dlsodes
This is the 12 November 2003 version of
DLSODES: Livermore Solver for Ordinary Differential Equations
with general Sparse Jacobian matrix.
This version is in double precision.
DLSODES 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).
DLSODES is a variant of the DLSODE package, and is intended for
problems in which the Jacobian matrix df/dy has an arbitrary
sparse structure (when the problem is stiff).
Authors: Alan C. Hindmarsh
Center for Applied Scientific Computing, L-561
Lawrence Livermore National Laboratory
Livermore, CA 94551
and
Andrew H. Sherman
J. S. Nolen and Associates
Houston, TX 77084
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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. 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., 18 (1982), pp. 1145-1151.
3. 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 DLSODES 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 F (NEQ, T, Y, YDOT)
DOUBLE PRECISION T, Y(*), YDOT(*)
which supplies the vector function f by loading YDOT(i) with f(i).
B. Next determine (or guess) whether or not the problem is stiff.
Stiffness occurs when the Jacobian matrix df/dy has an eigenvalue
whose real part is negative and large in magnitude, compared to the
reciprocal of the t span of interest. If the problem is nonstiff,
use a method flag MF = 10. If it is stiff, there are two standard
choices for the method flag, MF = 121 and MF = 222. In both cases,
DLSODES requires the Jacobian matrix in some form, and it treats this
matrix in general sparse form, with sparsity structure determined
internally. (For options where the user supplies the sparsity
structure, see the full description of MF below.)
C. If the problem is stiff, you are encouraged to supply the Jacobian
directly (MF = 121), but if this is not feasible, DLSODES will
compute it internally by difference quotients (MF = 222).
If you are supplying the Jacobian, provide a subroutine of the form:
SUBROUTINE JAC (NEQ, T, Y, J, IAN, JAN, PDJ)
DOUBLE PRECISION T, Y(*), IAN(*), JAN(*), PDJ(*)
Here NEQ, T, Y, and J are input arguments, and the JAC routine is to
load the array PDJ (of length NEQ) with the J-th column of df/dy.
I.e., load PDJ(i) with df(i)/dy(J) for all relevant values of i.
The arguments IAN and JAN should be ignored for normal situations.
DLSODES will call the JAC routine with J = 1,2,...,NEQ.
Only nonzero elements need be loaded. Usually, a crude approximation
to df/dy, possibly with fewer nonzero elements, will suffice.
D. Write a main program which calls Subroutine DLSODES 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
DLSODES. On the first call to DLSODES, 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 t.
TOUT = first point where output is desired (.ne. T).
ITOL = 1 or 2 according as ATOL (below) is a scalar or array.
RTOL = relative tolerance parameter (scalar).
ATOL = absolute tolerance parameter (scalar or array).
The estimated local error in Y(i) will be controlled so as
to be roughly less (in magnitude) than
EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
Thus the local error test passes if, in each component,
either the absolute error is less than ATOL (or ATOL(i)),
or the relative error is less than RTOL.
Use RTOL = 0.0 for pure absolute error control, and
use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error
control. Caution: actual (global) errors may exceed these
local tolerances, so choose them conservatively.
ITASK = 1 for normal computation of output values of Y at t = TOUT.
ISTATE = integer flag (input and output). Set ISTATE = 1.
IOPT = 0 to indicate no optional inputs used.
RWORK = real work array of length at least:
20 + 16*NEQ for MF = 10,
20 + (2 + 1./LENRAT)*NNZ + (11 + 9./LENRAT)*NEQ
for MF = 121 or 222,
where:
NNZ = the number of nonzero elements in the sparse
Jacobian (if this is unknown, use an estimate), and
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 if MF = 121 or 222, 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 dimension).
IWORK = integer work array of length at least 30.
LIW = declared length of IWORK (in user dimension).
JAC = name of subroutine for Jacobian matrix (MF = 121).
If used, this name must be declared External in calling
program. If not used, pass a dummy name.
MF = method flag. Standard values are:
10 for nonstiff (Adams) method, no Jacobian used
121 for stiff (BDF) method, user-supplied sparse Jacobian
222 for stiff method, internally generated sparse Jacobian
Note that the main program must declare arrays Y, RWORK, IWORK,
and possibly ATOL.
E. The output from the first call (or any call) is:
Y = array of computed values of y(t) vector.
T = corresponding value of independent variable (normally TOUT).
