dlsodi
This is the 18 November 2003 version of
DLSODI: Livermore Solver for Ordinary Differential Equations
(Implicit form).
This version is in double precision.
DLSODI 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.
DLSODI is a variant version of the DLSODE package.
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Reference:
Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE
Solvers, in Scientific Computing, R. S. Stepleman et al. (Eds.),
North-Holland, Amsterdam, 1983, pp. 55-64.
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Authors: Alan C. Hindmarsh and Jeffrey F. Painter
Center for Applied Scientific Computing, L-561
Lawrence Livermore National Laboratory
Livermore, CA 94551
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Summary of Usage.
Communication between the user and the DLSODI 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 DLSODI. (For uses of IRES, see the
paragraph on RES in the full description below.)
B. Next, decide whether full or banded form is more economical
for the storage of matrices. DLSODI must deal internally with the
matrices A and dr/dy, where r is the residual function defined above.
DLSODI generates a linear combination of these two matrices, and
this is treated in either full or banded form.
The matrix structure is communicated by a method flag MF,
which is 21 or 22 for the full case, and 24 or 25 in the band case.
In the banded case, DLSODI requires two half-bandwidth
parameters ML and MU. These are, respectively, the widths of the
lower and upper parts of the band, excluding the main diagonal.
Thus the band consists of the locations (i,j) with
i-ML .le. j .le. i+MU, and the full bandwidth is ML+MU+1.
Note that the band must accommodate the nonzero elements of
A(t,y), dg/dy, and d(A*s)/dy (s fixed). Alternatively, one
can define a band that encloses only the elements that are relatively
large in magnitude, and gain some economy in storage and possibly
also efficiency, although the appropriate threshhold for
retaining matrix elements is highly problem-dependent.
C. You must also provide a subroutine of the form:
SUBROUTINE ADDA (NEQ, T, Y, ML, MU, P, NROWP)
DOUBLE PRECISION T, Y(*), P(NROWP,*)
which adds the matrix A = A(t,y) to the contents of the array P.
T and the Y array are input and should not be altered.
In the full matrix case, this routine should add elements of
to P in the usual order. I.e., add A(i,j) to P(i,j). (Ignore the
ML and MU arguments in this case.)
In the band matrix case, this routine should add element A(i,j)
to P(i-j+MU+1,j). I.e., add the diagonal lines of A to the rows of
P from the top down (the top line of A added to the first row of P).
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 or 24, and provide a subroutine of the form:
SUBROUTINE JAC (NEQ, T, Y, S, ML, MU, P, NROWP)
DOUBLE PRECISION T, Y(*), S(*), P(NROWP,*)
which computes dr/dy as a function of t, y, and s. Here T, Y, and
S are inputs, and the routine is to load dr/dy into P as follows:
In the full matrix case (MF = 21), load P(i,j) with dr(i)/dy(j),
the partial derivative of r(i) with respect to y(j). (Ignore the
ML and MU arguments in this case.)
In the band matrix case (MF = 24), load P(i-j+mu+1,j) with
dr(i)/dy(j), i.e. load the diagonal lines of dr/dy into the rows of
P from the top down.
In either case, only nonzero elements need be loaded, and the
indexing of P is the same as in the ADDA routine.
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 or 25, and DLSODI 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 DLSODI compute
this vector (ISTATE = 0). (DLSODI 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 DLSODI 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 DLSODI. On the first call to DLSODI, 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 = 21 or 24). 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:
22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
22 + 10*NEQ + (2*ML + MU)*NEQ for MF = 24 or 25.
LRW = declared length of RWORK (in user's dimension).
IWORK = integer work array of length at least 20 + NEQ.
If MF = 24 or 25, input in IWORK(1),IWORK(2) the lower
and upper half-bandwidths ML,MU.
LIW = declared length of IWORK (in user's dimension).
MF = method flag. Standard values are:
21 for a user-supplied full Jacobian.
22 for an internally generated full Jacobian.
24 for a user-supplied banded Jacobian.
25 for an internally generated banded 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 DLSODI 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 DLSODI 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.
