dlsodis
This is the 18 November 2003 version of
DLSODIS: Livermore Solver for Ordinary Differential equations
(Implicit form) with general Sparse Jacobian matrices.
This version is in double precision.
DLSODIS solves the initial value problem for linearly implicit
systems of first order ODEs,
A(t,y) * dy/dt = g(t,y) , where A(t,y) is a square matrix,
or, in component form,
( a * ( dy / dt )) + ... + ( a * ( dy / dt )) =
i,1 1 i,NEQ NEQ
= g ( t, y , y ,..., y ) ( i = 1,...,NEQ )
i 1 2 NEQ
If A is singular, this is a differential-algebraic system.
DLSODIS is a variant version of the DLSODI package, and is intended
for stiff problems in which the matrix A and the Jacobian matrix
d(g - A*s)/dy have arbitrary sparse structures.
Authors: Alan C. Hindmarsh
Center for Applied Scientific Computing, L-561
Lawrence Livermore National Laboratory
Livermore, CA 94551
and
Sheila Balsdon
Zycor, Inc.
Austin, TX 78741
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References:
1. M. K. Seager and S. Balsdon, LSODIS, A Sparse Implicit
ODE Solver, in Proceedings of the IMACS 10th World Congress,
Montreal, August 8-13, 1982.
2. Alan C. Hindmarsh, LSODE and LSODI, Two New Initial Value
Ordinary Differential Equation Solvers,
ACM-SIGNUM Newsletter, vol. 15, no. 4 (1980), pp. 10-11.
3. S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman,
Yale Sparse Matrix Package: I. The Symmetric Codes,
Int. J. Num. Meth. Eng., vol. 18 (1982), pp. 1145-1151.
4. S. C. Eisenstat, M. C. Gursky, M. H. Schultz, and A. H. Sherman,
Yale Sparse Matrix Package: II. The Nonsymmetric Codes,
Research Report No. 114, Dept. of Computer Sciences, Yale
University, 1977.
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Summary of Usage.
Communication between the user and the DLSODIS 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 example
problem (with program and output) following this summary.
A. First, provide a subroutine of the form:
SUBROUTINE RES (NEQ, T, Y, S, R, IRES)
DOUBLE PRECISION T, Y(*), S(*), R(*)
which computes the residual function
r = g(t,y) - A(t,y) * s ,
as a function of t and the vectors y and s. (s is an internally
generated approximation to dy/dt.) The arrays Y and S are inputs
to the RES routine and should not be altered. The residual
vector is to be stored in the array R. The argument IRES should be
ignored for casual use of DLSODIS. (For uses of IRES, see the
paragraph on RES in the full description below.)
B. DLSODIS must deal internally with the matrices A and dr/dy, where
r is the residual function defined above. DLSODIS generates a linear
combination of these two matrices in sparse form.
The matrix structure is communicated by a method flag, MF:
MF = 21 or 22 when the user provides the structures of
matrix A and dr/dy,
MF = 121 or 222 when the user does not provide structure
information, and
MF = 321 or 422 when the user provides the structure
of matrix A.
C. You must also provide a subroutine of the form:
SUBROUTINE ADDA (NEQ, T, Y, J, IAN, JAN, P)
DOUBLE PRECISION T, Y(*), P(*)
INTEGER IAN(*), JAN(*)
which adds the matrix A = A(t,y) to the contents of the array P.
NEQ, T, Y, and J are input arguments and should not be altered.
This routine should add the J-th column of matrix A to the array
P (of length NEQ). I.e. add A(i,J) to P(i) for all relevant
values of i. The arguments IAN and JAN should be ignored for normal
situations. DLSODIS will call the ADDA routine with J = 1,2,...,NEQ.
D. For the sake of efficiency, you are encouraged to supply the
Jacobian matrix dr/dy in closed form, where r = g(t,y) - A(t,y)*s
(s = a fixed vector) as above. If dr/dy is being supplied,
use MF = 21, 121, or 321, and provide a subroutine of the form:
SUBROUTINE JAC (NEQ, T, Y, S, J, IAN, JAN, PDJ)
DOUBLE PRECISION T, Y(*), S(*), PDJ(*)
INTEGER IAN(*), JAN(*)
which computes dr/dy as a function of t, y, and s. Here NEQ, T, Y, S,
and J are input arguments, and the JAC routine is to load the array
PDJ (of length NEQ) with the J-th column of dr/dy. I.e. load PDJ(i)
with dr(i)/dy(J) for all relevant values of i. The arguments IAN and
JAN should be ignored for normal situations. DLSODIS will call the
JAC routine with J = 1,2,...,NEQ.
Only nonzero elements need be loaded. A crude approximation
to dr/dy, possibly with fewer nonzero elememts, will suffice.
Note that if A is independent of y (or this dependence
is weak enough to be ignored) then JAC is to compute dg/dy.
If it is not feasible to provide a JAC routine, use
MF = 22, 222, or 422 and DLSODIS will compute an approximate
Jacobian internally by difference quotients.
