NAG Library Routine Document

D02NEF

1  Purpose

2  Specification

3  Description

!     Declarations

      EXTERNAL RES, JAC
             .
             .
             .
!     Initialize the integrator
      CALL D02MWF(...)
!     Is the Jacobian matrix banded?
      IF (BANDED) CALL D02NPF(...)

!     Set DT to the required temporal resolution
!     Set TEND to the final time
!     Call the integrator for each temporal value:
1000  CALL D02NEF(...,RES,JAC,...)
!     Continue integration?
      IF (TOUT.LT.TEND .AND. ITASK.GE.0) THEN
        IF (ITASK.NE.1) TOUT = MIN(TOUT+DT,TEND)
!       Print solution
        CALL D02MCF(...)
        GO TO 1000
      ENDIF
             .
             .
             .

4  References

5  Parameters

1:     $\mathrm{NEQ}$ – INTEGERInput

On entry: the number of differential-algebraic equations to be solved.

Constraint: $\mathbf{NEQ}\ge 1$.

2:     $\mathrm{T}$ – REAL (KIND=nag_wp)Input/Output

On initial entry: the initial value of the independent variable, $t$.

On intermediate exit: $t$, the current value of the independent variable.

On final exit: the value of the independent variable at which the computed solution $y$ is returned (usually at TOUT).

3:     $\mathrm{TOUT}$ – REAL (KIND=nag_wp)Input

On entry: the next value of $t$ at which a computed solution is desired.

On initial entry: TOUT is used to determine the direction of integration. Integration is permitted in either direction (see also ITASK).

Constraint: $\mathbf{TOUT}\ne \mathbf{T}$.

4:     $\mathrm{Y}\left(\mathbf{NEQ}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: the vector of initial values of the dependent variables $y$.

On intermediate exit: the computed solution vector, $y$, evaluated at $t=T$.

On final exit: the computed solution vector, evaluated at $t$ (usually $t=\mathbf{TOUT}$).

5:     $\mathrm{YDOT}\left(\mathbf{NEQ}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: YDOT must contain approximations to the time derivatives $y\text{'}$ of the vector $y$ evaluated at the initial value of the independent variable.

On exit: the time derivatives $y\text{'}$ of the vector $y$ at the last integration point.

6:     $\mathrm{RTOL}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array RTOL depends on the value of ITOL as set in D02MWF; it  must be at least $\mathbf{NEQ}$ if $\mathbf{ITOL}=\mathrm{.TRUE.}$ and at least $1$ if $\mathbf{ITOL}=\mathrm{.FALSE.}$.

On entry: the relative local error tolerance.

Constraint: $\mathbf{RTOL}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$

where $n=\mathbf{NEQ}$ when $\mathbf{ITOL}=\mathrm{.TRUE.}$ and $n=1$ otherwise.

On exit: RTOL remains unchanged unless D02NEF exits with $\mathbf{IFAIL}=\mathbf{16}$ in which case the values may have been increased to values estimated to be appropriate for continuing the integration.

7:     $\mathrm{ATOL}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array ATOL depends on the value of ITOL as set in D02MWF; it  must be at least $\mathbf{NEQ}$ if $\mathbf{ITOL}=\mathrm{.TRUE.}$ and at least $1$ if $\mathbf{ITOL}=\mathrm{.FALSE.}$.

On entry: the absolute local error tolerance.

Constraint: $\mathbf{ATOL}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$

where $n=\mathbf{NEQ}$ when $\mathbf{ITOL}=\mathrm{.TRUE.}$ and $n=1$ otherwise.

On exit: ATOL remains unchanged unless D02NEF exits with $\mathbf{IFAIL}=\mathbf{16}$ in which case the values may have been increased to values estimated to be appropriate for continuing the integration.

8:     $\mathrm{ITASK}$ – INTEGERInput/Output

On initial entry: need not be set.

On exit: the task performed by the integrator on successful completion or an indicator that a problem occurred during integration.

$\mathbf{ITASK}=2$

The integration to TOUT was successfully completed ($\mathbf{T}=\mathbf{TOUT}$) by stepping exactly to TOUT.

$\mathbf{ITASK}=3$

The integration to TOUT was successfully completed ($\mathbf{T}=\mathbf{TOUT}$) by stepping past TOUT. Y and YDOT are obtained by interpolation.

$\mathbf{ITASK}<0$

Different negative values of ITASK returned correspond to different failure exits. IFAIL should always be checked in such cases and the corrective action taken where appropriate.

ITASK must remain unchanged between calls to D02NEF.

9:     $\mathrm{RES}$ – SUBROUTINE, supplied by the user.External Procedure

RES must evaluate the residual

$$R=F\left(t,y,y^{\prime }\right)\text{.}$$

The specification of RES is:

SUBROUTINE RES ( NEQ, T, Y, YDOT, R, IRES, IUSER, RUSER) INTEGER  NEQ, IRES, IUSER(*) REAL (KIND=nag_wp)  T, Y(NEQ), YDOT(NEQ), R(NEQ), RUSER(*)

1:     $\mathrm{NEQ}$ – INTEGERInput

On entry: the number of differential-algebraic equations being solved.

2:     $\mathrm{T}$ – REAL (KIND=nag_wp)Input

On entry: $t$, the current value of the independent variable.