ISTATE = 2 if DLSODES was successful, negative otherwise.
-1 means excess work done on this call (perhaps wrong MF).
-2 means excess accuracy requested (tolerances too small).
-3 means illegal input detected (see printed message).
-4 means repeated error test failures (check all inputs).
-5 means repeated convergence failures (perhaps bad Jacobian
supplied or wrong choice of MF or tolerances).
-6 means error weight became zero during problem. (Solution
component i vanished, and ATOL or ATOL(i) = 0.)
-7 means a fatal error return flag came from sparse solver
CDRV by way of DPRJS 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
DLSODES again with ISTATE = 3 to force the structure to be
reevaluated. See the full description of ISTATE below.
F. To continue the integration after a successful return, simply
reset TOUT and call DLSODES again. No other parameters need be reset.
-----------------------------------------------------------------------
Example Problem.
The following is a simple example problem, with the coding
needed for its solution by DLSODES. The problem is from chemical
kinetics, and consists of the following 12 rate equations:
dy1/dt = -rk1*y1
dy2/dt = rk1*y1 + rk11*rk14*y4 + rk19*rk14*y5
- rk3*y2*y3 - rk15*y2*y12 - rk2*y2
dy3/dt = rk2*y2 - rk5*y3 - rk3*y2*y3 - rk7*y10*y3
+ rk11*rk14*y4 + rk12*rk14*y6
dy4/dt = rk3*y2*y3 - rk11*rk14*y4 - rk4*y4
dy5/dt = rk15*y2*y12 - rk19*rk14*y5 - rk16*y5
dy6/dt = rk7*y10*y3 - rk12*rk14*y6 - rk8*y6
dy7/dt = rk17*y10*y12 - rk20*rk14*y7 - rk18*y7
dy8/dt = rk9*y10 - rk13*rk14*y8 - rk10*y8
dy9/dt = rk4*y4 + rk16*y5 + rk8*y6 + rk18*y7
dy10/dt = rk5*y3 + rk12*rk14*y6 + rk20*rk14*y7
+ rk13*rk14*y8 - rk7*y10*y3 - rk17*y10*y12
- rk6*y10 - rk9*y10
dy11/dt = rk10*y8
dy12/dt = rk6*y10 + rk19*rk14*y5 + rk20*rk14*y7
- rk15*y2*y12 - rk17*y10*y12
with rk1 = rk5 = 0.1, rk4 = rk8 = rk16 = rk18 = 2.5,
rk10 = 5.0, rk2 = rk6 = 10.0, rk14 = 30.0,
rk3 = rk7 = rk9 = rk11 = rk12 = rk13 = rk19 = rk20 = 50.0,
rk15 = rk17 = 100.0.
The t interval is from 0 to 1000, and the initial conditions
are y1 = 1, y2 = y3 = ... = y12 = 0. The problem is stiff.
The following coding solves this problem with DLSODES, using MF = 121
and printing results at t = .1, 1., 10., 100., 1000. It uses
ITOL = 1 and mixed relative/absolute tolerance controls.
During the run and at the end, statistical quantities of interest
are printed (see optional outputs in the full description below).