E. Next decide whether or not to provide the initial value of the
derivative vector dy/dt. If the initial value of A(t,y) is
nonsingular (and not too ill-conditioned), you may let DLSODIS compute
this vector (ISTATE = 0). (DLSODIS will solve the system A*s = g for
s, with initial values of A and g.) If A(t,y) is initially
singular, then the system is a differential-algebraic system, and
you must make use of the particular form of the system to compute the
initial values of y and dy/dt. In that case, use ISTATE = 1 and
load the initial value of dy/dt into the array YDOTI.
The input array YDOTI and the initial Y array must be consistent with
the equations A*dy/dt = g. This implies that the initial residual
r = g(t,y) - A(t,y)*YDOTI must be approximately zero.
F. Write a main program which calls Subroutine DLSODIS 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
DLSODIS. On the first call to DLSODIS, supply arguments as follows:
RES = name of user subroutine for residual function r.
ADDA = name of user subroutine for computing and adding A(t,y).
JAC = name of user subroutine for Jacobian matrix dr/dy
(MF = 121). If not used, pass a dummy name.
Note: The names for the RES and ADDA routines and (if used) the
JAC routine must be declared External in the calling program.
NEQ = number of scalar equations in the system.
Y = array of initial values, of length NEQ.
YDOTI = array of length NEQ (containing initial dy/dt if ISTATE = 1).
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 if the
initial dy/dt is supplied, and 0 otherwise.
IOPT = 0 to indicate no optional inputs used.
RWORK = real work array of length at least:
20 + (2 + 1./LENRAT)*NNZ + (11 + 9./LENRAT)*NEQ
where:
NNZ = the number of nonzero elements in the sparse
iteration matrix P = A - con*dr/dy (con = scalar)
(If NNZ is unknown, use an estimate of it.)
LENRAT = the real to integer wordlength ratio (usually 1 in
single precision and 2 in double precision).
In any case, the required size of RWORK cannot generally
be predicted in advance for any value of MF, and the
value above is a rough estimate of a crude lower bound.
Some experimentation with this size may be necessary.
(When known, the correct required length is an optional
output, available in IWORK(17).)
LRW = declared length of RWORK (in user's dimension).
IWORK = integer work array of length at least 30.
LIW = declared length of IWORK (in user's dimension).
MF = method flag. Standard values are:
121 for a user-supplied sparse Jacobian.
222 for an internally generated sparse Jacobian.
For other choices of MF, see the paragraph on MF in
the full description below.
Note that the main program must declare arrays Y, YDOTI, RWORK, IWORK,
and possibly ATOL.
G. 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 DLSODIS was successful, negative otherwise.
-1 means excess work done on this call (check all inputs).
-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 Jacobian
supplied or wrong choice of tolerances).
-6 means error weight became zero during problem. (Solution
component i vanished, and ATOL or ATOL(i) = 0.)
-7 cannot occur in casual use.
-8 means DLSODIS was unable to compute the initial dy/dt.
in casual use, this means A(t,y) is initially singular.
Supply YDOTI and use ISTATE = 1 on the first call.
-9 means a fatal error return flag came from sparse solver
CDRV by way of DPRJIS or DSOLSS. Should never happen.
A return with ISTATE = -1, -4, or -5, may result from using
an inappropriate sparsity structure, one that is quite
different from the initial structure. Consider calling
DLSODIS again with ISTATE = 3 to force the structure to be
reevaluated. See the full description of ISTATE below.
If DLSODIS returns ISTATE = -1, -4 or -5, then the output of
DLSODIS also includes YDOTI = array containing residual vector
r = g - A * dy/dt evaluated at the current t, y, and dy/dt.
H. To continue the integration after a successful return, simply
reset TOUT and call DLSODIS again. No other parameters need be reset.
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Example Problem.
The following is an example problem, with the coding needed
for its solution by DLSODIS. The problem comes from the partial
differential equation (the Burgers equation)
du/dt = - u * du/dx + eta * d**2 u/dx**2, eta = .05,
on -1 .le. x .le. 1. The boundary conditions are periodic:
u(-1,t) = u(1,t) and du/dx(-1,t) = du/dx(1,t)
The initial profile is a square wave,
u = 1 in ABS(x) .lt. .5, u = .5 at ABS(x) = .5, u = 0 elsewhere.
The PDE is discretized in x by a simplified Galerkin method,
using piecewise linear basis functions, on a grid of 40 intervals.
The result is a system A * dy/dt = g(y), of size NEQ = 40,
where y(i) is the approximation to u at x = x(i), with
x(i) = -1 + (i-1)*delx, delx = 2/NEQ = .05.
The individual equations in the system are (in order):
(1/6)dy(NEQ)/dt+(4/6)dy(1)/dt+(1/6)dy(2)/dt
= r4d*(y(NEQ)**2-y(2)**2)+eodsq*(y(2)-2*y(1)+y(NEQ))
for i = 2,3,...,nm1,
(1/6)dy(i-1)/dt+(4/6)dy(i)/dt+(1/6)dy(i+1)/dt
= r4d*(y(i-1)**2-y(i+1)**2)+eodsq*(y(i+1)-2*y(i)+y(i-1))
and finally
(1/6)dy(nm1)/dt+(4/6)dy(NEQ)/dt+(1/6)dy(1)/dt
= r4d*(y(nm1)**2-y(1)**2)+eodsq*(y(1)-2*y(NEQ)+y(nm1))
where r4d = 1/(4*delx), eodsq = eta/delx**2 and nm1 = NEQ-1.
