dlsoda
This is the 12 November 2003 version of
DLSODA: Livermore Solver for Ordinary Differential Equations, with
Automatic method switching for stiff and nonstiff problems.
This version is in double precision.
DLSODA 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).
This a variant version of the DLSODE package.
It switches automatically between stiff and nonstiff methods.
This means that the user does not have to determine whether the
problem is stiff or not, and the solver will automatically choose the
appropriate method. It always starts with the nonstiff method.
Authors: Alan C. Hindmarsh
Center for Applied Scientific Computing, L-561
Lawrence Livermore National Laboratory
Livermore, CA 94551
and
Linda R. Petzold
Univ. of California at Santa Barbara
Dept. of Computer Science
Santa Barbara, CA 93106
References:
1. 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.
2. Linda R. Petzold, Automatic Selection of Methods for Solving
Stiff and Nonstiff Systems of Ordinary Differential Equations,
Siam J. Sci. Stat. Comput. 4 (1983), pp. 136-148.
Summary of Usage
Communication between the user and the DLSODA 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 alternative treatment of the Jacobian matrix,
optional inputs and outputs, 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 F (NEQ, T, Y, YDOT)
DOUBLE PRECISION T, Y(*), YDOT(*)
which supplies the vector function f by loading YDOT(i) with f(i).
B. Write a main program which calls Subroutine DLSODA 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 DLSODA. On the first call to DLSODA, 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 less 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:
22 + NEQ * MAX(16, NEQ + 9).
See also Paragraph E below.
LRW = declared length of RWORK (in user's dimension).
IWORK = integer work array of length at least 20 + NEQ.
LIW = declared length of IWORK (in user's dimension).
JAC = name of subroutine for Jacobian matrix.
Use a dummy name. See also Paragraph E below.
JT = Jacobian type indicator. Set JT = 2.
See also Paragraph E below.
Note that the main program must declare arrays Y, RWORK, IWORK,
and possibly ATOL.
C. 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 DLSODA was successful, negative otherwise.
-1 means excess work done on this call (perhaps wrong JT).
-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 JT or tolerances).
-6 means error weight became zero during problem. (Solution
component i vanished, and ATOL or ATOL(i) = 0.)
-7 means work space insufficient to finish (see messages).
D. To continue the integration after a successful return, simply
reset TOUT and call DLSODA again. No other parameters need be reset.
E. Note: If and when DLSODA regards the problem as stiff, and
switches methods accordingly, it must make use of the NEQ by NEQ
Jacobian matrix, J = df/dy. For the sake of simplicity, the
inputs to DLSODA recommended in Paragraph B above cause DLSODA to
treat J as a full matrix, and to approximate it internally by
difference quotients. Alternatively, J can be treated as a band
matrix (with great potential reduction in the size of the RWORK
array). Also, in either the full or banded case, the user can supply
J in closed form, with a routine whose name is passed as the JAC
argument. These alternatives are described in the paragraphs on
RWORK, JAC, and JT in the full description of the call sequence below.
Example Problem
The following is a simple example problem, with the coding
needed for its solution by DLSODA. The problem is from chemical
kinetics, and consists of the following three rate equations:
dy1/dt = -.04*y1 + 1.e4*y2*y3
dy2/dt = .04*y1 - 1.e4*y2*y3 - 3.e7*y2**2
dy3/dt = 3.e7*y2**2
on the interval from t = 0.0 to t = 4.e10, with initial conditions
y1 = 1.0, y2 = y3 = 0. The problem is stiff.
The following coding solves this problem with DLSODA,
printing results at t = .4, 4., ..., 4.e10. It uses
ITOL = 2 and ATOL much smaller for y2 than y1 or y3 because
y2 has much smaller values.
At the end of the run, statistical quantities of interest are
printed (see optional outputs in the full description below).
EXTERNAL FEX
DOUBLE PRECISION ATOL, RTOL, RWORK, T, TOUT, Y
DIMENSION Y(3), ATOL(3), RWORK(70), IWORK(23)
NEQ = 3
Y(1) = 1.
Y(2) = 0.
Y(3) = 0.
T = 0.