EXTERNAL FEX, JEX
DOUBLE PRECISION ATOL, RTOL, RWORK, T, TOUT, Y
DIMENSION Y(12), RWORK(500), IWORK(30)
DATA LRW/500/, LIW/30/
NEQ = 12
DO 10 I = 1,NEQ
10 Y(I) = 0.0D0
Y(1) = 1.0D0
T = 0.0D0
TOUT = 0.1D0
ITOL = 1
RTOL = 1.0D-4
ATOL = 1.0D-6
ITASK = 1
ISTATE = 1
IOPT = 0
MF = 121
DO 40 IOUT = 1,5
CALL DLSODES (FEX, NEQ, Y, T, TOUT, ITOL, RTOL, ATOL,
1 ITASK, ISTATE, IOPT, RWORK, LRW, IWORK, LIW, JEX, MF)
WRITE(6,30)T,IWORK(11),RWORK(11),(Y(I),I=1,NEQ)
30 FORMAT(//' At t =',D11.3,4X,
1 ' No. steps =',I5,4X,' Last step =',D11.3/
2 ' Y array = ',4D14.5/13X,4D14.5/13X,4D14.5)
IF (ISTATE .LT. 0) GO TO 80
TOUT = TOUT*10.0D0
40 CONTINUE
LENRW = IWORK(17)
LENIW = IWORK(18)
NST = IWORK(11)
NFE = IWORK(12)
NJE = IWORK(13)
NLU = IWORK(21)
NNZ = IWORK(19)
NNZLU = IWORK(25) + IWORK(26) + NEQ
WRITE (6,70) LENRW,LENIW,NST,NFE,NJE,NLU,NNZ,NNZLU
70 FORMAT(//' Required RWORK size =',I4,' IWORK size =',I4/
1 ' No. steps =',I4,' No. f-s =',I4,' No. J-s =',I4,
2 ' No. LU-s =',I4/' No. of nonzeros in J =',I5,
3 ' No. of nonzeros in LU =',I5)
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
DOUBLE PRECISION RK1, RK2, RK3, RK4, RK5, RK6, RK7, RK8, RK9,
1 RK10, RK11, RK12, RK13, RK14, RK15, RK16, RK17
DIMENSION Y(12), YDOT(12)
DATA RK1/0.1D0/, RK2/10.0D0/, RK3/50.0D0/, RK4/2.5D0/, RK5/0.1D0/,
1 RK6/10.0D0/, RK7/50.0D0/, RK8/2.5D0/, RK9/50.0D0/, RK10/5.0D0/,
2 RK11/50.0D0/, RK12/50.0D0/, RK13/50.0D0/, RK14/30.0D0/,
3 RK15/100.0D0/, RK16/2.5D0/, RK17/100.0D0/, RK18/2.5D0/,
4 RK19/50.0D0/, RK20/50.0D0/
YDOT(1) = -RK1*Y(1)
YDOT(2) = RK1*Y(1) + RK11*RK14*Y(4) + RK19*RK14*Y(5)
1 - RK3*Y(2)*Y(3) - RK15*Y(2)*Y(12) - RK2*Y(2)
YDOT(3) = RK2*Y(2) - RK5*Y(3) - RK3*Y(2)*Y(3) - RK7*Y(10)*Y(3)
1 + RK11*RK14*Y(4) + RK12*RK14*Y(6)
YDOT(4) = RK3*Y(2)*Y(3) - RK11*RK14*Y(4) - RK4*Y(4)
YDOT(5) = RK15*Y(2)*Y(12) - RK19*RK14*Y(5) - RK16*Y(5)
YDOT(6) = RK7*Y(10)*Y(3) - RK12*RK14*Y(6) - RK8*Y(6)
YDOT(7) = RK17*Y(10)*Y(12) - RK20*RK14*Y(7) - RK18*Y(7)
YDOT(8) = RK9*Y(10) - RK13*RK14*Y(8) - RK10*Y(8)
YDOT(9) = RK4*Y(4) + RK16*Y(5) + RK8*Y(6) + RK18*Y(7)
YDOT(10) = RK5*Y(3) + RK12*RK14*Y(6) + RK20*RK14*Y(7)
1 + RK13*RK14*Y(8) - RK7*Y(10)*Y(3) - RK17*Y(10)*Y(12)
2 - RK6*Y(10) - RK9*Y(10)
YDOT(11) = RK10*Y(8)
YDOT(12) = RK6*Y(10) + RK19*RK14*Y(5) + RK20*RK14*Y(7)
1 - RK15*Y(2)*Y(12) - RK17*Y(10)*Y(12)
RETURN
END
SUBROUTINE JEX (NEQ, T, Y, J, IA, JA, PDJ)
DOUBLE PRECISION T, Y, PDJ
DOUBLE PRECISION RK1, RK2, RK3, RK4, RK5, RK6, RK7, RK8, RK9,
1 RK10, RK11, RK12, RK13, RK14, RK15, RK16, RK17
DIMENSION Y(12), IA(*), JA(*), PDJ(12)
DATA RK1/0.1D0/, RK2/10.0D0/, RK3/50.0D0/, RK4/2.5D0/, RK5/0.1D0/,
1 RK6/10.0D0/, RK7/50.0D0/, RK8/2.5D0/, RK9/50.0D0/, RK10/5.0D0/,
2 RK11/50.0D0/, RK12/50.0D0/, RK13/50.0D0/, RK14/30.0D0/,
3 RK15/100.0D0/, RK16/2.5D0/, RK17/100.0D0/, RK18/2.5D0/,
4 RK19/50.0D0/, RK20/50.0D0/