The following coding solves the problem with MF = 121, with output
of solution statistics at t = .1, .2, .3, and .4, and of the
solution vector at t = .4. Optional outputs (run statistics) are
also printed.
EXTERNAL RESID, ADDASP, JACSP
DOUBLE PRECISION ATOL, RTOL, RW, T, TOUT, Y, YDOTI, R4D, EODSQ, DELX
DIMENSION Y(40), YDOTI(40), RW(1409), IW(30)
COMMON /TEST1/ R4D, EODSQ, NM1
DATA ITOL/1/, RTOL/1.0D-3/, ATOL/1.0D-3/, ITASK/1/, IOPT/0/
DATA NEQ/40/, LRW/1409/, LIW/30/, MF/121/
DELX = 2.0/NEQ
R4D = 0.25/DELX
EODSQ = 0.05/DELX**2
NM1 = NEQ - 1
DO 10 I = 1,NEQ
10 Y(I) = 0.0
Y(11) = 0.5
DO 15 I = 12,30
15 Y(I) = 1.0
Y(31) = 0.5
T = 0.0
TOUT = 0.1
ISTATE = 0
DO 30 IO = 1,4
CALL DLSODIS (RESID, ADDASP, JACSP, NEQ, Y, YDOTI, T, TOUT,
1 ITOL, RTOL, ATOL, ITASK, ISTATE, IOPT, RW, LRW, IW, LIW, MF)
WRITE(6,20) T,IW(11),RW(11)
20 FORMAT(' At t =',F5.2,' No. steps =',I4,
1 ' Last step =',D12.4)
IF (ISTATE .NE. 2) GO TO 90
TOUT = TOUT + 0.1
30 CONTINUE
WRITE (6,40) (Y(I),I=1,NEQ)
40 FORMAT(/' Final solution values..'/8(5D12.4/))
WRITE(6,50) IW(17),IW(18),IW(11),IW(12),IW(13)
NNZLU = IW(25) + IW(26) + NEQ
WRITE(6,60) IW(19),NNZLU
50 FORMAT(/' Required RW size =',I5,' IW size =',I4/
1 ' No. steps =',I4,' No. r-s =',I4,' No. J-s =',i4)
60 FORMAT(' No. of nonzeros in P matrix =',I4,
1 ' No. of nonzeros in LU =',I4)
STOP
90 WRITE (6,95) ISTATE
95 FORMAT(///' Error halt.. ISTATE =',I3)
STOP
END
SUBROUTINE GFUN (N, T, Y, G)
DOUBLE PRECISION T, Y, G, R4D, EODSQ
DIMENSION G(N), Y(N)
COMMON /TEST1/ R4D, EODSQ, NM1
G(1) = R4D*(Y(N)**2-Y(2)**2) + EODSQ*(Y(2)-2.0*Y(1)+Y(N))
DO 10 I = 2,NM1
G(I) = R4D*(Y(I-1)**2 - Y(I+1)**2)
1 + EODSQ*(Y(I+1) - 2.0*Y(I) + Y(I-1))
10 CONTINUE
G(N) = R4D*(Y(NM1)**2-Y(1)**2) + EODSQ*(Y(1)-2.0*Y(N)+Y(NM1))
RETURN
END
SUBROUTINE RESID (N, T, Y, S, R, IRES)
DOUBLE PRECISION T, Y, S, R, R4D, EODSQ
DIMENSION Y(N), S(N), R(N)
COMMON /TEST1/ R4D, EODSQ, NM1
CALL GFUN (N, T, Y, R)
R(1) = R(1) - (S(N) + 4.0*S(1) + S(2))/6.0
DO 10 I = 2,NM1
10 R(I) = R(I) - (S(I-1) + 4.0*S(I) + S(I+1))/6.0
R(N) = R(N) - (S(NM1) + 4.0*S(N) + S(1))/6.0
RETURN
END
SUBROUTINE ADDASP (N, T, Y, J, IP, JP, P)
DOUBLE PRECISION T, Y, P
DIMENSION Y(N), IP(*), JP(*), P(N)
JM1 = J - 1
JP1 = J + 1
IF (J .EQ. N) JP1 = 1
IF (J .EQ. 1) JM1 = N
P(J) = P(J) + (2.0/3.0)
P(JP1) = P(JP1) + (1.0/6.0)
P(JM1) = P(JM1) + (1.0/6.0)
RETURN
END
SUBROUTINE JACSP (N, T, Y, S, J, IP, JP, PDJ)
DOUBLE PRECISION T, Y, S, PDJ, R4D, EODSQ
DIMENSION Y(N), S(N), IP(*), JP(*), PDJ(N)
COMMON /TEST1/ R4D, EODSQ, NM1
JM1 = J - 1
JP1 = J + 1
IF (J .EQ. 1) JM1 = N
IF (J .EQ. N) JP1 = 1
PDJ(JM1) = -2.0*R4D*Y(J) + EODSQ
PDJ(J) = -2.0*EODSQ
PDJ(JP1) = 2.0*R4D*Y(J) + EODSQ
RETURN
END
The output of this program (on a CDC-7600 in single precision)
is as follows:
At t = 0.10 No. steps = 15 Last step = 1.6863e-02
At t = 0.20 No. steps = 19 Last step = 2.4101e-02
At t = 0.30 No. steps = 22 Last step = 4.3143e-02
At t = 0.40 No. steps = 24 Last step = 5.7819e-02
Final solution values..
