DLSODA solves the initial value problem for stiff or nonstiff systems of first order ODEs of the form
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).
with automatic method switching for stiff and nonstiff problems.
This a variant version of the DLSODE package. It switches automatically between stiff and nonstiff methods. This means that the user does not have to determine whether the problem is stiff or not, and the solver will automatically choose the appropriate method. It always starts with the nonstiff method.
Communication between the user and the DLSODA package, for normal situations, is summarized here. This summary describes only a subset of the full set of options available. See the full description for details, including alternative treatment of the Jacobian matrix, optional inputs and outputs, nonstandard options, and instructions for special situations. See also the example problem (with program and output) following this summary.
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).
Write a main program which calls Subroutine DLSODA once for each point at which answers are desired. This should also provide for possible use of logical unit 6 for output of error messages by DLSODA. On the first call to DLSODA, supply arguments as follows:
name of subroutine for right-hand side vector f. This name must be declared External in calling program.
number of first order ODEs.
array of initial values, of length NEQ.
the initial value of the independent variable.
first point where output is desired (.ne. T).
1 or 2 according as ATOL (below) is a single or multi-value array.
relative tolerance parameter (scalar).
absolute tolerance parameter (array). the estimated local error in y(i) will be controlled so as to be less than
EWT(i) = RTOL*ABS(Y(i)) + ATOL if ITOL = 1, or
EWT(i) = RTOL*ABS(Y(i)) + ATOL(i) if ITOL = 2.
Thus the local error test passes if, in each component, either the absolute error is less than ATOL (or ATOL(i)), or the relative error is less than RTOL.
Use RTOL = 0.0 for pure absolute error control, and use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative error control. Caution: actual (global) errors may exceed these local tolerances, so choose them conservatively.
1 for normal computation of output values of y at t = TOUT.
integer flag (input and output). Set ISTATE = 1.
0 to indicate no optional inputs used.
real work array of length at least:
22 + NEQ * MAX(16, NEQ + 9).
See also Section E below.
declared length of RWORK (in user’s dimension).
integer work array of length at least 20 + NEQ.
declared length of IWORK (in user’s dimension).
name of subroutine for Jacobian matrix. Use a dummy name. See also Section E below.
Jacobian type indicator. Set JT = 2. See also Section E below. Note that the main program must declare arrays Y, RWORK, IWORK, and possibly ATOL.
The output from the first call (or any call) is:
array of computed values of y(t) vector.
corresponding value of independent variable (normally TOUT).
| values | descriptions |
|---|---|
| 2 | if DLSODA was successful, negative otherwise. |
| -1 | means excess work done on this call (perhaps wrong JT). |
| -2 | means excess accuracy requested (tolerances too small). |
| -3 | means illegal input detected (see printed message). |
| -4 | means repeated error test failures (check all inputs). |
| -5 | means repeated convergence failures (perhaps bad Jacobian |
| supplied or wrong choice of JT or tolerances). | |
| -6 | means error weight became zero during problem. (Solution |
| component i vanished, and ATOL or ATOL(i) = 0.) | |
| -7 | means work space insufficient to finish (see messages). |
To continue the integration after a successful return, simply reset TOUT and call DLSODA again. No other parameters need be reset.
Note: If and when DLSODA regards the problem as stiff, and switches methods accordingly, it must make use of the NEQ by NEQ Jacobian matrix, J = df/dy. For the sake of simplicity, the inputs to DLSODA recommended in Section B above cause DLSODA to treat J as a full matrix, and to approximate it internally by difference quotients. Alternatively, J can be treated as a band matrix (with great potential reduction in the size of the RWORK array). Also, in either the full or banded case, the user can supply J in closed form, with a routine whose name is passed as the JAC argument. These alternatives are described in the paragraphs on RWORK, JAC, and JT in the full description of the call sequence below.
Example Problem.
The following is a simple example problem, with the coding needed for its solution by DLSODA. The problem is from chemical kinetics, and consists of the following three rate equations:
dy1/dt = -.04*y1 + 1.e4*y2*y3
dy2/dt = .04*y1 - 1.e4*y2*y3 - 3.e7*y2**2
dy3/dt = 3.e7*y2**2
on the interval from t = 0.0 to t = 4.e10, with initial conditions
y1 = 1.0, y2 = y3 = 0. The problem is stiff.
The following coding solves this problem with DLSODA, printing results at t = .4, 4., …, 4.e10. It uses ITOL = 2, and ATOL much smaller for y2 than y1 or y3 because y2 has much smaller values. At the end of the run, statistical quantities of interest are printed (see optional outputs in the full description below).
program dlsoda_ex
use m_odepack
implicit none
external fex
external jdum
integer,parameter :: dp=kind(0.0d0)
real(kind=dp),dimension(3) :: atol,y
integer :: iopt,iout,istate,itask
integer :: itol,jt,liw,lrw,neq
integer,dimension(23) :: iwork
real(kind=dp) :: rtol,t,tout
real(kind=dp),dimension(70) :: rwork
neq = 3
y(1) = 1.
y(2) = 0.
y(3) = 0.
t = 0.
tout = .4
itol = 2
rtol = 1.D-4
atol(1) = 1.D-6
atol(2) = 1.D-10
atol(3) = 1.D-6
itask = 1
istate = 1
iopt = 0
lrw = 70
liw = 23
jt = 2
do iout = 1,12
call dlsoda(fex,[neq],y,t,tout,itol,[rtol],atol,itask,istate,iopt,&
& rwork,lrw,iwork,liw,jdum,jt)
write (6,99010) t,y(1),y(2),y(3)
99010 format (' At t =',d12.4,' Y =',3D14.6)
if ( istate<0 ) then
write (6,99020) istate
99020 format (///' Error halt.. ISTATE =',i3)
stop 1
else
tout = tout*10.
endif
enddo
write (6,99030) iwork(11),iwork(12),iwork(13),iwork(19), &
& rwork(15)
99030 format (/' No. steps =',i4,' No. f-s =',i4,' No. J-s =', &
&i4/' Method last used =',i2,' Last switch was at t =',&
& d12.4)
end program dlsoda_ex
subroutine jdum()
implicit none
end subroutine jdum
subroutine fex(Neq,T,Y,Ydot)
implicit none
integer,parameter :: dp=kind(0.0d0)
integer :: Neq
real(kind=dp) :: T
real(kind=dp),intent(in),dimension(3) :: Y
real(kind=dp),intent(inout),dimension(3) :: Ydot
Ydot(1) = -.04*Y(1) + 1.D4*Y(2)*Y(3)
Ydot(3) = 3.D7*Y(2)*Y(2)
Ydot(2) = -Ydot(1) - Ydot(3)
end subroutine fex
The output of this program (on a CDC-7600 in single precision) is as follows:
At t = 4.0000e-01 y = 9.851712e-01 3.386380e-05 1.479493e-02
At t = 4.0000e+00 Y = 9.055333e-01 2.240655e-05 9.444430e-02
At t = 4.0000e+01 Y = 7.158403e-01 9.186334e-06 2.841505e-01
At t = 4.0000e+02 Y = 4.505250e-01 3.222964e-06 5.494717e-01
At t = 4.0000e+03 Y = 1.831975e-01 8.941774e-07 8.168016e-01
At t = 4.0000e+04 Y = 3.898730e-02 1.621940e-07 9.610125e-01
At t = 4.0000e+05 Y = 4.936363e-03 1.984221e-08 9.950636e-01
At t = 4.0000e+06 Y = 5.161831e-04 2.065786e-09 9.994838e-01
At t = 4.0000e+07 Y = 5.179817e-05 2.072032e-10 9.999482e-01
At t = 4.0000e+08 Y = 5.283401e-06 2.113371e-11 9.999947e-01
At t = 4.0000e+09 Y = 4.659031e-07 1.863613e-12 9.999995e-01
At t = 4.0000e+10 Y = 1.404280e-08 5.617126e-14 1.000000e+00
No. steps = 361 No. f-s = 693 No. J-s = 64
Method last used = 2 Last switch was at t = 6.0092e-03
The user interface to DLSODA consists of the following parts.
The call sequence to Subroutine DLSODA, which is a driver routine for the solver. This includes descriptions of both the call sequence arguments and of user-supplied routines. following these descriptions is a description of optional inputs available through the call sequence, and then a description of optional outputs (in the work arrays).
Descriptions of other routines in the DLSODA package that may be (optionally) called by the user. These provide the ability to alter error message handling, save and restore the internal Common, and obtain specified derivatives of the solution y(t).
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.
Description of a subroutine in the DLSODA package, which the user may replace with his/her own version, if desired. this relates to the measurement of errors.
The call sequence parameters used for input only are
F, NEQ, TOUT, ITOL, RTOL, ATOL, ITASK, IOPT, LRW, LIW, JAC, JT,
and those used for both input and output are
Y, T, ISTATE.
The work arrays RWORK and IWORK are also used for conditional and optional inputs and optional outputs. (The term output here refers to the return from Subroutine DLSODA to the user’s calling program.)