GO TO (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), J
1 PDJ(1) = -RK1
PDJ(2) = RK1
RETURN
2 PDJ(2) = -RK3*Y(3) - RK15*Y(12) - RK2
PDJ(3) = RK2 - RK3*Y(3)
PDJ(4) = RK3*Y(3)
PDJ(5) = RK15*Y(12)
PDJ(12) = -RK15*Y(12)
RETURN
3 PDJ(2) = -RK3*Y(2)
PDJ(3) = -RK5 - RK3*Y(2) - RK7*Y(10)
PDJ(4) = RK3*Y(2)
PDJ(6) = RK7*Y(10)
PDJ(10) = RK5 - RK7*Y(10)
RETURN
4 PDJ(2) = RK11*RK14
PDJ(3) = RK11*RK14
PDJ(4) = -RK11*RK14 - RK4
PDJ(9) = RK4
RETURN
5 PDJ(2) = RK19*RK14
PDJ(5) = -RK19*RK14 - RK16
PDJ(9) = RK16
PDJ(12) = RK19*RK14
RETURN
6 PDJ(3) = RK12*RK14
PDJ(6) = -RK12*RK14 - RK8
PDJ(9) = RK8
PDJ(10) = RK12*RK14
RETURN
7 PDJ(7) = -RK20*RK14 - RK18
PDJ(9) = RK18
PDJ(10) = RK20*RK14
PDJ(12) = RK20*RK14
RETURN
8 PDJ(8) = -RK13*RK14 - RK10
PDJ(10) = RK13*RK14
PDJ(11) = RK10
9 RETURN
10 PDJ(3) = -RK7*Y(3)
PDJ(6) = RK7*Y(3)
PDJ(7) = RK17*Y(12)
PDJ(8) = RK9
PDJ(10) = -RK7*Y(3) - RK17*Y(12) - RK6 - RK9
PDJ(12) = RK6 - RK17*Y(12)
11 RETURN
12 PDJ(2) = -RK15*Y(2)
PDJ(5) = RK15*Y(2)
PDJ(7) = RK17*Y(10)
PDJ(10) = -RK17*Y(10)
PDJ(12) = -RK15*Y(2) - RK17*Y(10)
RETURN
END
The output of this program (on a Cray-1 in single precision)
is as follows:
At t = 1.000e-01 No. steps = 12 Last step = 1.515e-02
Y array = 9.90050e-01 6.28228e-03 3.65313e-03 7.51934e-07
1.12167e-09 1.18458e-09 1.77291e-12 3.26476e-07
5.46720e-08 9.99500e-06 4.48483e-08 2.76398e-06
At t = 1.000e+00 No. steps = 33 Last step = 7.880e-02
Y array = 9.04837e-01 9.13105e-03 8.20622e-02 2.49177e-05
1.85055e-06 1.96797e-06 1.46157e-07 2.39557e-05
3.26306e-05 7.21621e-04 5.06433e-05 3.05010e-03
At t = 1.000e+01 No. steps = 48 Last step = 1.239e+00
Y array = 3.67876e-01 3.68958e-03 3.65133e-01 4.48325e-05
6.10798e-05 4.33148e-05 5.90211e-05 1.18449e-04
3.15235e-03 3.56531e-03 4.15520e-03 2.48741e-01
At t = 1.000e+02 No. steps = 91 Last step = 3.764e+00
Y array = 4.44981e-05 4.42666e-07 4.47273e-04 -3.53257e-11
2.81577e-08 -9.67741e-11 2.77615e-07 1.45322e-07
1.56230e-02 4.37394e-06 1.60104e-02 9.52246e-01
At t = 1.000e+03 No. steps = 111 Last step = 4.156e+02
Y array = -2.65492e-13 2.60539e-14 -8.59563e-12 6.29355e-14
-1.78066e-13 5.71471e-13 -1.47561e-12 4.58078e-15
1.56314e-02 1.37878e-13 1.60184e-02 9.52719e-01
Required RWORK size = 442 IWORK size = 30
No. steps = 111 No. f-s = 142 No. J-s = 2 No. LU-s = 20
No. of nonzeros in J = 44 No. of nonzeros in LU = 50
-----------------------------------------------------------------------
Full Description of User Interface to DLSODES.
The user interface to DLSODES consists of the following parts.
1. The call sequence to Subroutine DLSODES, 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 DLSODES 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 DLSODES 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
F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, MF,
and those used for both input and output are
Y, T, ISTATE.