1.8371e-02 1.3578e-02 1.5864e-02 2.3805e-02 3.7245e-02
5.6630e-02 8.2538e-02 1.1538e-01 1.5522e-01 2.0172e-01
2.5414e-01 3.1150e-01 3.7259e-01 4.3608e-01 5.0060e-01
5.6482e-01 6.2751e-01 6.8758e-01 7.4415e-01 7.9646e-01
8.4363e-01 8.8462e-01 9.1853e-01 9.4500e-01 9.6433e-01
9.7730e-01 9.8464e-01 9.8645e-01 9.8138e-01 9.6584e-01
9.3336e-01 8.7497e-01 7.8213e-01 6.5315e-01 4.9997e-01
3.4672e-01 2.1758e-01 1.2461e-01 6.6208e-02 3.3784e-02
Required RW size = 1409 IW size = 30
No. steps = 24 No. r-s = 33 No. J-s = 8
No. of nonzeros in P matrix = 120 No. of nonzeros in LU = 194
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Full Description of User Interface to DLSODIS.
The user interface to DLSODIS consists of the following parts.
1. The call sequence to Subroutine DLSODIS, 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 DLSODIS 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 DLSODIS package, either of
which the user may replace with his/her own version, if desired.
These relate to the measurement of errors.
-----------------------------------------------------------------------
Part 1. Call Sequence.
The call sequence parameters used for input only are
RES, ADDA, JAC, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK,
IOPT, LRW, LIW, MF,
and those used for both input and output are
Y, T, ISTATE, YDOTI.
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 DLSODIS 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.
RES = the name of the user-supplied subroutine which supplies
the residual vector for the ODE system, defined by
r = g(t,y) - A(t,y) * s
as a function of the scalar t and the vectors
s and y (s approximates dy/dt). This subroutine
is to have the form
SUBROUTINE RES (NEQ, T, Y, S, R, IRES)
DOUBLE PRECISION T, Y(*), S(*), R(*)
where NEQ, T, Y, S, and IRES are input, and R and
IRES are output. Y, S, and R are arrays of length NEQ.
On input, IRES indicates how DLSODIS will use the
returned array R, as follows:
IRES = 1 means that DLSODIS needs the full residual,
r = g - A*s, exactly.
IRES = -1 means that DLSODIS is using R only to compute
the Jacobian dr/dy by difference quotients.
The RES routine can ignore IRES, or it can omit some terms
if IRES = -1. If A does not depend on y, then RES can
just return R = g when IRES = -1. If g - A*s contains other
additive terms that are independent of y, these can also be
dropped, if done consistently, when IRES = -1.
The subroutine should set the flag IRES if it
encounters a halt condition or illegal input.
Otherwise, it should not reset IRES. On output,
IRES = 1 or -1 represents a normal return, and
DLSODIS continues integrating the ODE. Leave IRES
unchanged from its input value.
IRES = 2 tells DLSODIS to immediately return control
to the calling program, with ISTATE = 3. This lets
the calling program change parameters of the problem
if necessary.
IRES = 3 represents an error condition (for example, an
illegal value of y). DLSODIS tries to integrate the system
without getting IRES = 3 from RES. If it cannot, DLSODIS
returns with ISTATE = -7 or -1.
On a return with ISTATE = 3, -1, or -7, the values
of T and Y returned correspond to the last point reached
successfully without getting the flag IRES = 2 or 3.
The flag values IRES = 2 and 3 should not be used to
handle switches or root-stop conditions. This is better
done by calling DLSODIS in a one-step mode and checking the
stopping function for a sign change at each step.
If quantities computed in the RES routine are needed
externally to DLSODIS, an extra call to RES should be made
for this purpose, for consistent and accurate results.
To get the current dy/dt for the S argument, use DINTDY.
RES must be declared External in the calling
program. See note below for more about RES.
ADDA = the name of the user-supplied subroutine which adds the
matrix A = A(t,y) to another matrix stored in sparse form.
This subroutine is to have the form
SUBROUTINE ADDA (NEQ, T, Y, J, IAN, JAN, P)
DOUBLE PRECISION T, Y(*), P(*)
INTEGER IAN(*), JAN(*)
where NEQ, T, Y, J, IAN, JAN, and P are input. This routine
should add the J-th column of matrix A to the array P, of
length NEQ. Thus a(i,J) is to be added to P(i) for all
relevant values of i. Here T and Y have the same meaning as
in Subroutine RES, and J is a column index (1 to NEQ).
IAN and JAN are undefined in calls to ADDA for structure
determination (MOSS .ne. 0). Otherwise, IAN and JAN are
structure descriptors, as defined under optional outputs
below, and so can be used to determine the relevant row
indices i, if desired.
Calls to ADDA are made with J = 1,...,NEQ, in that
order. ADDA must not alter its input arguments.
ADDA must be declared External in the calling program.
See note below for more information about ADDA.