The legality of input parameters will be thoroughly checked on the initial call for the problem, but not checked thereafter unless a change in input parameters is flagged by ISTATE = 3 on input.
The descriptions of the call arguments are as follows:
the name of the user-supplied subroutine defining the ODE system. The system must be put in the first-order form dy/dt = f(t,y), where f is a vector-valued function of the scalar t and the vector y. Subroutine F is to compute the function f. It is to have the form
SUBROUTINE F (NEQ, T, Y, YDOT)
DOUBLE PRECISION T, Y(*), YDOT(*)
where NEQ, T, and Y are input, and the array YDOT = f(t,y) is output. Y and YDOT are arrays of length NEQ. Subroutine F should not alter Y(1),…,Y(NEQ). F must be declared External in the calling program.
Subroutine F may access user-defined quantities in NEQ(2),… and/or in Y(NEQ(1)+1),… if NEQ is an array (dimensioned in F) and/or Y has length exceeding NEQ(1). See the descriptions of NEQ and Y below.
If quantities computed in the F routine are needed externally to DLSODA, an extra call to F should be made for this purpose, for consistent and accurate results. If only the derivative dy/dt is needed, use DINTDY instead.
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 single value, and it is generally referred to as a scalar in this user interface description. However, NEQ must be an array, with NEQ(1) set to the system size. (The DLSODA package accesses only NEQ(1).) In either case, this parameter is passed as the NEQ argument in all calls to F and JAC. Hence, if it is an multi-value 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.
a real array for the vector of dependent variables, of length NEQ or more. Used for both input and output on the first call (ISTATE = 1), and only for output on other calls. On the first call, Y must contain the vector of initial values. On output, Y contains the computed solution vector, evaluated at T. If desired, the Y array may be used for other purposes between calls to the solver.
This array is passed as the Y argument in all calls to F and JAC. Hence its length may exceed NEQ, and locations Y(NEQ+1),… may be used to store other real data and pass it to F and/or JAC. (The DLSODA package accesses only Y(1),…,Y(NEQ).)
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.
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).
an indicator for the type of error control. See description below under ATOL. Used only for input.
a relative error tolerance parameter, an array of length NEQ. See description below under ATOL. Input only.
an absolute error tolerance parameter, an array of length NEQ. Input only.
The input parameters ITOL, RTOL, and ATOL determine the error control performed by the solver. The solver will control the vector E = (E(i)) of estimated local errors in y, according to an inequality of the form
max-norm of ( E(i)/EWT(i) ) .le. 1,
where EWT = (EWT(i)) is a vector of positive error weights. The values of RTOL and ATOL should all be non-negative. The following table gives the types (single/mult-value array) of RTOL and ATOL, and the corresponding form of EWT(i).
| ITOL | RTOL | ATOL | EWT(i) |
|---|---|---|---|
| 1 | single | single | RTOL*ABS(Y(i)) + ATOL |
| 2 | single | array | RTOL*ABS(Y(i)) + ATOL(i) |
| 3 | array | single | RTOL(i)*ABS(Y(i)) + ATOL |
| 4 | array | array | RTOL(i)*ABS(Y(i)) + ATOL(i) |
Even when these parameters are a single value, it needs to be an array in the user’s calling program, or passed as a temporary array (ie. use “[NEQ]” on the call if NEQ is a scalar).
If none of the above choices (with ITOL, RTOL, and ATOL fixed throughout the problem) is suitable, more general error controls can be obtained by substituting a user-supplied routine for the setting of EWT. See Part 4 below.
If global errors are to be estimated by making a repeated run on the same problem with smaller tolerances, then all components of RTOL and ATOL (i.e. of EWT) should be scaled down uniformly.
an index specifying the task to be performed. Input only. ITASK has the following values and meanings.
| value | descriptions |
|---|---|
| 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).
an index used for input and output to specify the the state of the calculation.
On input, the values of ISTATE are as follows:
| value | descriptions |
|---|---|
| 1 | means this is the first call for the problem |
| (initializations will be done). See note below. | |
| 2 | means this is not the first call, and the calculation |
| is to continue normally, with no change in any input | |
| parameters except possibly TOUT and ITASK. | |
| (If ITOL, RTOL, and/or ATOL are changed between calls | |
| with ISTATE = 2, the new values will be used but not | |
| tested for legality.) | |
| 3 | means this is not the first call, and the |
| calculation is to continue normally, but with | |
| a change in input parameters other than | |
| TOUT and ITASK. Changes are allowed in | |
| NEQ, ITOL, RTOL, ATOL, IOPT, LRW, LIW, JT, ML, MU, | |
| and any optional inputs except H0, MXORDN, and MXORDS. | |
| (See IWORK description for ML and MU.) |
Note: A preliminary call with TOUT = T is not counted as a first call here, as no initialization or checking of input is done. (Such a call is sometimes useful for the purpose of outputting the initial conditions.) Thus the first call for which TOUT .ne. T requires ISTATE = 1 on input.
On output, ISTATE has the following values and meanings.
| value. | descriptions |
|---|---|
| 1 | means nothing was done; TOUT = T and ISTATE = 1 on input. |
| 2 | means the integration was performed successfully. |
| -1 | means an excessive amount of work (more than MXSTEP |
| steps) was done on this call, before completing the | |
| requested task, but the integration was otherwise | |
| successful as far as T. (MXSTEP is an optional input | |
| and is normally 500.) To continue, the user may | |
| simply reset ISTATE to a value .gt. 1 and call again | |
| (the excess work step counter will be reset to 0). | |
| In addition, the user may increase MXSTEP to avoid | |
| this error return (see below on optional inputs). | |
| -2 | means too much accuracy was requested for the precision |
| of the machine being used. This was detected before | |
| completing the requested task, but the integration | |
| was successful as far as T. To continue, the tolerance | |
| parameters must be reset, and ISTATE must be set | |
| to 3. The optional output TOLSF may be used for this | |
| purpose. (Note: If this condition is detected before | |
| taking any steps, then an illegal input return | |
| (ISTATE = -3) occurs instead.) | |
| -3 | means illegal input was detected, before taking any |
| integration steps. See written message for details. | |
| Note: If the solver detects an infinite loop of calls | |
| to the solver with illegal input, it will cause | |
| the run to stop. | |
| -4 | means there were repeated error test failures on |
| one attempted step, before completing the requested | |
| task, but the integration was successful as far as T. | |
| The problem may have a singularity, or the input | |
| may be inappropriate. | |
| -5 | means there were repeated convergence test failures on |
| one attempted step, before completing the requested | |
| task, but the integration was successful as far as T. | |
| This may be caused by an inaccurate Jacobian matrix, | |
| if one is being used. | |
| -6 | means EWT(i) became zero for some i during the |
| integration. Pure relative error control (ATOL(i)=0.0) | |
| was requested on a variable which has now vanished. | |
| The integration was successful as far as T. | |
| -7 | means the length of RWORK and/or IWORK was too small to |
| proceed, but the integration was successful as far as T. | |
| This happens when DLSODA chooses to switch methods | |
| but LRW and/or LIW is too small for the new method. |
Note: Since the normal output value of ISTATE is 2, it does not need to be reset for normal continuation. Also, since a negative input value of ISTATE will be regarded as illegal, a negative output value requires the user to change it, and possibly other inputs, before calling the solver again.
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.
a real array (double precision) for work space, and (in the first 20 words) for conditional and optional inputs and optional outputs.
As DLSODA switches automatically between stiff and nonstiff methods, the required length of RWORK can change during the problem. Thus the RWORK array passed to DLSODA can either have a static (fixed) length large enough for both methods, or have a dynamic (changing) length altered by the calling program in response to output from DLSODA.
If the RWORK length is to be fixed, it should be at least
MAX (LRN, LRS),
where LRN and LRS are the RWORK lengths required when the
current method is nonstiff or stiff, respectively.
The separate RWORK length requirements LRN and LRS are
as follows:
IF NEQ is constant and the maximum method orders have
their default values, then
LRN = 20 + 16*NEQ,
LRS = 22 + 9*NEQ + NEQ**2 if JT = 1 or 2,
LRS = 22 + 10*NEQ + (2*ML+MU)*NEQ if JT = 4 or 5.
Under any other conditions, LRN and LRS are given by:
LRN = 20 + NYH*(MXORDN+1) + 3*NEQ,
LRS = 20 + NYH*(MXORDS+1) + 3*NEQ + LMAT,
where
NYH = the initial value of NEQ,
MXORDN = 12, unless a smaller value is given as an
optional input,
MXORDS = 5, unless a smaller value is given as an
optional input,
LMAT = length of matrix work space:
LMAT = NEQ**2 + 2 if JT = 1 or 2,
LMAT = (2*ML + MU + 1)*NEQ + 2 if JT = 4 or 5.
If the length of RWORK is to be dynamic, then it should be at least LRN or LRS, as defined above, depending on the current method.
Initially, it must be at least LRN (since DLSODA starts with the nonstiff method).