TOUT = .4
ITOL = 2
RTOL = 1.D-4
ATOL(1) = 1.D-6
ATOL(2) = 1.D-10
ATOL(3) = 1.D-6
ITASK = 1
ISTATE = 1
IOPT = 0
LRW = 70
LIW = 23
JT = 2
DO 40 IOUT = 1,12
CALL DLSODA(FEX,NEQ,Y,T,TOUT,ITOL,RTOL,ATOL,ITASK,ISTATE,
1 IOPT,RWORK,LRW,IWORK,LIW,JDUM,JT)
WRITE(6,20)T,Y(1),Y(2),Y(3)
20 FORMAT(' At t =',D12.4,' Y =',3D14.6)
IF (ISTATE .LT. 0) GO TO 80
40 TOUT = TOUT*10.
WRITE(6,60)IWORK(11),IWORK(12),IWORK(13),IWORK(19),RWORK(15)
60 FORMAT(/' No. steps =',I4,' No. f-s =',I4,' No. J-s =',I4/
1 ' Method last used =',I2,' Last switch was at t =',D12.4)
STOP
80 WRITE(6,90)ISTATE
90 FORMAT(///' Error halt.. ISTATE =',I3)
STOP
END
SUBROUTINE FEX (NEQ, T, Y, YDOT)
DOUBLE PRECISION T, Y, YDOT
DIMENSION Y(3), YDOT(3)
YDOT(1) = -.04*Y(1) + 1.D4*Y(2)*Y(3)
YDOT(3) = 3.D7*Y(2)*Y(2)
YDOT(2) = -YDOT(1) - YDOT(3)
RETURN
END
The output of this program (on a CDC-7600 in single precision)
is as follows:
At t = 4.0000e-01 y = 9.851712e-01 3.386380e-05 1.479493e-02
At t = 4.0000e+00 Y = 9.055333e-01 2.240655e-05 9.444430e-02
At t = 4.0000e+01 Y = 7.158403e-01 9.186334e-06 2.841505e-01
At t = 4.0000e+02 Y = 4.505250e-01 3.222964e-06 5.494717e-01
At t = 4.0000e+03 Y = 1.831975e-01 8.941774e-07 8.168016e-01
At t = 4.0000e+04 Y = 3.898730e-02 1.621940e-07 9.610125e-01
At t = 4.0000e+05 Y = 4.936363e-03 1.984221e-08 9.950636e-01
At t = 4.0000e+06 Y = 5.161831e-04 2.065786e-09 9.994838e-01
At t = 4.0000e+07 Y = 5.179817e-05 2.072032e-10 9.999482e-01
At t = 4.0000e+08 Y = 5.283401e-06 2.113371e-11 9.999947e-01
At t = 4.0000e+09 Y = 4.659031e-07 1.863613e-12 9.999995e-01
At t = 4.0000e+10 Y = 1.404280e-08 5.617126e-14 1.000000e+00
No. steps = 361 No. f-s = 693 No. J-s = 64
Method last used = 2 Last switch was at t = 6.0092e-03
Full description of user interface to DLSODA
The user interface to DLSODA consists of the following parts.
1. The call sequence to Subroutine DLSODA, 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 DLSODA 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 a subroutine in the DLSODA package,
which the user may replace with his/her own version, if desired.
this relates to the measurement of errors.
Part 1. Call Sequence
The call sequence parameters used for input only are
F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, JT,
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 DLSODA 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 DLSODA, 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 F and/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 DLSODA package accesses only NEQ(1).) In either case,
this parameter is passed as the NEQ argument in all calls
to F and JAC. Hence, if it is an array, locations
NEQ(2),... may be used to store other integer data and pass
it to F and/or JAC. Subroutines F and/or JAC 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 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 F and/or JAC. (The DLSODA 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
max-norm of ( E(i)/EWT(i) ) .le. 1,
where EWT = (EWT(i)) is a vector of positive error weights.
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 a
user-supplied routine for the setting of EWT.
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, JT, ML, MU,
and any optional inputs except H0, MXORDN, and MXORDS.
(See IWORK description for ML and MU.)
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.
This may be caused by an inaccurate Jacobian matrix,
if one is being used.