JAC = the name of the user-supplied subroutine which supplies
the Jacobian matrix, dr/dy, where r = g - A*s. JAC is
required if MITER = 1, or MOSS = 1 or 3. Otherwise a dummy
name can be passed. This subroutine is to have the form
SUBROUTINE JAC (NEQ, T, Y, S, J, IAN, JAN, PDJ)
DOUBLE PRECISION T, Y(*), S(*), PDJ(*)
INTEGER IAN(*), JAN(*)
where NEQ, T, Y, S, J, IAN, and JAN are input. The
array PDJ, of length NEQ, is to be loaded with column J
of the Jacobian on output. Thus dr(i)/dy(J) is to be
loaded into PDJ(i) for all relevant values of i.
Here T, Y, and S have the same meaning as in Subroutine RES,
and J is a column index (1 to NEQ). IAN and JAN
are undefined in calls to JAC for structure determination
(MOSS .ne. 0). Otherwise, IAN and JAN are structure
descriptors, as defined under optional outputs below, and
so can be used to determine the relevant row indices i, if
desired.
JAC need not provide dr/dy exactly. A crude
approximation (possibly with greater sparsity) will do.
In any case, PDJ is preset to zero by the solver,
so that only the nonzero elements need be loaded by JAC.
Calls to JAC are made with J = 1,...,NEQ, in that order, and
each such set of calls is preceded by a call to RES with the
same arguments NEQ, T, Y, S, and IRES. Thus to gain some
efficiency intermediate quantities shared by both calculations
may be saved in a user Common block by RES and not recomputed
by JAC, if desired. JAC must not alter its input arguments.
JAC must be declared External in the calling program.
See note below for more about JAC.
Note on RES, ADDA, and JAC:
These subroutines may access user-defined quantities in
NEQ(2),... and/or in Y(NEQ(1)+1),... if NEQ is an array
(dimensioned in the subroutines) and/or Y has length
exceeding NEQ(1). However, these subroutines should not
alter NEQ(1), Y(1),...,Y(NEQ) or any other input variables.
See the descriptions of NEQ and Y below.
NEQ = the size of the system (number of first order ordinary
differential equations or scalar algebraic 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 RES, ADDA, or JAC.
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 DLSODIS package accesses only NEQ(1).) In either case,
this parameter is passed as the NEQ argument in all calls
to RES, ADDA, and JAC. Hence, if it is an array,
locations NEQ(2),... may be used to store other integer data
and pass it to RES, ADDA, or JAC. Each such subroutine
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 = 0 or 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 RES,
ADDA, and JAC. Hence its length may exceed NEQ,
and locations Y(NEQ+1),... may be used to store other real
data and pass it to RES, ADDA, or JAC. (The DLSODIS
package accesses only Y(1),...,Y(NEQ). )
YDOTI = a real array for the initial value of the vector
dy/dt and for work space, of dimension at least NEQ.
On input:
If ISTATE = 0 then DLSODIS will compute the initial value
of dy/dt, if A is nonsingular. Thus YDOTI will
serve only as work space and may have any value.
If ISTATE = 1 then YDOTI must contain the initial value
of dy/dt.
If ISTATE = 2 or 3 (continuation calls) then YDOTI
may have any value.
Note: If the initial value of A is singular, then
DLSODIS cannot compute the initial value of dy/dt, so
it must be provided in YDOTI, with ISTATE = 1.
On output, when DLSODIS terminates abnormally with ISTATE =
-1, -4, or -5, YDOTI will contain the residual
r = g(t,y) - A(t,y)*(dy/dt). If r is large, t is near
its initial value, and YDOTI is supplied with ISTATE = 1,
there may have been an incorrect input value of
YDOTI = dy/dt, or the problem (as given to DLSODIS)
may not have a solution.
If desired, the YDOTI array may be used for other
purposes between calls to the solver.
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 = 0 or 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 scalar 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
state of the calculation.
On input, the values of ISTATE are as follows.
0 means this is the first call for the problem, and
DLSODIS is to compute the initial value of dy/dt
(while doing other initializations). See note below.
1 means this is the first call for the problem, and
the initial value of dy/dt has been supplied in
YDOTI (DLSODIS will do other initializations).
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,
the conditional inputs IA, JA, IC, and JC,
and any of the optional inputs except H0.
A call with ISTATE = 3 will cause the sparsity
structure of the problem to be recomputed.
(Structure information is reread from IA and JA if
MOSS = 0, 3, or 4 and from IC and JC if MOSS = 0).
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 = 0 or 1 on input.
On output, ISTATE has the following values and meanings.
0 or 1 means nothing was done; TOUT = T and
ISTATE = 0 or 1 on input.
2 means that the integration was performed successfully.
3 means that the user-supplied Subroutine RES signalled
DLSODIS to halt the integration and return (IRES = 2).
Integration as far as T was achieved with no occurrence
of IRES = 2, but this flag was set on attempting the
next step.
-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.
This may be caused by an inaccurate Jacobian matrix.
-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 that the user-supplied Subroutine RES set
its error flag (IRES = 3) despite repeated tries by
DLSODIS to avoid that condition.
-8 means that ISTATE was 0 on input but DLSODIS was unable
to compute the initial value of dy/dt. See the
printed message for details.