On any return from DLSODA, the optional output MCUR indicates the current method. If MCUR differs from the value it had on the previous return, or if there has only been one call to DLSODA and MCUR is now 2, then DLSODA has switched methods during the last call, and the length of RWORK should be reset (to LRN if MCUR = 1, or to LRS if MCUR = 2). (An increase in the RWORK length is required if DLSODA returned ISTATE = -7, but not otherwise.)
After resetting the length, call DLSODA with ISTATE = 3 to signal that change.
the length of the array RWORK, as declared by the user. (This will be checked by the solver.)
an integer array for work space. As DLSODA switches automatically between stiff and nonstiff methods, the required length of IWORK can change during problem, between LIS = 20 + NEQ and LIN = 20, respectively. Thus the IWORK array passed to DLSODA can either have a fixed length of at least 20 + NEQ, or have a dynamic length of at least LIN or LIS, depending on the current method. The comments on dynamic length under RWORK above apply here. Initially, this length need only be at least LIN = 20 .
The first few words of IWORK are used for conditional and optional inputs and optional outputs.
The following 2 words in IWORK are conditional inputs:
IWORK(1) = ML
IWORK(2) = MU
These are the lower and upper half-bandwidths, respectively, of the banded Jacobian, excluding the main diagonal. The band is defined by the matrix locations (i,j) with i-ML .le. j .le. i+MU. ML and MU must satisfy 0 .le. ML,MU .le. NEQ-1. These are required if JT is 4 or 5, and ignored otherwise. ML and MU may in fact be the band parameters for a matrix to which df/dy is only approximately equal.
the length of the array IWORK, as declared by the user. (This will be checked by the solver.)
Note: The base addresses of the work arrays must not be altered between calls to DLSODA for the same problem. The contents of the work arrays must not be altered between calls, except possibly for the conditional and optional inputs, and except for the last 3*NEQ words of RWORK. The latter space is used for internal scratch space, and so is available for use by the user outside DLSODA between calls, if desired (but not for use by F or JAC).
the name of the user-supplied routine to compute the Jacobian matrix, df/dy, if JT = 1 or 4. The JAC routine is optional, but if the problem is expected to be stiff much of the time, you are encouraged to supply JAC, for the sake of efficiency. (Alternatively, set JT = 2 or 5 to have DLSODA compute df/dy internally by difference quotients.) If and when DLSODA uses df/dy, it treats this NEQ by NEQ matrix either as full (JT = 1 or 2), or as banded (JT = 4 or 5) with half-bandwidths ML and MU (discussed under IWORK above). In either case, if JT = 1 or 4, the JAC routine must compute df/dy as a function of the scalar t and the vector y. It is to have the form
SUBROUTINE JAC (NEQ, T, Y, ML, MU, PD, NROWPD)
DOUBLE PRECISION T, Y(*), PD(NROWPD,*)
where NEQ, T, Y, ML, MU, and NROWPD are input and the array PD is to be loaded with partial derivatives (elements of the Jacobian matrix) on output. PD must be given a first dimension of NROWPD. T and Y have the same meaning as in Subroutine F.
In the full matrix case (JT = 1), ML and MU are ignored, and the Jacobian is to be loaded into PD in columnwise manner, with df(i)/dy(j) loaded into PD(i,j).
In the band matrix case (JT = 4), the elements within the band are to be loaded into PD in columnwise manner, with diagonal lines of df/dy loaded into the rows of PD. Thus df(i)/dy(j) is to be loaded into PD(i-j+MU+1,j). ML and MU are the half-bandwidth parameters (see IWORK). The locations in PD in the two triangular areas which correspond to nonexistent matrix elements can be ignored or loaded arbitrarily, as they are overwritten by DLSODA.
JAC need not provide df/dy exactly. A crude approximation (possibly with a smaller bandwidth) will do.
In either case, PD is preset to zero by the solver, so that only the nonzero elements need be loaded by JAC. Each call to JAC is preceded by a call to F with the same arguments NEQ, T, and Y. Thus to gain some efficiency, intermediate quantities shared by both calculations may be saved in a user Common block by F and not recomputed by JAC, if desired. Also, JAC may alter the Y array, if desired. JAC must be declared External in the calling program.
Subroutine JAC may access user-defined quantities in NEQ(2),… and/or in Y(NEQ(1)+1),… if NEQ is an array (dimensioned in JAC) and/or Y has length exceeding NEQ(1). See the descriptions of NEQ and Y above.
Jacobian type indicator. Used only for input. JT specifies how the Jacobian matrix df/dy will be treated, if and when DLSODA requires this matrix. JT has the following values and meanings:
| value | description |
|---|---|
| 1 | means a user-supplied full (NEQ by NEQ) Jacobian. |
| 2 | means an internally generated (difference quotient) full |
| Jacobian (using NEQ extra calls to F per df/dy value). | |
| 4 | means a user-supplied banded Jacobian. |
| 5 | means an internally generated banded Jacobian (using |
| ML+MU+1 extra calls to F per df/dy evaluation). |
If JT = 1 or 4, the user must supply a Subroutine JAC (the name is arbitrary) as described above under JAC. If JT = 2 or 5, a dummy argument can be used.
The following is a list of the optional inputs provided for in the call sequence. (See also Part 2.) For each such input variable, this table lists its name as used in this documentation, its location in the call sequence, its meaning, and the default value. The use of any of these inputs requires IOPT = 1, and in that case all of these inputs are examined. A value of zero for any of these optional inputs will cause the default value to be used. Thus to use a subset of the optional inputs, simply preload locations 5 to 10 in RWORK and IWORK to 0.0 and 0 respectively, and then set those of interest to nonzero values.
| Name | Location | Meaning and Default Value |
|---|---|---|
| H0 | RWORK(5) | the step size to be attempted on the first step. |
| The default value is determined by the solver. | ||
| HMAX | RWORK(6) | the maximum absolute step size allowed. |
| The default value is infinite. | ||
| HMIN | RWORK(7) | the minimum absolute step size allowed. |
| The default value is 0. (This lower bound is not | ||
| enforced on the final step before reaching TCRIT | ||
| when ITASK = 4 or 5.) | ||
| IXPR | IWORK(5) | flag to generate extra printing at method switches. |
| IXPR = 0 means no extra printing (the default). | ||
| IXPR = 1 means print data on each switch. | ||
| T, H, and NST will be printed on the same logical | ||
| unit as used for error messages. | ||
| MXSTEP | IWORK(6) | maximum number of (internally defined) steps |
| allowed during one call to the solver. | ||
| The default value is 500. | ||
| MXHNIL | IWORK(7) | maximum number of messages printed (per problem) |
| warning that T + H = T on a step (H = step size). | ||
| This must be positive to result in a non-default | ||
| value. The default value is 10. | ||
| MXORDN | IWORK(8) | the maximum order to be allowed for the nonstiff |
| (Adams) method. the default value is 12. | ||
| if MXORDN exceeds the default value, it will | ||
| be reduced to the default value. | ||
| MXORDN is held constant during the problem. | ||
| MXORDS | IWORK(9) | the maximum order to be allowed for the stiff |
| (BDF) method. The default value is 5. | ||
| If MXORDS exceeds the default value, it will | ||
| be reduced to the default value. | ||
| MXORDS is held constant during the problem. |
As optional additional output from DLSODA, the variables listed below are quantities related to the performance of DLSODA which are available to the user. These are communicated by way of the work arrays, but also have internal mnemonic names as shown. except where stated otherwise, all of these outputs are defined on any successful return from DLSODA, and on any return with ISTATE = -1, -2, -4, -5, or -6. On an illegal input return (ISTATE = -3), they will be unchanged from their existing values (if any), except possibly for TOLSF, LENRW, and LENIW. On any error return, outputs relevant to the error will be defined, as noted below.
| Name | Location | Meaning |
|---|---|---|
| HU | RWORK(11) | the step size in t last used (successfully). |
| HCUR | RWORK(12) | the step size to be attempted on the next step. |
| TCUR | RWORK(13) | the current value of the independent variable |
| which the solver has actually reached, i.e. the | ||
| current internal mesh point in t. On output, TCUR | ||
| will always be at least as far as the argument | ||
| T, but may be farther (if interpolation was done). | ||
| TOLSF | RWORK(14) | a tolerance scale factor, greater than 1.0, |
| computed when a request for too much accuracy was | ||
| detected (ISTATE = -3 if detected at the start of | ||
| the problem, ISTATE = -2 otherwise). If ITOL is | ||
| left unaltered but RTOL and ATOL are uniformly | ||
| scaled up by a factor of TOLSF for the next call, | ||
| then the solver is deemed likely to succeed. | ||
| (The user may also ignore TOLSF and alter the | ||
| tolerance parameters in any other way appropriate.) | ||
| TSW | RWORK(15) | the value of t at the time of the last method |
| switch, if any. | ||
| NST | IWORK(11) | the number of steps taken for the problem so far. |
| NFE | IWORK(12) | the number of f evaluations for the problem so far. |
| NJE | IWORK(13) | the number of Jacobian evaluations (and of matrix |
| LU decompositions) for the problem so far. | ||
| NQU | IWORK(14) | the method order last used (successfully). |
| NQCUR | IWORK(15) | the order to be attempted on the next step. |
| IMXER | IWORK(16) | the index of the component of largest magnitude in |
| the weighted local error vector ( E(i)/EWT(i) ), | ||
| on an error return with ISTATE = -4 or -5. | ||
| LENRW | IWORK(17) | the length of RWORK actually required, assuming |
| that the length of RWORK is to be fixed for the | ||
| rest of the problem, and that switching may occur. | ||
| This is defined on normal returns and on an illegal | ||
| input return for insufficient storage. | ||
| LENIW | IWORK(18) | the length of IWORK actually required, assuming |
| that the length of IWORK is to be fixed for the | ||
| rest of the problem, and that switching may occur. | ||
| This is defined on normal returns and on an illegal | ||
| input return for insufficient storage. | ||
| MUSED | IWORK(19) | the method indicator for the last successful step: |
| 1 means Adams (nonstiff), 2 means BDF (stiff). | ||
| MCUR | IWORK(20) | the current method indicator: |
| 1 means Adams (nonstiff), 2 means BDF (stiff). | ||
| This is the method to be attempted | ||
| on the next step. Thus it differs from MUSED | ||
| only if a method switch has just been made. |
The following two arrays are segments of the RWORK array which may also be of interest to the user as optional outputs. For each array, the table below gives its internal name, its base address in RWORK, and its description.