-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 length of RWORK and/or IWORK was too small to
proceed, but the integration was successful as far as T.
This happens when DLSODA chooses to switch methods
but LRW and/or LIW is too small for the new method.
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 array (double precision) for work space, and (in the
first 20 words) for conditional and optional inputs and
optional outputs.
As DLSODA switches automatically between stiff and nonstiff
methods, the required length of RWORK can change during the
problem. Thus the RWORK array passed to DLSODA can either
have a static (fixed) length large enough for both methods,
or have a dynamic (changing) length altered by the calling
program in response to output from DLSODA.
--- Fixed Length Case ---
If the RWORK length is to be fixed, it should be at least
MAX (LRN, LRS),
where LRN and LRS are the RWORK lengths required when the
current method is nonstiff or stiff, respectively.
The separate RWORK length requirements LRN and LRS are
as follows:
IF NEQ is constant and the maximum method orders have
their default values, then
LRN = 20 + 16*NEQ,
LRS = 22 + 9*NEQ + NEQ**2 if JT = 1 or 2,
LRS = 22 + 10*NEQ + (2*ML+MU)*NEQ if JT = 4 or 5.
Under any other conditions, LRN and LRS are given by:
LRN = 20 + NYH*(MXORDN+1) + 3*NEQ,
LRS = 20 + NYH*(MXORDS+1) + 3*NEQ + LMAT,
where
NYH = the initial value of NEQ,
MXORDN = 12, unless a smaller value is given as an
optional input,
MXORDS = 5, unless a smaller value is given as an
optional input,
LMAT = length of matrix work space:
LMAT = NEQ**2 + 2 if JT = 1 or 2,
LMAT = (2*ML + MU + 1)*NEQ + 2 if JT = 4 or 5.
--- Dynamic Length Case ---
If the length of RWORK is to be dynamic, then it should
be at least LRN or LRS, as defined above, depending on the
current method. Initially, it must be at least LRN (since
DLSODA starts with the nonstiff method). On any return
from DLSODA, the optional output MCUR indicates the current
method. If MCUR differs from the value it had on the
previous return, or if there has only been one call to
DLSODA and MCUR is now 2, then DLSODA has switched
methods during the last call, and the length of RWORK
should be reset (to LRN if MCUR = 1, or to LRS if
MCUR = 2). (An increase in the RWORK length is required
if DLSODA returned ISTATE = -7, but not otherwise.)
After resetting the length, call DLSODA with ISTATE = 3
to signal that change.
LRW = the length of the array RWORK, as declared by the user.
(This will be checked by the solver.)
IWORK = an integer array for work space.
As DLSODA switches automatically between stiff and nonstiff
methods, the required length of IWORK can change during
problem, between
LIS = 20 + NEQ and LIN = 20,
respectively. Thus the IWORK array passed to DLSODA can
either have a fixed length of at least 20 + NEQ, or have a
dynamic length of at least LIN or LIS, depending on the
current method. The comments on dynamic length under
RWORK above apply here. Initially, this length need
only be at least LIN = 20.
The first few words of IWORK are used for conditional and
optional inputs and optional outputs.
The following 2 words in IWORK are conditional inputs:
IWORK(1) = ML these are the lower and upper
IWORK(2) = MU half-bandwidths, respectively, of the
banded Jacobian, excluding the main diagonal.
The band is defined by the matrix locations
(i,j) with i-ML .le. j .le. i+MU. ML and MU
must satisfy 0 .le. ML,MU .le. NEQ-1.
These are required if JT is 4 or 5, and
ignored otherwise. ML and MU may in fact be
the band parameters for a matrix to which
df/dy is only approximately equal.
LIW = the length of the array IWORK, as declared by the user.
(This will be checked by the solver.)
Note: The base addresses of the work arrays must not be
altered between calls to DLSODA for the same problem.
The contents of the work arrays must not be altered
between calls, 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 DLSODA between calls, if
desired (but not for use by F or JAC).