The work arrays RWORK and IWORK are also used for conditional and
optional inputs and optional outputs. (The term output here refers
to the return from Subroutine DLSODES 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 DLSODES, 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 DLSODES 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 DLSODES package accesses only
Y(1),...,Y(NEQ).)
T = the independent variable. On input, T is used only on the
first call, as the initial point of the integration.
on output, after each call, T is the value at which a
computed solution Y is evaluated (usually the same as TOUT).
On an error return, T is the farthest point reached.
TOUT = the next value of t at which a computed solution is desired.
Used only for input.
When starting the problem (ISTATE = 1), TOUT may be equal
to T for one call, then should .ne. T for the next call.
For the initial T, an input value of TOUT .ne. T is used
in order to determine the direction of the integration
(i.e. the algebraic sign of the step sizes) and the rough
scale of the problem. Integration in either direction
(forward or backward in t) is permitted.
If ITASK = 2 or 5 (one-step modes), TOUT is ignored after
the first call (i.e. the first call with TOUT .ne. T).
Otherwise, TOUT is required on every call.
If ITASK = 1, 3, or 4, the values of TOUT need not be
monotone, but a value of TOUT which backs up is limited
to the current internal T interval, whose endpoints are
TCUR - HU and TCUR (see optional outputs, below, for
TCUR and HU).
ITOL = an indicator for the type of error control. See
description below under ATOL. Used only for input.
RTOL = a relative error tolerance parameter, either a scalar or
an array of length NEQ. See description below under ATOL.
Input only.
ATOL = an absolute error tolerance parameter, either a scalar or
an array of length NEQ. Input only.
The input parameters ITOL, RTOL, and ATOL determine
the error control performed by the solver. The solver will
control the vector E = (E(i)) of estimated local errors
in y, according to an inequality of the form
RMS-norm of ( E(i)/EWT(i) ) .le. 1,
where EWT(i) = RTOL(i)*ABS(Y(i)) + ATOL(i),
and the RMS-norm (root-mean-square norm) here is
RMS-norm(v) = SQRT(sum v(i)**2 / NEQ). Here EWT = (EWT(i))
is a vector of weights which must always be positive, and
the values of RTOL and ATOL should all be non-negative.
The following table gives the types (scalar/array) of
RTOL and ATOL, and the corresponding form of EWT(i).
ITOL RTOL ATOL EWT(i)
1 scalar scalar RTOL*ABS(Y(i)) + ATOL
2 scalar array RTOL*ABS(Y(i)) + ATOL(i)
3 array scalar RTOL(i)*ABS(Y(i)) + ATOL
4 array array RTOL(i)*ABS(Y(i)) + ATOL(i)
When either of these parameters is a scalar, it need not
be dimensioned in the user's calling program.
If none of the above choices (with ITOL, RTOL, and ATOL
fixed throughout the problem) is suitable, more general
error controls can be obtained by substituting
user-supplied routines for the setting of EWT and/or for
the norm calculation. See Part 4 below.
If global errors are to be estimated by making a repeated
run on the same problem with smaller tolerances, then all
components of RTOL and ATOL (i.e. of EWT) should be scaled
down uniformly.
ITASK = an index specifying the task to be performed.
Input only. ITASK has the following values and meanings.
1 means normal computation of output values of y(t) at
t = TOUT (by overshooting and interpolating).
2 means take one step only and return.
3 means stop at the first internal mesh point at or
beyond t = TOUT and return.
4 means normal computation of output values of y(t) at
t = TOUT but without overshooting t = TCRIT.
TCRIT must be input as RWORK(1). TCRIT may be equal to
or beyond TOUT, but not behind it in the direction of
integration. This option is useful if the problem
has a singularity at or beyond t = TCRIT.
5 means take one step, without passing TCRIT, and return.
TCRIT must be input as RWORK(1).
Note: If ITASK = 4 or 5 and the solver reaches TCRIT
(within roundoff), it will return T = TCRIT (exactly) to
indicate this (unless ITASK = 4 and TOUT comes before TCRIT,
in which case answers at t = TOUT are returned first).
ISTATE = an index used for input and output to specify the
the state of the calculation.
On input, the values of ISTATE are as follows.
1 means this is the first call for the problem
(initializations will be done). See note below.