| Name | Base Address | Description |
|---|---|---|
| YH | 21 | the Nordsieck history array, of size NYH by |
| (NQCUR + 1), where NYH is the initial value | ||
| of NEQ. For j = 0,1,…,NQCUR, column j+1 | ||
| of YH contains HCUR**j/factorial(j) times | ||
| the j-th derivative of the interpolating | ||
| polynomial currently representing the solution, | ||
| evaluated at T = TCUR. | ||
| ACOR | LACOR | array of size NEQ used for the accumulated |
| (from Common | corrections on each step, scaled on output | |
| as noted) | to represent the estimated local error in y | |
| on the last step. This is the vector E in | ||
| the description of the error control. It is | ||
| defined only on a successful return from | ||
| DLSODA. The base address LACOR is obtained by | ||
| including in the user’s program the | ||
| following 2 lines: | ||
| COMMON /DLS001/ RLS(218), ILS(37) | ||
| LACOR = ILS(22) |
The following are optional calls which the user may make to gain additional capabilities in conjunction with DLSODA. (The routines XSETUN and XSETF are designed to conform to the SLATEC error handling package.)
| Form of Call | Function |
|---|---|
| CALL XSETUN(LUN) | set the logical unit number, LUN, for |
| output of messages from DLSODA, if | |
| the default is not desired. | |
| The default value of LUN is 6. | |
| CALL XSETF(MFLAG) | set a flag to control the printing of |
| messages by DLSODA. | |
| MFLAG = 0 means do not print. (Danger: | |
| This risks losing valuable information.) | |
| MFLAG = 1 means print (the default). | |
| Either of the above calls may be made at | |
| any time and will take effect immediately. | |
| CALL DSRCMA(RSAV,ISAV,JOB) | saves and restores the contents of |
| the internal Common blocks used by | |
| DLSODA (see Part 3 below). | |
| RSAV must be a real array of length 240 | |
| or more, and ISAV must be an integer | |
| array of length 46 or more. | |
| JOB=1 means save Common into RSAV/ISAV. | |
| JOB=2 means restore Common from RSAV/ISAV. | |
| DSRCMA is useful if one is | |
| interrupting a run and restarting | |
| later, or alternating between two or | |
| more problems solved with DLSODA. | |
| CALL DINTDY(,,,,,) | provide derivatives of y, of various |
| (see below) | orders, at a specified point t, if |
| desired. It may be called only after | |
| a successful return from DLSODA. |
The detailed instructions for using DINTDY are as follows:
The form of the call is:
CALL DINTDY (T, K, RWORK(21), NYH, DKY, IFLAG)
The input parameters are:
value of independent variable where answers are desired (normally the same as the T last returned by DLSODA). For valid results, T must lie between TCUR - HU and TCUR. (See optional outputs for TCUR and HU.)
integer order of the derivative desired. K must satisfy 0 .le. K .le. NQCUR, where NQCUR is the current order (see optional outputs). The capability corresponding to K = 0, i.e. computing y(T), is already provided by DLSODA directly. Since NQCUR .ge. 1, the first derivative dy/dt is always available with DINTDY.
the base address of the history array YH.
The output parameters are:
If the solution of a given problem by DLSODA is to be interrupted and then later continued, such as when restarting an interrupted run or alternating between two or more problems, the user should save, following the return from the last DLSODA call prior to the interruption, the contents of the call sequence variables and state and later restore these values before the next DLSODA call for that problem. To save and restore the current state, use Subroutine DSRCMA (see Part 2 above).
Below is a description of a routine in the DLSODA package which relates to the measurement of errors, and can be replaced by a user-supplied version, if desired. However, since such a replacement may have a major impact on performance, it should be done only when absolutely necessary, and only with great caution. (Note: The means by which the package version of a routine is superseded by the user’s version may be system-dependent.)
The following subroutine is called just before each internal integration step, and sets the array of error weights, EWT, as described under ITOL/RTOL/ATOL above:
Subroutine DEWSET (NEQ, ITOL, RTOL, ATOL, YCUR, EWT)
where NEQ, ITOL, RTOL, and ATOL are as in the DLSODA call sequence, YCUR contains the current dependent variable vector, and EWT is the array of weights set by DEWSET.
If the user supplies this subroutine, it must return in EWT(i) (i = 1,…,NEQ) a positive quantity suitable for comparing errors in y(i) to. The EWT array returned by DEWSET is passed to the DMNORM routine, and also used by DLSODA in the computation of the optional output IMXER, and the increments for difference quotient Jacobians.
In the user-supplied version of DEWSET, it may be desirable to use the current values of derivatives of y. Derivatives up to order NQ are available from the history array YH, described above under optional outputs. In DEWSET, YH is identical to the YCUR array, extended to NQ + 1 columns with a column length of NYH and scale factors of H**j/factorial(j). On the first call for the problem, given by NST = 0, NQ is 1 and H is temporarily set to 1.0. NYH is the initial value of NEQ. The quantities NQ, H, and NST can be obtained by including in DEWSET the statements:
DOUBLE PRECISION RLS
COMMON /DLS001/ RLS(218),ILS(37)
NQ = ILS(33)
NST = ILS(34)
H = RLS(212)
Thus, for example, the current value of dy/dt can be obtained as YCUR(NYH+i)/H (i=1,…,NEQ) (and the division by H is unnecessary when NST = 0).
DLSODA is derived from the Livermore Solver for Ordinary Differential Equations package ODEPACK, and is based on the 12 November 2003 version in double precision.
Alan C. Hindmarsh
Center for Applied Scientific Computing, L-561
Lawrence Livermore National Laboratory
Livermore, CA 94551
and
Linda R. Petzold
Univ. of California at Santa Barbara
Dept. of Computer Science
Santa Barbara, CA 93106
In addition to Subroutine DLSODA, the DLSODA package includes the following subroutines and function routines:
computes an interpolated value of the y vector at t = TOUT.
is the core integrator, which does one step of the integration and the associated error control.
sets all method coefficients and test constants.
computes and preprocesses the Jacobian matrix J = df/dy and the Newton iteration matrix P = I - h*l0*J.
manages solution of linear system in chord iteration.
sets the error weight vector EWT before each step.
computes the weighted max-norm of a vector.
computes the norm of a full matrix consistent with the weighted max-norm on vectors.
computes the norm of a band matrix consistent with the weighted max-norm on vectors.
is a user-callable routine to save and restore the contents of the internal Common blocks.
are routines from LINPACK for solving full systems of linear algebraic equations.
are routines from LINPACK for solving banded linear systems.
handle the printing of all error messages and warnings. XERRWD is machine-dependent.