-9 means a fatal error return flag came from the sparse
solver CDRV by way of DPRJIS or DSOLSS (numerical
factorization or backsolve). This should never happen.
The integration was successful as far as T.
Note: An error return with ISTATE = -1, -4, or -5
may mean that the sparsity structure of the
problem has changed significantly since it was last
determined (or input). In that case, one can attempt to
complete the integration by setting ISTATE = 3 on the next
call, so that a new structure determination is done.
Note: Since the normal output value of ISTATE is 2,
it does not need to be reset for normal continuation.
similarly, ISTATE (= 3) need not be reset if RES told
DLSODIS to return because the calling program must change
the parameters of the problem.
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 work array used for a mixture of real (double precision)
and integer work space.
The length of RWORK (in real words) must be at least
20 + NYH*(MAXORD + 1) + 3*NEQ + LWM 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),
LWM = 2*NNZ + 2*NEQ + (NNZ+9*NEQ)/LENRAT if MITER = 1,
LWM = 2*NNZ + 2*NEQ + (NNZ+10*NEQ)/LENRAT if MITER = 2.
in the above formulas,
NNZ = number of nonzero elements in the iteration matrix
P = A - con*J (con is a constant and J is the
Jacobian matrix dr/dy).
LENRAT = the real to integer wordlength ratio (usually 1 in
single precision and 2 in double precision).
(See the MF description for METH and MITER.)
Thus if MAXORD has its default value and NEQ is constant,
the minimum length of RWORK is:
20 + 16*NEQ + LWM for MF = 11, 111, 311, 12, 212, 412,
20 + 9*NEQ + LWM for MF = 21, 121, 321, 22, 222, 422.
The above formula for LWM is only a crude lower bound.
The required length of RWORK cannot be readily predicted
in general, as it depends on the sparsity structure
of the problem. Some experimentation may be necessary.
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
32 + 2*NEQ + NZA + NZC for MOSS = 0,
30 for MOSS = 1 or 2,
31 + NEQ + NZA for MOSS = 3 or 4.
(NZA is the number of nonzero elements in matrix A, and
NZC is the number of nonzero elements in dr/dy.)
In DLSODIS, IWORK is used for conditional and
optional inputs and optional outputs.
The following two blocks of words in IWORK are conditional
inputs, required if MOSS = 0, 3, or 4, but not otherwise
(see the description of MF for MOSS).
IWORK(30+j) = IA(j) (j=1,...,NEQ+1)
IWORK(31+NEQ+k) = JA(k) (k=1,...,NZA)
The two arrays IA and JA describe the sparsity structure
to be assumed for the matrix A. JA contains the row
indices where nonzero elements occur, reading in columnwise
order, and IA contains the starting locations in JA of the
descriptions of columns 1,...,NEQ, in that order, with
IA(1) = 1. Thus, for each column index j = 1,...,NEQ, the
values of the row index i in column j where a nonzero
element may occur are given by
i = JA(k), where IA(j) .le. k .lt. IA(j+1).
If NZA is the total number of nonzero locations assumed,
then the length of the JA array is NZA, and IA(NEQ+1) must
be NZA + 1. Duplicate entries are not allowed.
The following additional blocks of words are required
if MOSS = 0, but not otherwise. If LC = 31 + NEQ + NZA, then
IWORK(LC+j) = IC(j) (j=1,...,NEQ+1), and
IWORK(LC+NEQ+1+k) = JC(k) (k=1,...,NZC)
The two arrays IC and JC describe the sparsity
structure to be assumed for the Jacobian matrix dr/dy.
They are used in the same manner as the above IA and JA
arrays. If NZC is the number of nonzero locations
assumed, then the length of the JC array is NZC, and
IC(NEQ+1) must be NZC + 1. Duplicate entries are not
allowed.
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 DLSODIS
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 DLSODIS between calls, if
desired (but not for use by RES, ADDA, or JAC).
MF = the method flag. Used only for input.
MF has three decimal digits-- MOSS, METH, and MITER.
For standard options:
MF = 100*MOSS + 10*METH + MITER.
MOSS indicates the method to be used to obtain the sparsity
structure of the Jacobian matrix:
MOSS = 0 means the user has supplied IA, JA, IC, and JC
(see descriptions under IWORK above).
MOSS = 1 means the user has supplied JAC (see below) and
the structure will be obtained from NEQ initial
calls to JAC and NEQ initial calls to ADDA.
MOSS = 2 means the structure will be obtained from NEQ+1
initial calls to RES and NEQ initial calls to ADDA
MOSS = 3 like MOSS = 1, except user has supplied IA and JA.
MOSS = 4 like MOSS = 2, except user has supplied IA and JA.
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).
The BDF method is strongly preferred for stiff problems,
while the Adams method is preferred when the problem is
not stiff. If the matrix A(t,y) is nonsingular,
stiffness here can be taken to mean that of the explicit
ODE system dy/dt = A-inverse * g. If A is singular,
the concept of stiffness is not well defined.
If you do not know whether the problem is stiff, we
recommend using METH = 2. If it is stiff, the advantage
of METH = 2 over METH = 1 will be great, while if it is
not stiff, the advantage of METH = 1 will be slight.
If maximum efficiency is important, some experimentation
with METH may be necessary.