2 means this is not the first call, and the calculation
is to continue normally, with no change in any input
parameters except possibly TOUT and ITASK.
(If ITOL, RTOL, and/or ATOL are changed between calls
with ISTATE = 2, the new values will be used but not
tested for legality.)
3 means this is not the first call, and the
calculation is to continue normally, but with
a change in input parameters other than
TOUT and ITASK. Changes are allowed in
NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, MF,
the conditional inputs IA and JA,
and any of the optional inputs except H0.
In particular, if MITER = 1 or 2, a call with ISTATE = 3
will cause the sparsity structure of the problem to be
recomputed (or reread from IA and JA 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 = 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 a fatal error return flag came from the sparse
solver CDRV by way of DPRJS 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 and with
MITER = 1 or 2 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.
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 = 0 if MITER = 0,
LWM = 2*NNZ + 2*NEQ + (NNZ+9*NEQ)/LENRAT if MITER = 1,
LWM = 2*NNZ + 2*NEQ + (NNZ+10*NEQ)/LENRAT if MITER = 2,
LWM = NEQ + 2 if MITER = 3.
In the above formulas,
NNZ = number of nonzero elements in the Jacobian matrix.
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 for MF = 10,
20 + 16*NEQ + LWM for MF = 11, 111, 211, 12, 112, 212,
22 + 17*NEQ for MF = 13,
20 + 9*NEQ for MF = 20,
20 + 9*NEQ + LWM for MF = 21, 121, 221, 22, 122, 222,
22 + 10*NEQ for MF = 23.
If MITER = 1 or 2, 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
31 + NEQ + NNZ if MOSS = 0 and MITER = 1 or 2, or
30 otherwise.
(NNZ is the number of nonzero elements in df/dy.)
In DLSODES, IWORK is used only for conditional and
optional inputs and optional outputs.
The following two blocks of words in IWORK are conditional
inputs, required if MOSS = 0 and MITER = 1 or 2, 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,...,NNZ)
The two arrays IA and JA describe the sparsity structure
to be assumed for the Jacobian matrix. 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 NNZ is the total number of nonzero locations assumed,
then the length of the JA array is NNZ, and IA(NEQ+1) must
be NNZ + 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 DLSODES
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 DLSODES between calls, if
desired (but not for use by F or JAC).
JAC = name of user-supplied routine (MITER = 1 or MOSS = 1) to
compute the Jacobian matrix, df/dy, as a function of
the scalar t and the vector y. It is to have the form
SUBROUTINE JAC (NEQ, T, Y, J, IAN, JAN, PDJ)
DOUBLE PRECISION T, Y(*), IAN(*), JAN(*), PDJ(*)
where NEQ, T, Y, J, IAN, and JAN are input, and the array
PDJ, of length NEQ, is to be loaded with column J
of the Jacobian on output. Thus df(i)/dy(J) is to be
loaded into PDJ(i) for all relevant values of i.
Here T and Y have the same meaning as in Subroutine F,
and J is a column index (1 to NEQ). IAN and JAN are
undefined in calls to JAC for structure determination
(MOSS = 1). 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 df/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 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. JAC must not alter its input arguments.
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.
MF = the method flag. Used only for input.
MF has three decimal digits-- MOSS, METH, MITER--
MF = 100*MOSS + 10*METH + MITER.
MOSS indicates the method to be used to obtain the sparsity
structure of the Jacobian matrix if MITER = 1 or 2:
MOSS = 0 means the user has supplied IA and JA
(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.
MOSS = 2 means the structure will be obtained from NEQ+1
initial calls to F.
METH indicates the basic linear multistep method:
METH = 1 means the implicit Adams method.
METH = 2 means the method based on Backward
Differentiation Formulas (BDFs).
MITER indicates the corrector iteration method:
MITER = 0 means functional iteration (no Jacobian matrix
is involved).
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 F per df/dy value,
where NGP is an optional output described below.)
MITER = 3 means chord iteration with an internally
generated diagonal Jacobian approximation
(using 1 extra call to F per df/dy evaluation).
If MITER = 1 or MOSS = 1, 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 = 10 for a nonstiff problem,
MF = 21 or 22 for a stiff problem with IA/JA supplied
(21 if JAC is supplied, 22 if not),
MF = 121 for a stiff problem with JAC supplied,
but not IA/JA,
MF = 222 for a stiff problem with neither IA/JA nor
JAC supplied.