Note: DMNORM, DFNORM, DBNORM, and IXSAV are function routines. All the others are subroutines.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| real | :: | f | ||||
| integer | :: | Neq(*) | ||||
| real(kind=dp) | :: | Y(*) | ||||
| real(kind=dp), | intent(inout) | :: | T | |||
| real(kind=dp), | intent(inout) | :: | Tout | |||
| integer | :: | Itol | ||||
| real(kind=dp) | :: | Rtol(*) | ||||
| real(kind=dp) | :: | Atol(*) | ||||
| integer | :: | Itask | ||||
| integer | :: | Istate | ||||
| integer | :: | Iopt | ||||
| real(kind=dp), | intent(inout) | :: | Rwork(Lrw) | |||
| integer | :: | Lrw | ||||
| integer, | intent(inout) | :: | Iwork(Liw) | |||
| integer | :: | Liw | ||||
| integer | :: | jac | ||||
| integer | :: | Jt |
| Type | Visibility | Attributes | Name | Initial | |||
|---|---|---|---|---|---|---|---|
| real(kind=dp), | public | :: | atoli | ||||
| real(kind=dp), | public | :: | ayi | ||||
| real(kind=dp), | public | :: | big | ||||
| real(kind=dp), | public | :: | ewti | ||||
| real(kind=dp), | public | :: | h0 | ||||
| real(kind=dp), | public | :: | hmax | ||||
| real(kind=dp), | public | :: | hmx | ||||
| integer, | public | :: | i | ||||
| integer, | public | :: | i1 | ||||
| integer, | public | :: | i2 | ||||
| integer, | public | :: | iflag | ||||
| logical, | public | :: | ihit | ||||
| integer, | public | :: | imxer | ||||
| integer, | public | :: | kgo | ||||
| integer, | public | :: | len1 | ||||
| integer, | public | :: | len1c | ||||
| integer, | public | :: | len1n | ||||
| integer, | public | :: | len1s | ||||
| integer, | public | :: | len2 | ||||
| integer, | public | :: | leniw | ||||
| integer, | public | :: | leniwc | ||||
| integer, | public | :: | lenrw | ||||
| integer, | public | :: | lenrwc | ||||
| integer, | public | :: | lenwm | ||||
| integer, | public | :: | lf0 | ||||
| integer, | public | :: | ml | ||||
| integer, | public, | dimension(2), save | :: | mord | |||
| character(len=60), | public | :: | msg | ||||
| integer, | public | :: | mu | ||||
| integer, | public, | save | :: | mxhnl0 | |||
| integer, | public, | save | :: | mxstp0 | |||
| real(kind=dp), | public | :: | rh | ||||
| real(kind=dp), | public | :: | rtoli | ||||
| real(kind=dp), | public | :: | size | ||||
| real(kind=dp), | public | :: | sum | ||||
| real(kind=dp), | public | :: | tcrit | ||||
| real(kind=dp), | public | :: | tdist | ||||
| real(kind=dp), | public | :: | tnext | ||||
| real(kind=dp), | public | :: | tol | ||||
| real(kind=dp), | public | :: | tolsf | ||||
| real(kind=dp), | public | :: | tp | ||||
| real(kind=dp), | public | :: | w0 |
subroutine dlsoda(f,Neq,Y,T,Tout,Itol,Rtol,Atol,Itask,Istate,Iopt,Rwork,Lrw,Iwork,Liw,jac,Jt) ! external :: f integer :: Neq(*) real(kind=dp) :: Y(*) real(kind=dp),intent(inout) :: T real(kind=dp),intent(inout) :: Tout integer :: Itol real(kind=dp) :: Rtol(*) real(kind=dp) :: Atol(*) integer :: Itask integer :: Istate integer :: Iopt integer :: Lrw real(kind=dp),intent(inout) :: Rwork(Lrw) integer :: Liw integer,intent(inout) :: Iwork(Liw) external :: jac integer :: Jt real(kind=dp) :: atoli, ayi, big, ewti, h0, hmax, hmx, rh, rtoli, & & size, sum, tcrit, tdist, tnext, tol, tolsf, tp, w0 integer :: i, i1, i2, iflag, imxer, kgo, len1, len1c, len1n, len1s, & & len2, leniw, leniwc, lenrw, lenrwc, lenwm, lf0 , ml, mu logical :: ihit integer,dimension(2),save :: mord character(60) :: msg integer,save :: mxhnl0, mxstp0 ! ! ----------------------------------------------------------------------- ! The following two internal Common blocks contain ! (a) variables which are local to any subroutine but whose values must ! be preserved between calls to the routine ("own" variables), and ! (b) variables which are communicated between subroutines. ! The block DLS001 is declared in subroutines DLSODA, DINTDY, DSTODA, ! DPRJA, and DSOLSY. ! The type(DLSA01)::DLSA is declared in subroutines DLSODA, DSTODA, and DPRJA. ! Groups of variables are replaced by dummy arrays in the Common ! declarations in routines where those variables are not used. ! ----------------------------------------------------------------------- ! data mord(1), mord(2)/12, 5/, mxstp0/500/, mxhnl0/10/ ihit=.false. ! ----------------------------------------------------------------------- ! Block A. ! This code block is executed on every call. ! It tests ISTATE and ITASK for legality and branches appropriately. ! If ISTATE .gt. 1 but the flag INIT shows that initialization has ! not yet been done, an error return occurs. ! If ISTATE = 1 and TOUT = T, return immediately. ! ----------------------------------------------------------------------- if ( Istate<1 .or. Istate>3 ) then msg = 'DLSODA- ISTATE (=I1) illegal.' call xerrwd(msg,30,1,0,1,Istate,0,0,0.0D0,0.0D0) if ( Istate<0 ) then msg = 'DLSODA- Run aborted.. apparent infinite loop. ' call xerrwd(msg,50,303,2,0,0,0,0,0.0D0,0.0D0) return endif Istate = -3 return endif if ( Itask<1 .or. Itask>5 ) then msg = 'DLSODA- ITASK (=I1) illegal. ' call xerrwd(msg,30,2,0,1,Itask,0,0,0.0D0,0.0D0) Istate = -3 return endif if ( Istate==1 ) then dls1%init = 0 if ( Tout==T ) return else if ( dls1%init==0 ) then msg = 'DLSODA- ISTATE .gt. 1 but DLSODA not initialized.' call xerrwd(msg,50,3,0,0,0,0,0,0.0D0,0.0D0) Istate = -3 return endif if ( Istate==2 ) goto 100 endif ! ----------------------------------------------------------------------- ! Block B. ! The next code block is executed for the initial call (ISTATE = 1), ! or for a continuation call with parameter changes (ISTATE = 3). ! It contains checking of all inputs and various initializations. ! ! First check legality of the non-optional inputs NEQ, ITOL, IOPT, ! JT, ML, and MU. ! ----------------------------------------------------------------------- if ( Neq(1)<=0 ) then msg = 'DLSODA- NEQ (=I1) .lt. 1 ' call xerrwd(msg,30,4,0,1,Neq(1),0,0,0.0D0,0.0D0) Istate = -3 return endif if ( Istate/=1 ) then if ( Neq(1)>dls1%n ) then msg = 'DLSODA- ISTATE = 3 and NEQ increased (I1 to I2). ' call xerrwd(msg,50,5,0,2,dls1%n,Neq(1),0,0.0D0,0.0D0) Istate = -3 return endif endif dls1%n = Neq(1) if ( Itol<1 .or. Itol>4 ) then msg = 'DLSODA- ITOL (=I1) illegal. ' call xerrwd(msg,30,6,0,1,Itol,0,0,0.0D0,0.0D0) Istate = -3 return endif if ( Iopt<0 .or. Iopt>1 ) then msg = 'DLSODA- IOPT (=I1) illegal. ' call xerrwd(msg,30,7,0,1,Iopt,0,0,0.0D0,0.0D0) Istate = -3 return endif if ( Jt==3 .or. Jt<1 .or. Jt>5 ) then msg = 'DLSODA- JT (=I1) illegal. ' call xerrwd(msg,30,8,0,1,Jt,0,0,0.0D0,0.0D0) Istate = -3 return endif dlsa%jtyp = Jt if ( Jt>2 ) then ml = Iwork(1) mu = Iwork(2) if ( ml<0 .or. ml>=dls1%n ) then msg = 'DLSODA- ML (=I1) illegal: .lt.0 or .ge.NEQ (=I2) ' call xerrwd(msg,50,9,0,2,ml,Neq(1),0,0.0D0,0.0D0) Istate = -3 return endif if ( mu<0 .or. mu>=dls1%n ) then msg = 'DLSODA- MU (=I1) illegal: .lt.0 or .ge.NEQ (=I2) ' call xerrwd(msg,50,10,0,2,mu,Neq(1),0,0.0D0,0.0D0) Istate = -3 return endif endif ! Next process and check the