If DLSODI returns ISTATE = -1, -4, or -5, then the output of
DLSODI 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 DLSODI again. No other parameters need be reset.
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Example Problem.
The following is a simple example problem, with the coding
needed for its solution by DLSODI. The problem is from chemical
kinetics, and consists of the following three equations:
dy1/dt = -.04*y1 + 1.e4*y2*y3
dy2/dt = .04*y1 - 1.e4*y2*y3 - 3.e7*y2**2
0. = y1 + y2 + y3 - 1.
on the interval from t = 0.0 to t = 4.e10, with initial conditions
y1 = 1.0, y2 = y3 = 0.
The following coding solves this problem with DLSODI, using MF = 21
and 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. dy/dt is supplied in YDOTI. We had
obtained the initial value of dy3/dt by differentiating the
third equation and evaluating the first two at t = 0.
At the end of the run, statistical quantities of interest are
printed (see optional outputs in the full description below).
EXTERNAL RESID, APLUSP, DGBYDY
DOUBLE PRECISION ATOL, RTOL, RWORK, T, TOUT, Y, YDOTI
DIMENSION Y(3), YDOTI(3), ATOL(3), RWORK(58), IWORK(23)
NEQ = 3
Y(1) = 1.
Y(2) = 0.
Y(3) = 0.
YDOTI(1) = -.04
YDOTI(2) = .04
YDOTI(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 = 58
LIW = 23
MF = 21
DO 40 IOUT = 1,12
CALL DLSODI(RESID, APLUSP, DGBYDY, NEQ, Y, YDOTI, T, TOUT, ITOL,
1 RTOL, ATOL, ITASK, ISTATE, IOPT, RWORK, LRW, IWORK, LIW, MF)
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)
60 FORMAT(/' No. steps =',I4,' No. r-s =',I4,' No. J-s =',I4)
STOP
80 WRITE (6,90) ISTATE
90 FORMAT(///' Error halt.. ISTATE =',I3)
STOP
END
SUBROUTINE RESID(NEQ, T, Y, S, R, IRES)
DOUBLE PRECISION T, Y, S, R
DIMENSION Y(3), S(3), R(3)
R(1) = -.04*Y(1) + 1.D4*Y(2)*Y(3) - S(1)
R(2) = .04*Y(1) - 1.D4*Y(2)*Y(3) - 3.D7*Y(2)*Y(2) - S(2)
R(3) = Y(1) + Y(2) + Y(3) - 1.
RETURN
END
SUBROUTINE APLUSP(NEQ, T, Y, ML, MU, P, NROWP)
DOUBLE PRECISION T, Y, P
DIMENSION Y(3), P(NROWP,3)
P(1,1) = P(1,1) + 1.
P(2,2) = P(2,2) + 1.
RETURN
END
SUBROUTINE DGBYDY(NEQ, T, Y, S, ML, MU, P, NROWP)
DOUBLE PRECISION T, Y, S, P
DIMENSION Y(3), S(3), P(NROWP,3)
P(1,1) = -.04
P(1,2) = 1.D4*Y(3)
P(1,3) = 1.D4*Y(2)
P(2,1) = .04
P(2,2) = -1.D4*Y(3) - 6.D7*Y(2)
P(2,3) = -1.D4*Y(2)
P(3,1) = 1.
P(3,2) = 1.
P(3,3) = 1.
RETURN
END
The output of this program (on a CDC-7600 in single precision)
is as follows:
At t = 4.0000e-01 Y = 9.851726e-01 3.386406e-05 1.479357e-02
At t = 4.0000e+00 Y = 9.055142e-01 2.240418e-05 9.446344e-02
At t = 4.0000e+01 Y = 7.158050e-01 9.184616e-06 2.841858e-01
At t = 4.0000e+02 Y = 4.504846e-01 3.222434e-06 5.495122e-01
At t = 4.0000e+03 Y = 1.831701e-01 8.940379e-07 8.168290e-01
At t = 4.0000e+04 Y = 3.897016e-02 1.621193e-07 9.610297e-01
At t = 4.0000e+05 Y = 4.935213e-03 1.983756e-08 9.950648e-01
At t = 4.0000e+06 Y = 5.159269e-04 2.064759e-09 9.994841e-01
At t = 4.0000e+07 Y = 5.306413e-05 2.122677e-10 9.999469e-01
At t = 4.0000e+08 Y = 5.494532e-06 2.197826e-11 9.999945e-01
At t = 4.0000e+09 Y = 5.129457e-07 2.051784e-12 9.999995e-01
At t = 4.0000e+10 Y = -7.170472e-08 -2.868188e-13 1.000000e+00
No. steps = 330 No. r-s = 404 No. J-s = 69
-----------------------------------------------------------------------
Full Description of User Interface to DLSODI.