The sparseness structure can be changed during the
problem by making a call to DLSODES 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.)
SETH RWORK(8) the element threshhold for sparsity determination
when MOSS = 1 or 2. If the absolute value of
an estimated Jacobian element is .le. SETH, it
will be assumed to be absent in the structure.
The default value of SETH is 0.
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 DLSODES, the variables listed
below are quantities related to the performance of DLSODES
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 DLSODES, 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.)
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,
excluding those for structure determination
(MOSS = 2).
NJE IWORK(13) the number of Jacobian evaluations for the problem
so far, excluding those for structure determination
(MOSS = 1).
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 Jacobian
matrix, including the diagonal (MITER = 1 or 2).
(This may differ from that given by IA(NEQ+1)-1
if MOSS = 0, because of added diagonal entries.)
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 f
evaluations needed for each Jacobian evaluation.
NLU IWORK(21) the number of sparse LU decompositions for the
problem so far.
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 (MITER = 1 or 2).
NZU IWORK(26) the number of nonzero elements in the strict upper
triangle of the LU factorization used in the chord
iteration (MITER = 1 or 2).
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 Jacobian matrix, as used by
DLSODES when MITER = 1 or 2.
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.
(If MOSS = 0, IAN/JAN may differ from the
input IA/JA because of a different ordering
in each column, and added diagonal entries.)
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 successful return from
DLSODES.
-----------------------------------------------------------------------
Part 2. Other Routines Callable.
The following are optional calls which the user may make to
gain additional capabilities in conjunction with DLSODES.
(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 DLSODES, 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 DLSODES.
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
DLSODES (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 DLSODES.
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 DLSODES.
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 DLSODES).
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 DLSODES 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 DLSODES is to be used in an overlay situation, the user
must declare, in the primary overlay, the variables in:
(1) the call sequence to DLSODES, 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 DLSODES 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 DLSODES 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 DLSODES 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 DLSODES call for that problem. To save and restore the Common
blocks, use Subroutine DSRCMS (see Part 2 above).
-----------------------------------------------------------------------
Part 4. Optionally Replaceable Solver Routines.
Below are descriptions of two routines in the DLSODES 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 DLSODES 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 DLSODES 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 DLSODES.
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)
19810120 DATE WRITTEN
19820315 Upgraded MDI in ODRV package: operates on M + M-transpose.
19820426 Numerous revisions in use of work arrays;
use wordlength ratio LENRAT; added IPISP & LRAT to Common;
added optional outputs IPIAN/IPJAN;
numerous corrections to comments.
19830503 Added routine CNTNZU; added NZL and NZU to /LSS001/;
changed ADJLR call logic; added optional outputs NZL & NZU;
revised counter initializations; revised PREP stmt. numbers;
corrections to comments throughout.
19870320 Corrected jump on test of umax in CDRV routine;
added ISTATE = -7 return.
19870330 Major update: corrected comments throughout;
removed TRET from Common; rewrote EWSET with 4 loops;
fixed t test in INTDY; added Cray directives in STODE;
in STODE, 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 blocks, to enable
interrupt/restart feature.
20031112 Added SAVE statements for data-loaded constants.
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Other routines in the DLSODES package.
In addition to Subroutine DLSODES, the DLSODES package includes the
following subroutines and function routines:
DIPREP acts as an iterface between DLSODES and DPREP, and also does
adjusting of work space pointers and work arrays.
DPREP is called by DIPREP to compute sparsity and do sparse matrix
preprocessing if MITER = 1 or 2.
JGROUP is called by DPREP to compute groups of Jacobian column
indices for use when MITER = 2.
ADJLR adjusts the length of required sparse matrix work space.
It is called by DPREP.
CNTNZU is called by DPREP and counts the nonzero elements in the
strict upper triangle of J + J-transpose, where J = df/dy.
DINTDY computes an interpolated value of the y vector at t = TOUT.
DSTODE is the core integrator, which does one step of the
integration and the associated error control.
DCFODE sets all method coefficients and test constants.
DPRJS computes and preprocesses the Jacobian matrix J = df/dy
and the Newton iteration matrix P = I - 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.
DLSODES uses CDRV to solve linear systems in which the
coefficient matrix is P = I - con*J, where I is the
identity, con is a scalar, and J is an approximation to
the Jacobian df/dy. Because CDRV deals with rowwise
sparsity descriptions, CDRV works with P-transpose, not P.
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.
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