optional inputs. -------------------------- if ( Iopt==1 ) then dlsa%ixpr = Iwork(5) if ( dlsa%ixpr<0 .or. dlsa%ixpr>1 ) then msg = 'DLSODA- IXPR (=I1) illegal. ' call xerrwd(msg,30,11,0,1,dlsa%ixpr,0,0,0.0D0,0.0D0) Istate = -3 return endif dls1%mxstep = Iwork(6) if ( dls1%mxstep<0 ) then msg = 'DLSODA- MXSTEP (=I1) .lt. 0 ' call xerrwd(msg,30,12,0,1,dls1%mxstep,0,0,0.0D0,0.0D0) Istate = -3 return endif if ( dls1%mxstep==0 ) dls1%mxstep = mxstp0 dls1%mxhnil = Iwork(7) if ( dls1%mxhnil<0 ) then msg = 'DLSODA- MXHNIL (=I1) .lt. 0 ' call xerrwd(msg,30,13,0,1,dls1%mxhnil,0,0,0.0D0,0.0D0) Istate = -3 return endif if ( dls1%mxhnil==0 ) dls1%mxhnil = mxhnl0 if ( Istate==1 ) then h0 = Rwork(5) dlsa%mxordn = Iwork(8) if ( dlsa%mxordn<0 ) then msg = 'DLSODA- MXORDN (=I1) .lt. 0 ' call xerrwd(msg,30,28,0,1,dlsa%mxordn,0,0,0.0D0,0.0D0) Istate = -3 return endif if ( dlsa%mxordn==0 ) dlsa%mxordn = 100 dlsa%mxordn = min(dlsa%mxordn,mord(1)) dlsa%mxords = Iwork(9) if ( dlsa%mxords<0 ) then msg = 'DLSODA- MXORDS (=I1) .lt. 0 ' call xerrwd(msg,30,29,0,1,dlsa%mxords,0,0,0.0D0,0.0D0) Istate = -3 return endif if ( dlsa%mxords==0 ) dlsa%mxords = 100 dlsa%mxords = min(dlsa%mxords,mord(2)) if ( (Tout-T)*h0<0.0D0 ) then msg = 'DLSODA- TOUT (=R1) behind T (=R2) ' call xerrwd(msg,40,14,0,0,0,0,2,Tout,T) msg = ' Integration direction is given by H0 (=R1) ' call xerrwd(msg,50,14,0,0,0,0,1,h0,0.0D0) Istate = -3 return endif endif hmax = Rwork(6) if ( hmax<0.0D0 ) then msg = 'DLSODA- HMAX (=R1) .lt. 0.0 ' call xerrwd(msg,30,15,0,0,0,0,1,hmax,0.0D0) Istate = -3 return endif dls1%hmxi = 0.0D0 if ( hmax>0.0D0 ) dls1%hmxi = 1.0D0/hmax dls1%hmin = Rwork(7) if ( dls1%hmin<0.0D0 ) then msg = 'DLSODA- HMIN (=R1) .lt. 0.0 ' call xerrwd(msg,30,16,0,0,0,0,1,dls1%hmin,0.0D0) Istate = -3 return endif else dlsa%ixpr = 0 dls1%mxstep = mxstp0 dls1%mxhnil = mxhnl0 dls1%hmxi = 0.0D0 dls1%hmin = 0.0D0 if ( Istate==1 ) then h0 = 0.0D0 dlsa%mxordn = mord(1) dlsa%mxords = mord(2) endif endif ! ----------------------------------------------------------------------- ! Set work array pointers and check lengths LRW and LIW. ! If ISTATE = 1, METH is initialized to 1 here to facilitate the ! checking of work space lengths. ! Pointers to segments of RWORK and IWORK are named by prefixing L to ! the name of the segment. E.g., the segment YH starts at RWORK(LYH). ! Segments of RWORK (in order) are denoted YH, WM, EWT, SAVF, ACOR. ! If the lengths provided are insufficient for the current method, ! an error return occurs. This is treated as illegal input on the ! first call, but as a problem interruption with ISTATE = -7 on a ! continuation call. If the lengths are sufficient for the current ! method but not for both methods, a warning message is sent. ! ----------------------------------------------------------------------- if ( Istate==1 ) dls1%meth = 1 if ( Istate==1 ) dls1%nyh = dls1%n dls1%lyh = 21 len1n = 20 + (dlsa%mxordn+1)*dls1%nyh len1s = 20 + (dlsa%mxords+1)*dls1%nyh dls1%lwm = len1s + 1 lenwm=0 if ( Jt<=2 ) lenwm = dls1%n*dls1%n + 2 if ( Jt>=4 ) lenwm = (2*ml+mu+1)*dls1%n + 2 len1s = len1s + lenwm len1c = len1n if ( dls1%meth==2 ) len1c = len1s len1 = max(len1n,len1s) len2 = 3*dls1%n lenrw = len1 + len2 lenrwc = len1c + len2 Iwork(17) = lenrw dls1%liwm = 1 leniw = 20 + dls1%n leniwc = 20 if ( dls1%meth==2 ) leniwc = leniw Iwork(18) = leniw if ( Istate==1 .and. Lrw<lenrwc ) then msg = 'DLSODA- RWORK length needed, LENRW (=I1), exceeds LRW (=I2)' call xerrwd(msg,60,17,0,2,lenrw,Lrw,0,0.0D0,0.0D0) Istate = -3 return endif if ( Istate==1 .and. Liw<leniwc ) then msg = 'DLSODA- IWORK length needed, LENIW (=I1), exceeds LIW (=I2)' call xerrwd(msg,60,18,0,2,leniw,Liw,0,0.0D0,0.0D0) Istate = -3 return endif if ( Istate==3 .and. Lrw<lenrwc ) goto 600 if ( Istate==3 .and. Liw<leniwc ) goto 700 dls1%lewt = len1 + 1 dlsa%insufr = 0 if ( Lrw<lenrw ) then dlsa%insufr = 2 dls1%lewt = len1c + 1 msg = 'DLSODA- Warning.. RWORK length is sufficient for now, but ' call xerrwd(msg,60,103,0,0,0,0,0,0.0D0,0.0D0) msg = ' may not be later. Integration will proceed anyway. ' call xerrwd(msg,60,103,0,0,0,0,0,0.0D0,0.0D0) msg = ' Length needed is LENRW = I1, while LRW = I2.' call xerrwd(msg,50,103,0,2,lenrw,Lrw,0,0.0D0,0.0D0) endif dls1%lsavf = dls1%lewt + dls1%n dls1%lacor = dls1%lsavf + dls1%n dlsa%insufi = 0 if ( Liw<leniw ) then dlsa%insufi = 2 msg = 'DLSODA- Warning.. IWORK length is sufficient for now, but ' call xerrwd(msg,60,104,0,0,0,0,0,0.0D0,0.0D0) msg = ' may not be later. Integration will proceed anyway. ' call xerrwd(msg,60,104,0,0,0,0,0,0.0D0,0.0D0) msg = ' Length needed is LENIW = I1, while LIW = I2.' call xerrwd(msg,50,104,0,2,leniw,Liw,0,0.0D0,0.0D0) endif ! Check RTOL and ATOL for legality. ------------------------------------ rtoli = Rtol(1) atoli = Atol(1) do i = 1, dls1%n if ( Itol>=3 ) rtoli = Rtol(i) if ( Itol==2 .or. Itol==4 ) atoli = Atol(i) if ( rtoli<0.0D0 ) then msg = 'DLSODA- RTOL(I1) is R1 .lt. 0.0 ' call xerrwd(msg,40,19,0,1,i,0,1,rtoli,0.0D0) Istate = -3 return endif if ( atoli<0.0D0 ) then msg = 'DLSODA- ATOL(I1) is R1 .lt. 0.0 ' call xerrwd(msg,40,20,0,1,i,0,1,atoli,0.0D0) Istate = -3 return endif enddo if ( Istate==1 ) then ! ----------------------------------------------------------------------- ! Block C. ! The next block is for the initial call only (ISTATE = 1). ! It contains all remaining initializations, the initial call to F, ! and the calculation of the initial step size. ! The error weights in EWT are inverted after being loaded. ! ----------------------------------------------------------------------- dls1%uround = epsilon(0.0d0) dls1%tn = T dlsa%tsw = T dls1%maxord = dlsa%mxordn if ( Itask==4 .or. Itask==5 ) then tcrit = Rwork(1) if ( (tcrit-Tout)*(Tout-T)<0.0D0 ) then msg = 'DLSODA- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) ' call xerrwd(msg,60,25,0,0,0,0,2,tcrit,Tout) Istate = -3 return endif if ( h0/=0.0D0 .and. (T+h0-tcrit)*h0>0.0D0 ) h0 = tcrit - T endif dls1%jstart = 0 dls1%nhnil = 0 dls1%nst = 0 dls1%nje = 0 dls1%nslast = 0 dls1%hu = 0.0D0 dls1%nqu = 0 dlsa%mused = 0 dls1%miter = 0 dls1%ccmax = 0.3D0 dls1%maxcor = 3 dls1%msbp = 20 dls1%mxncf = 10 ! Initial call to F. (LF0 points to YH(*,2).) ------------------------- lf0 = dls1%lyh + dls1%nyh call f(Neq,T,Y,Rwork(lf0)) dls1%nfe = 1 ! Load the initial value vector in YH. --------------------------------- do i = 1, dls1%n Rwork(i+dls1%lyh-1) = Y(i) enddo ! Load and invert the EWT array. (H is temporarily set to 1.0.) ------- dls1%nq = 1 dls1%h = 1.0D0 call dewset(dls1%n,Itol,Rtol,Atol,Rwork(dls1%lyh),Rwork(dls1%lewt)) do i = 1, dls1%n if ( Rwork(i+dls1%lewt-1)<=0.0D0 ) then ewti = Rwork(dls1%lewt+i-1) msg = 'DLSODA- EWT(I1) is R1 .le. 0.0 ' call xerrwd(msg,40,21,0,1,i,0,1,ewti,0.0D0) Istate = -3 return endif Rwork(i+dls1%lewt-1) = 1.0D0/Rwork(i+dls1%lewt-1) enddo ! ----------------------------------------------------------------------- ! The coding below computes the step size, H0, to be