MITER indicates the corrector iteration method:
MITER = 1 means chord iteration with a user-supplied
sparse Jacobian, given by Subroutine JAC.
MITER = 2 means chord iteration with an internally
generated (difference quotient) sparse
Jacobian (using NGP extra calls to RES per
dr/dy value, where NGP is an optional
output described below.)
If MITER = 1 or MOSS = 1 or 3 the user must supply a
Subroutine JAC (the name is arbitrary) as described above
under JAC. Otherwise, a dummy argument can be used.
The standard choices for MF are:
MF = 21 or 22 for a stiff problem with IA/JA and IC/JC
supplied,
MF = 121 for a stiff problem with JAC supplied, but not
IA/JA or IC/JC,
MF = 222 for a stiff problem with neither IA/JA, IC/JC/,
nor JAC supplied,
MF = 321 for a stiff problem with IA/JA and JAC supplied,
but not IC/JC,
MF = 422 for a stiff problem with IA/JA supplied, but not
IC/JC or JAC.
The sparseness structure can be changed during the problem
by making a call to DLSODIS with ISTATE = 3.
-----------------------------------------------------------------------
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.)
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.
-----------------------------------------------------------------------
Optional Outputs.
As optional additional output from DLSODIS, the variables listed
below are quantities related to the performance of DLSODIS
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 DLSODIS, and on any return with
ISTATE = -1, -2, -4, -5, -6, or -7. On a return with -3 (illegal
input) or -8, 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.
NRE IWORK(12) the number of residual evaluations (RES calls)
for the problem so far, excluding those for
structure determination (MOSS = 2 or 4).
NJE IWORK(13) the number of Jacobian evaluations (each involving
an evaluation of A and dr/dy) for the problem so
far, excluding those for structure determination
(MOSS = 1 or 3). This equals the number of calls
to ADDA and (if MITER = 1) JAC.
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.
NNZ IWORK(19) the number of nonzero elements in the iteration
matrix P = A - con*J (con is a constant and
J is the Jacobian matrix dr/dy).
NGP IWORK(20) the number of groups of column indices, used in
difference quotient Jacobian aproximations if
MITER = 2. This is also the number of extra RES
evaluations needed for each Jacobian evaluation.
NLU IWORK(21) the number of sparse LU decompositions for the
problem so far. (Excludes the LU decomposition
necessary when ISTATE = 0.)
LYH IWORK(22) the base address in RWORK of the history array YH,
described below in this list.
IPIAN IWORK(23) the base address of the structure descriptor array
IAN, described below in this list.
IPJAN IWORK(24) the base address of the structure descriptor array
JAN, described below in this list.
NZL IWORK(25) the number of nonzero elements in the strict lower
triangle of the LU factorization used in the chord
iteration.
NZU IWORK(26) the number of nonzero elements in the strict upper
triangle of the LU factorization used in the chord
iteration. The total number of nonzeros in the
factorization is therefore NZL + NZU + NEQ.
The following four 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, and its description.
For YH and ACOR, the base addresses are in RWORK (a real array).
The integer arrays IAN and JAN are to be obtained by declaring an
integer array IWK and identifying IWK(1) with RWORK(21), using either
an equivalence statement or a subroutine call. Then the base
addresses IPIAN (of IAN) and IPJAN (of JAN) in IWK are to be obtained
as optional outputs IWORK(23) and IWORK(24), respectively.
Thus IAN(1) is IWK(ipian), etc.
Name Base Address Description
IAN IPIAN (in IWK) structure descriptor array of size NEQ + 1.
JAN IPJAN (in IWK) structure descriptor array of size NNZ.
(see above) IAN and JAN together describe the sparsity
structure of the iteration matrix
P = A - con*J, as used by DLSODIS.
JAN contains the row indices of the nonzero
locations, reading in columnwise order, and
IAN contains the starting locations in JAN of
the descriptions of columns 1,...,NEQ, in
that order, with IAN(1) = 1. Thus for each
j = 1,...,NEQ, the row indices i of the
nonzero locations in column j are
i = JAN(k), IAN(j) .le. k .lt. IAN(j+1).
Note that IAN(NEQ+1) = NNZ + 1.
YH LYH the Nordsieck history array, of size NYH by
(optional (NQCUR + 1), where NYH is the initial value
output) 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. The base address LYH
is another optional output, listed above.
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 return from DLSODIS with ISTATE = 2.
-----------------------------------------------------------------------
Part 2. Other Routines Callable.
The following are optional calls which the user may make to
gain additional capabilities in conjunction with DLSODIS.
(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 DLSODIS, 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 DLSODIS.
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 DSRCMS(RSAV,ISAV,JOB) saves and restores the contents of
the internal Common blocks used by
DLSODIS (see Part 3 below).
RSAV must be a real array of length 224
or more, and ISAV must be an integer
array of length 71 or more.
JOB=1 means save Common into RSAV/ISAV.
JOB=2 means restore Common from RSAV/ISAV.
DSRCMS is useful if one is
interrupting a run and restarting
later, or alternating between two or
more problems solved with DLSODIS.
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 DLSODIS.
The detailed instructions for using DINTDY are as follows.
The form of the call is:
LYH = IWORK(22)
CALL DINTDY (T, K, RWORK(LYH), 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 DLSODIS).
