obtained as
YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
unnecessary when NST = 0).
(b) DVNORM.
The following is a real function routine which computes the weighted
root-mean-square norm of a vector v:
D = DVNORM (N, V, W)
where:
N = the length of the vector,
V = real array of length N containing the vector,
W = real array of length N containing weights,
D = SQRT( (1/N) * sum(V(i)*W(i))**2 ).
DVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
EWT is as set by Subroutine DEWSET.
If the user supplies this function, it should return a non-negative
value of DVNORM suitable for use in the error control in DLSODPK.
None of the arguments should be altered by DVNORM.
For example, a user-supplied DVNORM routine might:
-substitute a max-norm of (V(i)*W(i)) for the RMS-norm, or
-ignore some components of V in the norm, with the effect of
suppressing the error control on those components of y.
-----------------------------------------------------------------------
***REVISION HISTORY (YYYYMMDD)
19860901 DATE WRITTEN
19861010 Numerous minor revisions to SPIOM and SPGMR routines;
minor corrections to prologues and comments.
19870114 Changed name SPGMR to SPIGMR; revised residual norm
calculation in SPIGMR (for incomplete case);
revised error return logic in SPIGMR;
19870330 Major update: corrected comments throughout;
removed TRET from Common; rewrote EWSET with 4 loops;
fixed t test in INTDY; added Cray directives in STODPK;
in STODPK, fixed DELP init. and logic around PJAC call;
combined routines to save/restore Common;
passed LEVEL = 0 in error message calls (except run abort).
19871130 Added option MITER = 9; shortened WM array by 2;
revised early return from SPIOM and SPIGMR;
replaced copy loops with SCOPY/DCOPY calls;
minor corrections/revisions to SOLPK, SPIGMR, ATV, ATP;
corrections to main prologue and internal comments.
19880304 Corrections to type declarations in SOLPK, SPIOM, USOL.
19891025 Added ISTATE = -7 return; minor revisions to USOL;
added initialization of JACFLG in main driver;
removed YH and NYH from PKSET call list;
minor revisions to SPIOM and SPIGMR;
corrections to main prologue and internal comments.
19900803 Added YSV to JAC call list; minor comment corrections.
20010425 Major update: convert source lines to upper case;
added *DECK lines; changed from 1 to * in dummy dimensions;
changed names R1MACH/D1MACH to RUMACH/DUMACH;
renamed routines for uniqueness across single/double prec.;
converted intrinsic names to generic form;
removed ILLIN and NTREP (data loaded) from Common;
removed all 'own' variables from Common;
changed error messages to quoted strings;
replaced XERRWV/XERRWD with 1993 revised version;
converted prologues, comments, error messages to mixed case;
numerous corrections to prologues and internal comments.
20010507 Converted single precision source to double precision.
20020502 Corrected declarations in descriptions of user routines.
20030603 Corrected duplicate type declaration for DUMACH.
20031105 Restored 'own' variables to Common blocks, to enable
interrupt/restart feature.
20031112 Added SAVE statements for data-loaded constants.
20031117 Changed internal name NPE to NJE.
-----------------------------------------------------------------------
Other routines in the DLSODPK package.
In addition to Subroutine DLSODPK, the DLSODPK package includes the
following subroutines and function routines:
DINTDY computes an interpolated value of the y vector at t = TOUT.
DEWSET sets the error weight vector EWT before each step.
DVNORM computes the weighted RMS-norm of a vector.
DSTODPK is the core integrator, which does one step of the
integration and the associated error control.
DCFODE sets all method coefficients and test constants.
DPKSET interfaces between DSTODPK and the JAC routine.
DSOLPK manages solution of linear system in Newton iteration.
DSPIOM performs the SPIOM algorithm.
DATV computes a scaled, preconditioned product (I-hl0*J)*v.
DORTHOG orthogonalizes a vector against previous basis vectors.
DHEFA generates an LU factorization of a Hessenberg matrix.
DHESL solves a Hessenberg square linear system.
DSPIGMR performs the SPIGMR algorithm.
DHEQR generates a QR factorization of a Hessenberg matrix.
DHELS finds the least squares solution of a Hessenberg system.
DPCG performs Preconditioned Conjugate Gradient algorithm (PCG).
DPCGS performs the PCGS algorithm.
DATP computes the product A*p, where A = I - hl0*df/dy.
DUSOL interfaces to the user's PSOL routine (MITER = 9).
DSRCPK is a user-callable routine to save and restore
the contents of the internal Common blocks.
DAXPY, DCOPY, DDOT, DNRM2, and DSCAL are basic linear
algebra modules (from the BLAS collection).
DUMACH computes the unit roundoff in a machine-independent manner.
XERRWD, XSETUN, XSETF, IXSAV, and IUMACH handle the printing of all
error messages and warnings. XERRWD is machine-dependent.
Note: DVNORM, DDOT, DNRM2, DUMACH, IXSAV, and IUMACH are function
routines. All the others are subroutines.