attempted on the ! first step, unless the user has supplied a value for this. ! First check that TOUT - T differs significantly from zero. ! A scalar tolerance quantity TOL is computed, as MAX(RTOL(i)) ! if this is positive, or MAX(ATOL(i)/ABS(Y(i))) otherwise, adjusted ! so as to be between 100*UROUND and 1.0E-3. ! Then the computed value H0 is given by: ! ! H0**(-2) = 1./(TOL * w0**2) + TOL * (norm(F))**2 ! ! where w0 = MAX ( ABS(T), ABS(TOUT) ), ! F = the initial value of the vector f(t,y), and ! norm() = the weighted vector norm used throughout, given by ! the DMNORM function routine, and weighted by the ! tolerances initially loaded into the EWT array. ! The sign of H0 is inferred from the initial values of TOUT and T. ! ABS(H0) is made .le. ABS(TOUT-T) in any case. ! ----------------------------------------------------------------------- if ( h0==0.0D0 ) then tdist = abs(Tout-T) w0 = max(abs(T),abs(Tout)) if ( tdist<2.0D0*dls1%uround*w0 ) then msg = 'DLSODA- TOUT(=R1) too close to T(=R2) to start integration.' call xerrwd(msg,60,22,0,0,0,0,2,Tout,T) Istate = -3 return endif tol = Rtol(1) if ( Itol>2 ) then do i = 1, dls1%n tol = max(tol,Rtol(i)) enddo endif if ( tol<=0.0D0 ) then atoli = Atol(1) do i = 1, dls1%n if ( Itol==2 .or. Itol==4 ) atoli = Atol(i) ayi = abs(Y(i)) if ( ayi/=0.0D0 ) tol = max(tol,atoli/ayi) enddo endif tol = max(tol,100.0D0*dls1%uround) tol = min(tol,0.001D0) sum = dmnorm(dls1%n,Rwork(lf0),Rwork(dls1%lewt)) sum = 1.0D0/(tol*w0*w0) + tol*sum**2 h0 = 1.0D0/sqrt(sum) h0 = min(h0,tdist) h0 = sign(h0,Tout-T) endif ! Adjust H0 if necessary to meet HMAX bound. --------------------------- rh = abs(h0)*dls1%hmxi if ( rh>1.0D0 ) h0 = h0/rh ! Load H with H0 and scale YH(*,2) by H0. ------------------------------ dls1%h = h0 do i = 1, dls1%n Rwork(i+lf0-1) = h0*Rwork(i+lf0-1) enddo goto 300 else ! If ISTATE = 3, set flag to signal parameter changes to DSTODA. ------- dls1%jstart = -1 if ( dls1%n/=dls1%nyh ) then ! NEQ was reduced. Zero part of YH to avoid undefined references. ----- i1 = dls1%lyh + dls1%l*dls1%nyh i2 = dls1%lyh + (dls1%maxord+1)*dls1%nyh - 1 if ( i1<=i2 ) then do i = i1, i2 Rwork(i) = 0.0D0 enddo endif endif endif ! ----------------------------------------------------------------------- ! Block D. ! The next code block is for continuation calls only (ISTATE = 2 or 3) ! and is to check stop conditions before taking a step. ! ----------------------------------------------------------------------- 100 continue dls1%nslast = dls1%nst select case (Itask) case (2) goto 200 case (3) tp = dls1%tn - dls1%hu*(1.0D0+100.0D0*dls1%uround) if ( (tp-Tout)*dls1%h>0.0D0 ) then msg = 'DLSODA- ITASK = I1 and TOUT (=R1) behind TCUR - HU (= R2) ' call xerrwd(msg,60,23,0,1,Itask,0,2,Tout,tp) Istate = -3 return endif if ( (dls1%tn-Tout)*dls1%h<0.0D0 ) goto 200 T = dls1%tn goto 400 case (4) tcrit = Rwork(1) if ( (dls1%tn-tcrit)*dls1%h>0.0D0 ) then msg = 'DLSODA- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) ' call xerrwd(msg,60,24,0,0,0,0,2,tcrit,dls1%tn) Istate = -3 return endif if ( (tcrit-Tout)*dls1%h<0.0D0 ) then msg = 'DLSODA- ITASK = 4 or 5 and TCRIT (=R1) behind TOUT (=R2) ' call xerrwd(msg,60,25,0,0,0,0,2,tcrit,Tout) Istate = -3 return endif if ( (dls1%tn-Tout)*dls1%h>=0.0D0 ) then call dintdy(Tout,0,Rwork(dls1%lyh),dls1%nyh,Y,iflag) if ( iflag/=0 ) then msg = 'DLSODA- Trouble in DINTDY. ITASK = I1, TOUT = R1' call xerrwd(msg,50,27,0,1,Itask,0,1,Tout,0.0D0) Istate = -3 return endif T = Tout goto 500 endif case (5) tcrit = Rwork(1) if ( (dls1%tn-tcrit)*dls1%h>0.0D0 ) then msg = 'DLSODA- ITASK = 4 or 5 and TCRIT (=R1) behind TCUR (=R2) ' call xerrwd(msg,60,24,0,0,0,0,2,tcrit,dls1%tn) Istate = -3 return endif case default if ( (dls1%tn-Tout)*dls1%h<0.0D0 ) goto 200 call dintdy(Tout,0,Rwork(dls1%lyh),dls1%nyh,Y,iflag) if ( iflag/=0 ) then msg = 'DLSODA- Trouble in DINTDY. ITASK = I1, TOUT = R1' call xerrwd(msg,50,27,0,1,Itask,0,1,Tout,0.0D0) Istate = -3 return endif T = Tout goto 500 endselect hmx = abs(dls1%tn) + abs(dls1%h) ihit = abs(dls1%tn-tcrit)<=100.0D0*dls1%uround*hmx if ( ihit ) T = tcrit if ( ihit ) goto 400 tnext = dls1%tn + dls1%h*(1.0D0+4.0D0*dls1%uround) if ( (tnext-tcrit)*dls1%h>0.0D0 ) then dls1%h = (tcrit-dls1%tn)*(1.0D0-4.0D0*dls1%uround) if ( Istate==2 .and. dls1%jstart>=0 ) dls1%jstart = -2 endif ! ----------------------------------------------------------------------- ! Block E. ! The next block is normally executed for all calls and contains ! the call to the one-step core integrator DSTODA. ! ! This is a looping point for the integration steps. ! ! First check for too many steps being taken, update EWT (if not at ! start of problem), check for too much accuracy being requested, and ! check for H below the roundoff level in T. ! ----------------------------------------------------------------------- 200 continue if ( dls1%meth/=dlsa%mused ) then if ( dlsa%insufr==1 ) goto 600 if ( dlsa%insufi==1 ) goto 700 endif if ( (dls1%nst-dls1%nslast)>=dls1%mxstep ) then !=================================================================================================================================== ! ----------------------------------------------------------------------- ! Block H. ! The following block handles all unsuccessful returns other than ! those for illegal input. First the error message routine is called. ! If there was an error test or convergence test failure, IMXER is set. ! Then Y is loaded from YH and T is set to TN. ! The optional outputs are loaded into the work arrays before returning. ! ----------------------------------------------------------------------- ! The maximum number of steps was taken before reaching TOUT. ---------- ! Block H. msg = 'DLSODA- At current T (=R1), MXSTEP (=I1) steps ' call xerrwd(msg,50,201,0,0,0,0,0,0.0D0,0.0D0) msg = ' taken on this call before reaching TOUT ' call xerrwd(msg,50,201,0,1,dls1%mxstep,0,1,dls1%tn,0.0D0) Istate = -1 goto 900 else call dewset(dls1%n,Itol,Rtol,Atol,Rwork(dls1%lyh),Rwork(dls1%lewt)) do i = 1, dls1%n if ( Rwork(i+dls1%lewt-1)<=0.0D0 ) then ! EWT(i) .le. 0.0 for some i (not at start of problem). ---------------- ! Block H. ewti = Rwork(dls1%lewt+i-1) msg = 'DLSODA- At T (=R1), EWT(I1) has become R2 .le. 0.' call xerrwd(msg,50,202,0,1,i,0,2,dls1%tn,ewti) Istate = -6 goto 900 else Rwork(i+dls1%lewt-1) = 1.0D0/Rwork(i+dls1%lewt-1) endif enddo endif 300 continue tolsf = dls1%uround*dmnorm(dls1%n,Rwork(dls1%lyh),Rwork(dls1%lewt)) if ( tolsf<=1.0D0 ) then if ( (dls1%tn+dls1%h)==dls1%tn ) then dls1%nhnil = dls1%nhnil + 1 if ( dls1%nhnil<=dls1%mxhnil ) then msg = 'DLSODA- Warning..Internal T (=R1) and H (=R2) are' call xerrwd(msg,50,101,0,0,0,0,0,0.0D0,0.0D0) msg = ' such that in the machine, T + H = T on the next step ' call xerrwd(msg,60,101,0,0,0,0,0,0.0D0,0.0D0) msg = ' (H = step size). Solver will continue anyway.' call xerrwd(msg,50,101,0,0,0,0,2,dls1%tn,dls1%h) if ( dls1%nhnil>=dls1%mxhnil ) then msg = 'DLSODA- Above