JAC = the name of the user-supplied routine to compute the
Jacobian matrix, df/dy, if JT = 1 or 4. The JAC routine
is optional, but if the problem is expected to be stiff much
of the time, you are encouraged to supply JAC, for the sake
of efficiency. (Alternatively, set JT = 2 or 5 to have
DLSODA compute df/dy internally by difference quotients.)
If and when DLSODA uses df/dy, it treats this NEQ by NEQ
matrix either as full (JT = 1 or 2), or as banded (JT =
4 or 5) with half-bandwidths ML and MU (discussed under
IWORK above). In either case, if JT = 1 or 4, the JAC
routine must compute df/dy as a function of the scalar t
and the vector y. It is to have the form
SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
where NEQ, T, Y, ML, MU, and NROWPD are input and the array
PD is to be loaded with partial derivatives (elements of
the Jacobian matrix) on output. PD must be given a first
dimension of NROWPD. T and Y have the same meaning as in
Subroutine F.
In the full matrix case (JT = 1), ML and MU are
ignored, and the Jacobian is to be loaded into PD in
columnwise manner, with df(i)/dy(j) loaded into PD(i,j).
In the band matrix case (JT = 4), the elements
within the band are to be loaded into PD in columnwise
manner, with diagonal lines of df/dy loaded into the rows
of PD. Thus df(i)/dy(j) is to be loaded into PD(i-j+MU+1,j).
ML and MU are the half-bandwidth parameters (see IWORK).
The locations in PD in the two triangular areas which
correspond to nonexistent matrix elements can be ignored
or loaded arbitrarily, as they are overwritten by DLSODA.
JAC need not provide df/dy exactly. A crude
approximation (possibly with a smaller bandwidth) will do.
In either case, PD is preset to zero by the solver,
so that only the nonzero elements need be loaded by JAC.
Each call to JAC is preceded by a call to F with the same
arguments NEQ, T, and Y. Thus to gain some efficiency,
intermediate quantities shared by both calculations may be
saved in a user Common block by F and not recomputed by JAC,
if desired. Also, JAC may alter the Y array, if desired.
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.
JT = Jacobian type indicator. Used only for input.
JT specifies how the Jacobian matrix df/dy will be
treated, if and when DLSODA requires this matrix.
JT has the following values and meanings:
1 means a user-supplied full (NEQ by NEQ) Jacobian.
2 means an internally generated (difference quotient) full
Jacobian (using NEQ extra calls to F per df/dy value).
4 means a user-supplied banded Jacobian.
5 means an internally generated banded Jacobian (using
ML+MU+1 extra calls to F per df/dy evaluation).
If JT = 1 or 4, the user must supply a Subroutine JAC
(the name is arbitrary) as described above under JAC.
If JT = 2 or 5, a dummy argument 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.)
IXPR IWORK(5) flag to generate extra printing at method switches.
IXPR = 0 means no extra printing (the default).
IXPR = 1 means print data on each switch.
T, H, and NST will be printed on the same logical
unit as used for error messages.
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.
MXORDN IWORK(8) the maximum order to be allowed for the nonstiff
(Adams) method. the default value is 12.
if MXORDN exceeds the default value, it will
be reduced to the default value.
MXORDN is held constant during the problem.
MXORDS IWORK(9) the maximum order to be allowed for the stiff
(BDF) method. The default value is 5.
If MXORDS exceeds the default value, it will
be reduced to the default value.
MXORDS is held constant during the problem.
Optional Outputs
As optional additional output from DLSODA, the variables listed
below are quantities related to the performance of DLSODA
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 DLSODA, and on any return with
ISTATE = -1, -2, -4, -5, or -6. 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.)
TSW RWORK(15) the value of t at the time of the last method
switch, if any.
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.
NJE IWORK(13) the number of Jacobian evaluations (and of matrix
LU decompositions) for the problem so far.
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, assuming
that the length of RWORK is to be fixed for the
rest of the problem, and that switching may occur.
This is defined on normal returns and on an illegal
input return for insufficient storage.
LENIW IWORK(18) the length of IWORK actually required, assuming
that the length of IWORK is to be fixed for the
rest of the problem, and that switching may occur.
This is defined on normal returns and on an illegal
input return for insufficient storage.
MUSED IWORK(19) the method indicator for the last successful step:
1 means Adams (nonstiff), 2 means BDF (stiff).