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 DLSODIS directly. Since NQCUR .ge. 1, the first
derivative dy/dt is always available with DINTDY.
LYH = the base address of the history array YH, obtained
as an optional output as shown above.
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 DLSODIS is to be used in an overlay situation, the user
must declare, in the primary overlay, the variables in:
(1) the call sequence to DLSODIS, and
(2) the two internal Common blocks
/DLS001/ of length 255 (218 double precision words
followed by 37 integer words),
/DLSS01/ of length 40 (6 double precision words
followed by 34 integer words).
If DLSODIS 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 DLSODIS 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 DLSODIS 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 DLSODIS call for that problem. To save and restore the Common
blocks, use Subroutines DSRCMS (see Part 2 above).
-----------------------------------------------------------------------
Part 4. Optionally Replaceable Solver Routines.
Below are descriptions of two routines in the DLSODIS 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 DLSODIS 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 DLSODIS in the computation
of the optional output IMXER, 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 DLSODIS.
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)
19820714 DATE WRITTEN
19830812 Major update, based on recent LSODI and LSODES revisions:
Upgraded MDI in ODRV package: operates on M + M-transpose.
Numerous revisions in use of work arrays;
use wordlength ratio LENRAT; added IPISP & LRAT to Common;
added optional outputs IPIAN/IPJAN;
Added routine CNTNZU; added NZL and NZU to /LSS001/;
changed ADJLR call logic; added optional outputs NZL & NZU;
revised counter initializations; revised PREPI stmt. nos.;
revised difference quotient increment;
eliminated block /LSI001/, using IERPJ flag;
revised STODI logic after PJAC return;
revised tuning of H change and step attempts in STODI;
corrections to main prologue and comments throughout.
19870320 Corrected jump on test of umax in CDRV routine.
20010125 Numerous revisions: corrected comments throughout;
removed TRET from Common; rewrote EWSET with 4 loops;
fixed t test in INTDY; added Cray directives in STODI;
in STODI, fixed DELP init. and logic around PJAC call;
combined routines to save/restore Common;
passed LEVEL = 0 in error message calls (except run abort).
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;
converted arithmetic IF statements to logical IF statements;
numerous corrections to prologues and internal comments.
20010507 Converted single precision source to double precision.
20020502 Corrected declarations in descriptions of user routines.
20031021 Fixed address offset bugs in Subroutine DPREPI.
20031027 Changed 0. to 0.0D0 in Subroutine DPREPI.
20031105 Restored 'own' variables to Common blocks, to enable
interrupt/restart feature.
20031112 Added SAVE statements for data-loaded constants.
20031117 Changed internal names NRE, LSAVR to NFE, LSAVF resp.
-----------------------------------------------------------------------
Other routines in the DLSODIS package.
In addition to Subroutine DLSODIS, the DLSODIS package includes the
following subroutines and function routines:
DIPREPI acts as an interface between DLSODIS and DPREPI, and also
does adjusting of work space pointers and work arrays.
DPREPI is called by DIPREPI to compute sparsity and do sparse
matrix preprocessing.
DAINVGS computes the initial value of the vector
dy/dt = A-inverse * g
ADJLR adjusts the length of required sparse matrix work space.
It is called by DPREPI.
CNTNZU is called by DPREPI and counts the nonzero elements in the
strict upper triangle of P + P-transpose.
JGROUP is called by DPREPI to compute groups of Jacobian column
indices for use when MITER = 2.
DINTDY computes an interpolated value of the y vector at t = TOUT.
DSTODI is the core integrator, which does one step of the
integration and the associated error control.
DCFODE sets all method coefficients and test constants.
DPRJIS computes and preprocesses the Jacobian matrix J = dr/dy
and the Newton iteration matrix P = A - h*l0*J.
DSOLSS manages solution of linear system in chord iteration.
DEWSET sets the error weight vector EWT before each step.
DVNORM computes the weighted RMS-norm of a vector.
DSRCMS is a user-callable routine to save and restore
the contents of the internal Common blocks.
ODRV constructs a reordering of the rows and columns of
a matrix by the minimum degree algorithm. ODRV is a
driver routine which calls Subroutines MD, MDI, MDM,
MDP, MDU, and SRO. See Ref. 2 for details. (The ODRV
module has been modified since Ref. 2, however.)
CDRV performs reordering, symbolic factorization, numerical
factorization, or linear system solution operations,
depending on a path argument IPATH. CDRV is a
driver routine which calls Subroutines NROC, NSFC,
NNFC, NNSC, and NNTC. See Ref. 3 for details.
DLSODIS uses CDRV to solve linear systems in which the
coefficient matrix is P = A - con*J, where A is the
matrix for the linear system A(t,y)*dy/dt = g(t,y),
con is a scalar, and J is an approximation to
the Jacobian dr/dy. Because CDRV deals with rowwise
sparsity descriptions, CDRV works with P-transpose, not P.
DLSODIS also uses CDRV to solve the linear system
A(t,y)*dy/dt = g(t,y) for dy/dt when ISTATE = 0.
(For this, CDRV works with A-transpose, not A.)
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, DUMACH, IXSAV, and IUMACH are function routines.
All the others are subroutines.