The user interface to DLSODI consists of the following parts.
1. The call sequence to Subroutine DLSODI, 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 DLSODI 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 DLSODI 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 DLSODI 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 DLSODI will use the
returned array R, as follows:
IRES = 1 means that DLSODI needs the full residual,
r = g - A*s, exactly.
IRES = -1 means that DLSODI 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
DLSODI continues integrating the ODE. Leave IRES
unchanged from its input value.
IRES = 2 tells DLSODI 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). DLSODI tries to integrate the system
without getting IRES = 3 from RES. If it cannot, DLSODI
returns with ISTATE = -7 or -1.
On an DLSODI 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 DLSODI 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 DLSODI, 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 the same form
as A. The storage form is determined by MITER (see MF).
This subroutine is to have the form
SUBROUTINE ADDA (NEQ, T, Y, ML, MU, P, NROWP)
DOUBLE PRECISION T, Y(*), P(NROWP,*)
where NEQ, T, Y, ML, MU, and NROWP are input and P is
output. Y is an array of length NEQ, and the matrix P is
stored in an NROWP by NEQ array.
In the full matrix case ( MITER = 1 or 2) ADDA should
add A to P(i,j). ML and MU are ignored.
i,j
In the band matrix case ( MITER = 4 or 5) ADDA should
add A to P(i-j+MU+1,j).
i,j
See JAC for details on this band storage form.
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. The form of the
Jacobian matrix is determined by MITER. JAC is required
if MITER = 1 or 4 -- otherwise a dummy name can be
passed. This subroutine is to have the form
SUBROUTINE JAC ( NEQ, T, Y, S, ML, MU, P, NROWP )
DOUBLE PRECISION T, Y(*), S(*), P(NROWP,*)
where NEQ, T, Y, S, ML, MU, and NROWP are input and P
is output. Y and S are arrays of length NEQ, and the
matrix P is stored in an NROWP by NEQ array.
P is to be loaded with partial derivatives (elements
of the Jacobian matrix) on output.
In the full matrix case (MITER = 1), ML and MU
are ignored and the Jacobian is to be loaded into P
by columns-- i.e., dr(i)/dy(j) is loaded into P(i,j).
In the band matrix case (MITER = 4), the elements
within the band are to be loaded into P by columns,
with diagonal lines of dr/dy loaded into the
rows of P. Thus dr(i)/dy(j) is to be loaded
into P(i-j+MU+1,j). The locations in P in the two
triangular areas which correspond to nonexistent matrix
elements can be ignored or loaded arbitrarily, as they
they are overwritten by DLSODI. ML and MU are the
half-bandwidth parameters (see IWORK).
In either case, P 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 RES with the same
arguments NEQ, T, Y, and S. 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. Also, JAC may alter the Y array, if desired.
JAC need not provide dr/dy exactly. A crude
approximation (possibly with a smaller bandwidth) will do.
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 routines 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 DLSODI 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 DLSODI
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 DLSODI 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
DLSODI cannot compute the initial value of dy/dt, so
it must be provided in YDOTI, with ISTATE = 1.
On output, when DLSODI 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,
then there may have been an incorrect input value of
YDOTI = dy/dt, or the problem (as given to DLSODI)
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
DLSODI 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 (DLSODI 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, ML, MU,
and any of the optional inputs except H0.
(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 = 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
DLSODI 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
DLSODI to avoid that condition.