MCUR IWORK(20) the current method indicator:
1 means Adams (nonstiff), 2 means BDF (stiff).
This is the method to be attempted
on the next step. Thus it differs from MUSED
only if a method switch has just been made.
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 LACOR array of size NEQ used for the accumulated
(from Common corrections on each step, scaled on output
as noted) 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
DLSODA. The base address LACOR is obtained by
including in the user's program the
following 2 lines:
COMMON /DLS001/ RLS(218), ILS(37)
LACOR = ILS(22)
Part 2. Other Routines Callable
The following are optional calls which the user may make to
gain additional capabilities in conjunction with DLSODA.
(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 DLSODA, 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 DLSODA.
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 DSRCMA(RSAV,ISAV,JOB) saves and restores the contents of
the internal Common blocks used by
DLSODA (see Part 3 below).
RSAV must be a real array of length 240
or more, and ISAV must be an integer
array of length 46 or more.
JOB=1 means save Common into RSAV/ISAV.
JOB=2 means restore Common from RSAV/ISAV.
DSRCMA is useful if one is
interrupting a run and restarting
later, or alternating between two or
more problems solved with DLSODA.
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 DLSODA.
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 DLSODA).
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 DLSODA 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 DLSODA is to be used in an overlay situation, the user
must declare, in the primary overlay, the variables in:
(1) the call sequence to DLSODA, and
(2) the two internal Common blocks
/DLS001/ of length 255 (218 double precision words
followed by 37 integer words),
/DLSA01/ of length 31 (22 double precision words
followed by 9 integer words).
If DLSODA 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 DLSODA 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 DLSODA 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 DLSODA call for that problem. To save and restore the Common
blocks, use Subroutine DSRCMA (see Part 2 above).
-----------------------------------------------------------------------
Part 4. Optionally Replaceable Solver Routines
Below is a description of a routine in the DLSODA package which
relates to the measurement of errors, and 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 DLSODA 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
DMNORM routine, and also used by DLSODA 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).
REVISION HISTORY (YYYYMMDD)
19811102 DATE WRITTEN
19820126 Fixed bug in tests of work space lengths;
minor corrections in main prologue and comments.
19870330 Major update: corrected comments throughout;
removed TRET from Common; rewrote EWSET with 4 loops;
fixed t test in INTDY; added Cray directives in STODA;
in STODA, fixed DELP init. and logic around PJAC call;
combined routines to save/restore Common;
passed LEVEL = 0 in error message calls (except run abort).
19970225 Fixed lines setting JSTART = -2 in Subroutine LSODA.
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.
20010613 Revised excess accuracy test (to match rest of ODEPACK).
20010808 Fixed bug in DPRJA (matrix in DBNORM call).
20020502 Corrected declarations in descriptions of user routines.
20031105 Restored 'own' variables to Common blocks, to enable
interrupt/restart feature.
20031112 Added SAVE statements for data-loaded constants.
Other routines in the DLSODA package.
In addition to Subroutine DLSODA, the DLSODA package includes the
following subroutines and function routines:
DINTDY computes an interpolated value of the y vector at t = TOUT.
DSTODA is the core integrator, which does one step of the
integration and the associated error control.
DCFODE sets all method coefficients and test constants.
DPRJA computes and preprocesses the Jacobian matrix J = df/dy
and the Newton iteration matrix P = I - h*l0*J.
DSOLSY manages solution of linear system in chord iteration.
DEWSET sets the error weight vector EWT before each step.
DMNORM computes the weighted max-norm of a vector.
DFNORM computes the norm of a full matrix consistent with the
weighted max-norm on vectors.
DBNORM computes the norm of a band matrix consistent with the
weighted max-norm on vectors.
DSRCMA is a user-callable routine to save and restore
the contents of the internal Common blocks.
DGEFA and DGESL are routines from LINPACK for solving full
systems of linear algebraic equations.
DGBFA and DGBSL are routines from LINPACK for solving banded
linear systems.
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: DMNORM, DFNORM, DBNORM, DUMACH, IXSAV, and IUMACH are
function routines. All the others are subroutines.