warning has been issued I1 times. ' call xerrwd(msg,50,102,0,0,0,0,0,0.0D0,0.0D0) msg = ' It will not be issued again for this problem.' call xerrwd(msg,50,102,0,1,dls1%mxhnil,0,0,0.0D0,0.0D0) endif endif endif ! ----------------------------------------------------------------------- ! CALL DSTODA(NEQ,Y,YH,NYH,YH,EWT,SAVF,ACOR,WM,IWM,f,JAC,DPRJA,DSOLSY) ! ----------------------------------------------------------------------- call dstoda(Neq,Y,Rwork(dls1%lyh),dls1%nyh,Rwork(dls1%lyh),Rwork(dls1%lewt), & & Rwork(dls1%lsavf),Rwork(dls1%lacor),Rwork(dls1%lwm), & & Iwork(dls1%liwm),f,jac,dprja,dsolsy) kgo = 1 - dls1%kflag select case (kgo) case (2) ! KFLAG = -1. Error test failed repeatedly or with ABS(H) = HMIN. ----- ! Block H. msg = 'DLSODA- At T(=R1) and step size H(=R2), the error' call xerrwd(msg,50,204,0,0,0,0,0,0.0D0,0.0D0) msg = ' test failed repeatedly or with ABS(H) = HMIN' call xerrwd(msg,50,204,0,0,0,0,2,dls1%tn,dls1%h) Istate = -4 goto 800 case (3) ! KFLAG = -2. Convergence failed repeatedly or with ABS(H) = HMIN. ---- ! Block H. msg = 'DLSODA- At T (=R1) and step size H (=R2), the ' call xerrwd(msg,50,205,0,0,0,0,0,0.0D0,0.0D0) msg = ' corrector convergence failed repeatedly ' call xerrwd(msg,50,205,0,0,0,0,0,0.0D0,0.0D0) msg = ' or with ABS(H) = HMIN ' call xerrwd(msg,30,205,0,0,0,0,2,dls1%tn,dls1%h) Istate = -5 goto 800 case default ! ----------------------------------------------------------------------- ! Block F. ! The following block handles the case of a successful return from the ! core integrator (KFLAG = 0). ! If a method switch was just made, record TSW, reset MAXORD, ! set JSTART to -1 to signal DSTODA to complete the switch, ! and do extra printing of data if IXPR = 1. ! Then, in any case, check for stop conditions. ! ----------------------------------------------------------------------- dls1%init = 1 if ( dls1%meth/=dlsa%mused ) then dlsa%tsw = dls1%tn dls1%maxord = dlsa%mxordn if ( dls1%meth==2 ) dls1%maxord = dlsa%mxords if ( dls1%meth==2 ) Rwork(dls1%lwm) = sqrt(dls1%uround) dlsa%insufr = min(dlsa%insufr,1) dlsa%insufi = min(dlsa%insufi,1) dls1%jstart = -1 if ( dlsa%ixpr/=0 ) then if ( dls1%meth==2 ) then msg = 'DLSODA- A switch to the BDF (stiff) method has occurred ' call xerrwd(msg,60,105,0,0,0,0,0,0.0D0,0.0D0) endif if ( dls1%meth==1 ) then msg = 'DLSODA- A switch to the Adams (nonstiff) method has occurred' call xerrwd(msg,60,106,0,0,0,0,0,0.0D0,0.0D0) endif msg = ' at T = R1, tentative step size H = R2, step NST = I1 ' call xerrwd(msg,60,107,0,1,dls1%nst,0,2,dls1%tn,dls1%h) endif endif select case (Itask) case (2) case (3) ! ITASK = 3. Jump to exit if TOUT was reached. ------------------------ if ( (dls1%tn-Tout)*dls1%h<0.0D0 ) goto 200 case (4) ! ITASK = 4. See if TOUT or TCRIT was reached. Adjust H if necessary. if ( (dls1%tn-Tout)*dls1%h<0.0D0 ) then hmx = abs(dls1%tn) + abs(dls1%h) ihit = abs(dls1%tn-tcrit)<=100.0D0*dls1%uround*hmx if ( .not.(ihit) ) then tnext = dls1%tn + dls1%h*(1.0D0+4.0D0*dls1%uround) if ( (tnext-tcrit)*dls1%h>0.0D0 ) then dls1%h = (tcrit-dls1%tn)*(1.0D0-4.0D0*dls1%uround) if ( dls1%jstart>=0 ) dls1%jstart = -2 endif goto 200 endif else call dintdy(Tout,0,Rwork(dls1%lyh),dls1%nyh,Y,iflag) T = Tout goto 500 endif case (5) ! ITASK = 5. See if TCRIT was reached and jump to exit. --------------- hmx = abs(dls1%tn) + abs(dls1%h) ihit = abs(dls1%tn-tcrit)<=100.0D0*dls1%uround*hmx case default ! ITASK = 1. If TOUT has been reached, interpolate. ------------------- if ( (dls1%tn-Tout)*dls1%h<0.0D0 ) goto 200 call dintdy(Tout,0,Rwork(dls1%lyh),dls1%nyh,Y,iflag) T = Tout goto 500 endselect endselect else tolsf = tolsf*2.0D0 if ( dls1%nst==0 ) then msg = 'DLSODA- At start of problem, too much accuracy ' call xerrwd(msg,50,26,0,0,0,0,0,0.0D0,0.0D0) msg = ' requested for precision of machine.. See TOLSF (=R1) ' call xerrwd(msg,60,26,0,0,0,0,1,tolsf,0.0D0) Rwork(14) = tolsf Istate = -3 return endif ! Too much accuracy requested for machine precision. ------------------- ! Block H. msg = 'DLSODA- At T (=R1), too much accuracy requested ' call xerrwd(msg,50,203,0,0,0,0,0,0.0D0,0.0D0) msg = ' for precision of machine.. See TOLSF (=R2) ' call xerrwd(msg,50,203,0,0,0,0,2,dls1%tn,tolsf) Rwork(14) = tolsf Istate = -2 goto 900 endif ! ----------------------------------------------------------------------- ! Block G. ! The following block handles all successful returns from DLSODA. ! If ITASK .ne. 1, Y is loaded from YH and T is set accordingly. ! ISTATE is set to 2, and the optional outputs are loaded into the ! work arrays before returning. ! ----------------------------------------------------------------------- 400 continue do i = 1, dls1%n Y(i) = Rwork(i+dls1%lyh-1) enddo T = dls1%tn if ( Itask==4 .or. Itask==5 ) then if ( ihit ) T = tcrit endif 500 continue Istate = 2 Rwork(11) = dls1%hu Rwork(12) = dls1%h Rwork(13) = dls1%tn Rwork(15) = dlsa%tsw Iwork(11) = dls1%nst Iwork(12) = dls1%nfe Iwork(13) = dls1%nje Iwork(14) = dls1%nqu Iwork(15) = dls1%nq Iwork(19) = dlsa%mused Iwork(20) = dls1%meth return ! RWORK length too small to proceed. ----------------------------------- ! Block H. 600 continue msg = 'DLSODA- At current T(=R1), RWORK length too small' call xerrwd(msg,50,206,0,0,0,0,0,0.0D0,0.0D0) msg = ' to proceed. The integration was otherwise successful.' call xerrwd(msg,60,206,0,0,0,0,1,dls1%tn,0.0D0) Istate = -7 goto 900 ! IWORK length too small to proceed. ----------------------------------- ! Block H. 700 continue msg = 'DLSODA- At current T(=R1), IWORK length too small' call xerrwd(msg,50,207,0,0,0,0,0,0.0D0,0.0D0) msg = ' to proceed. The integration was otherwise successful.' call xerrwd(msg,60,207,0,0,0,0,1,dls1%tn,0.0D0) Istate = -7 goto 900 !=================================================================================================================================== ! Compute IMXER if relevant. ------------------------------------------- ! Block H. 800 continue big = 0.0D0 imxer = 1 do i = 1, dls1%n size = abs(Rwork(i+dls1%lacor-1)*Rwork(i+dls1%lewt-1)) if ( big<size ) then big = size imxer = i endif enddo Iwork(16) = imxer ! Set Y vector, T, and optional outputs. ------------------------------- 900 continue do i = 1, dls1%n Y(i) = Rwork(i+dls1%lyh-1) enddo T = dls1%tn Rwork(11) = dls1%hu Rwork(12) = dls1%h Rwork(13) = dls1%tn Rwork(15) = dlsa%tsw Iwork(11) = dls1%nst Iwork(12) = dls1%nfe Iwork(13) = dls1%nje Iwork(14) = dls1%nqu Iwork(15) = dls1%nq Iwork(19) = dlsa%mused Iwork(20) = dls1%meth ! ----------------------------------------------------------------------- ! Block I. Many returns with ISTATE=-3 ! The following block handles all error returns due to illegal input ! (ISTATE = -3), as detected before calling the core integrator. ! First the error message routine is called. If the illegal input ! is a negative ISTATE, the run is aborted (apparent infinite loop). ! ----------------------------------------------------------------------- ! end subroutine dlsoda