-8 means that ISTATE was 0 on input but DLSODI was unable
to compute the initial value of dy/dt. See the
printed message for details.
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
DLSODI 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 real working array (double precision).
The length of RWORK must be at least
20 + NYH*(MAXORD + 1) + 3*NEQ + LENWM 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),
LENWM = NEQ**2 + 2 if MITER is 1 or 2, and
LENWM = (2*ML+MU+1)*NEQ + 2 if MITER is 4 or 5.
(See MF description for the definition of METH and MITER.)
Thus if MAXORD has its default value and NEQ is constant,
this length is
22 + 16*NEQ + NEQ**2 for MF = 11 or 12,
22 + 17*NEQ + (2*ML+MU)*NEQ for MF = 14 or 15,
22 + 9*NEQ + NEQ**2 for MF = 21 or 22,
22 + 10*NEQ + (2*ML+MU)*NEQ for MF = 24 or 25.
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
20 + NEQ . 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
matrices in the problem-- the Jacobian dr/dy
and the left-hand side matrix A. These
half-bandwidths exclude the main diagonal,
so the total bandwidth is ML + MU + 1 .
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 MITER is 4 or 5, and
ignored otherwise.
ML and MU may in fact be the band parameters
for matrices to which dr/dy and A are only
approximately equal.
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 DLSODI
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 DLSODI between calls, if
desired (but not for use by RES, ADDA, or JAC).
MF = the method flag. Used only for input. The legal values of
MF are 11, 12, 14, 15, 21, 22, 24, and 25.
MF has decimal digits METH and MITER: MF = 10*METH + MITER.
METH indicates the basic linear multistep method:
METH = 1 means the implicit Adams method.
METH = 2 means the method based on Backward
Differentiation Formulas (BDFs).
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
full (NEQ by NEQ) Jacobian.
MITER = 2 means chord iteration with an internally
generated (difference quotient) full Jacobian.
This uses NEQ+1 extra calls to RES per dr/dy
evaluation.
MITER = 4 means chord iteration with a user-supplied
banded Jacobian.
MITER = 5 means chord iteration with an internally
generated banded Jacobian (using ML+MU+2
extra calls to RES per dr/dy evaluation).
If MITER = 1 or 4, the user must supply a Subroutine JAC
(the name is arbitrary) as described above under JAC.
For other values of MITER, 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.)
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 DLSODI, the variables listed
below are quantities related to the performance of DLSODI
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 DLSODI, 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.
NJE IWORK(13) the number of Jacobian evaluations (each involving
an evaluation of A and dr/dy) for the problem so
far. This equals the number of calls to ADDA and
(if MITER = 1 or 4) JAC, and the number of matrix
LU decompositions.
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.
The following two arrays are segments of the RWORK array which
may also be of interest to the user as optional outputs.
For each array, the table below gives its internal name,
its base address in RWORK, and its description.
Name Base Address Description
YH 21 the Nordsieck history array, of size NYH by
(NQCUR + 1), where NYH is the initial value
of NEQ. For j = 0,1,...,NQCUR, column j+1
of YH contains HCUR**j/factorial(j) times
the j-th derivative of the interpolating
polynomial currently representing the solution,
evaluated at t = TCUR.
ACOR LENRW-NEQ+1 array of size NEQ used for the accumulated
corrections on each step, scaled on output to
represent the estimated local error in y on the
last step. This is the vector E in the descrip-
tion of the error control. It is defined only
on a return from DLSODI 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 DLSODI.
(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 DLSODI, 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 DLSODI.
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 DSRCOM(RSAV,ISAV,JOB) saves and restores the contents of
the internal Common blocks used by
DLSODI (see Part 3 below).
RSAV must be a real array of length 218
or more, and ISAV must be an integer
array of length 37 or more.
JOB=1 means save Common into RSAV/ISAV.
JOB=2 means restore Common from RSAV/ISAV.
DSRCOM is useful if one is
interrupting a run and restarting
later, or alternating between two or
more problems solved with DLSODI.
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 DLSODI.
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 DLSODI).
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 DLSODI 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 DLSODI is to be used in an overlay situation, the user
must declare, in the primary overlay, the variables in:
(1) the call sequence to DLSODI, and
(2) the internal Common block
/DLS001/ of length 255 (218 double precision words
followed by 37 integer words),
If DLSODI 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 block in the calling program to insure
that their contents are preserved.
If the solution of a given problem by DLSODI 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 DLSODI 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 DLSODI call for that problem. To save and restore the Common
blocks, use Subroutine DSRCOM (see Part 2 above).
-----------------------------------------------------------------------
Part 4. Optionally Replaceable Solver Routines.
Below are descriptions of two routines in the DLSODI 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 DLSODI 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 DLSODI in the computation
of the optional output IMXER, the diagonal Jacobian approximation,
and the increments for difference quotient Jacobians.
In the user-supplied version of DEWSET, it may be desirable to use
the current values of derivatives of y. Derivatives up to order NQ
are available from the history array YH, described above under
optional outputs. In DEWSET, YH is identical to the YCUR array,
extended to NQ + 1 columns with a column length of NYH and scale
factors of H**j/factorial(j). On the first call for the problem,
given by NST = 0, NQ is 1 and H is temporarily set to 1.0.
NYH is the initial value of NEQ. The quantities NQ, H, and NST
can be obtained by including in DEWSET the statements:
DOUBLE PRECISION RLS
COMMON /DLS001/ RLS(218),ILS(37)
NQ = ILS(33)
NST = ILS(34)
H = RLS(212)
Thus, for example, the current value of dy/dt can be obtained as
YCUR(NYH+i)/H (i=1,...,NEQ) (and the division by H is
unnecessary when NST = 0).
(b) DVNORM.
The following is a real function routine which computes the weighted
root-mean-square norm of a vector v:
D = DVNORM (N, V, W)
where:
N = the length of the vector,
V = real array of length N containing the vector,
W = real array of length N containing weights,
D = SQRT( (1/N) * sum(V(i)*W(i))**2 ).
DVNORM is called with N = NEQ and with W(i) = 1.0/EWT(i), where
EWT is as set by Subroutine DEWSET.
If the user supplies this function, it should return a non-negative
value of DVNORM suitable for use in the error control in DLSODI.
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)
19800424 DATE WRITTEN
19800519 Corrected access of YH on forced order reduction;
numerous corrections to prologues and other comments.
19800617 In main driver, added loading of SQRT(UROUND) in RWORK;
minor corrections to main prologue.
19800903 Corrected ISTATE logic; minor changes in prologue.
19800923 Added zero initialization of HU and NQU.
19801028 Reorganized RES calls in AINVG, STODI, and PREPJI;
in LSODI, corrected NRE increment and reset LDY0 at 580;
numerous corrections to main prologue.
19801218 Revised XERRWD routine; minor corrections to main prologue.
19810330 Added Common block /LSI001/; use LSODE's INTDY and SOLSY;
minor corrections to XERRWD and error message at 604;
minor corrections to declarations; corrections to prologues.
19810818 Numerous revisions: replaced EWT by 1/EWT; used flags
JCUR, ICF, IERPJ, IERSL between STODI and subordinates;
added tuning parameters CCMAX, MAXCOR, MSBP, MXNCF;
reorganized returns from STODI; reorganized type decls.;
fixed message length in XERRWD; changed default LUNIT to 6;
changed Common lengths; changed comments throughout.
19820906 Corrected use of ABS(H) in STODI; minor comment fixes.
19830510 Numerous revisions: revised diff. 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 internal 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 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.
20031105 Restored 'own' variables to Common block, 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 DLSODI package.
In addition to Subroutine DLSODI, the DLSODI package includes the
following subroutines and function routines:
DAINVG computes the initial value of the vector
dy/dt = A-inverse * g
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.
DPREPJI computes and preprocesses the Jacobian matrix
and the Newton iteration matrix P.
DSOLSY 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.
DSRCOM 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: DVNORM, DUMACH, IXSAV, and IUMACH are function routines.
All the others are subroutines.