3:     $\mathrm{Y}\left(\mathbf{NEQ}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{NEQ}$, the current solution component.

4:     $\mathrm{YDOT}\left(\mathbf{NEQ}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the derivative of the solution at the current point $t$.

5:     $\mathrm{R}\left(\mathbf{NEQ}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{R}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $R$, for $\mathit{i}=1,2,\dots ,\mathbf{NEQ}$ where

$$R=F\left(\mathbf{T},\mathbf{Y},\mathbf{YDOT}\right)\text{.}$$

6:     $\mathrm{IRES}$ – INTEGERInput/Output

On entry: is always equal to zero.

On exit: IRES should normally be left unchanged. However, if an illegal value of Y is encountered, IRES should be set to $-1$; D02NEF will then attempt to resolve the problem so that illegal values of Y are not encountered. IRES should be set to $-2$ if you wish to return control to the calling (sub)routine; this will cause D02NEF to exit with $\mathbf{IFAIL}=\mathbf{23}$.

7:     $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace

8:     $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

RES is called with the parameters IUSER and RUSER as supplied to D02NEF. You are free to use the arrays IUSER and RUSER to supply information to RES as an alternative to using COMMON global variables.

RES must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02NEF is called. Parameters denoted as Input must not be changed by this procedure.

10:   $\mathrm{JAC}$ – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

Evaluates the matrix of partial derivatives, $J$, where

$$J_{ij}=\frac{\partial F_i}{\partial y_j}+\mathbf{CJ}\times \frac{\partial F_i}{\partial {y^{\prime }}_j}\text{,\hspace{1em}}i,j=1,2,\dots ,\mathbf{NEQ}\text{.}$$

If this option is not required, the actual argument for JAC must be the dummy routine D02NEZ. (D02NEZ is included in the NAG Library.) You must indicate to the integrator whether this option is to be used by setting the parameter JCEVAL appropriately in a call to the setup routine D02MWF.

The specification of JAC is:

SUBROUTINE JAC ( NEQ, T, Y, YDOT, PD, CJ, IUSER, RUSER) INTEGER  NEQ, IUSER(*) REAL (KIND=nag_wp)  T, Y(NEQ), YDOT(NEQ), PD(*), CJ, RUSER(*)

1:     $\mathrm{NEQ}$ – INTEGERInput

On entry: the number of differential-algebraic equations being solved.

2:     $\mathrm{T}$ – REAL (KIND=nag_wp)Input

On entry: $t$, the current value of the independent variable.

3:     $\mathrm{Y}\left(\mathbf{NEQ}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{NEQ}$, the current solution component.

4:     $\mathrm{YDOT}\left(\mathbf{NEQ}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the derivative of the solution at the current point $t$.

5:     $\mathrm{PD}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: PD is preset to zero before the call to JAC.

On exit: if the Jacobian is full then $\mathbf{PD}\left(\left(\mathit{j}-1\right)\times \mathbf{NEQ}+\mathit{i}\right)=J_{\mathit{i}\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathbf{NEQ}$ and $\mathit{j}=1,2,\dots ,\mathbf{NEQ}$; if the Jacobian is banded then $\mathbf{PD}\left(\left(j-1\right)\times \left(2\mathbf{ML}+\mathbf{MU}+1\right)+\mathbf{ML}+\mathbf{MU}+i-j+1\right)=J_{ij}$, for $\max \left(1,j-\mathbf{MU}\right)\le i\le \min \left(n,j+\mathbf{ML}\right)$; (see also in F07BDF (DGBTRF)).

6:     $\mathrm{CJ}$ – REAL (KIND=nag_wp)Input

On entry: CJ is a scalar constant which will be defined in D02NEF.

7:     $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace

8:     $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

JAC is called with the parameters IUSER and RUSER as supplied to D02NEF. You are free to use the arrays IUSER and RUSER to supply information to JAC as an alternative to using COMMON global variables.

JAC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02NEF is called. Parameters denoted as Input must not be changed by this procedure.

11:   $\mathrm{ICOM}\left(50+\mathbf{NEQ}\right)$ – INTEGER arrayCommunication Array

ICOM contains information which is usually of no interest, but is necessary for subsequent calls. However you may find the following useful:

$\mathbf{ICOM}\left(22\right)$

The order of the method to be attempted on the next step.

$\mathbf{ICOM}\left(23\right)$

The order of the method used on the last step.

$\mathbf{ICOM}\left(26\right)$

The number of steps taken so far.

$\mathbf{ICOM}\left(27\right)$

The number of calls to RES so far.

$\mathbf{ICOM}\left(28\right)$

The number of evaluations of the matrix of partial derivatives needed so far.

$\mathbf{ICOM}\left(29\right)$

The total number of error test failures so far.

$\mathbf{ICOM}\left(30\right)$

The total number of convergence test failures so far.

12:   $\mathrm{COM}\left(\mathbf{LCOM}\right)$ – REAL (KIND=nag_wp) arrayCommunication Array

COM contains information which is usually of no interest, but is necessary for subsequent calls. However you may find the following useful:

$\mathbf{COM}\left(3\right)$

The step size to be attempted on the next step.

$\mathbf{COM}\left(4\right)$

The current value of the independent variable, i.e., the farthest point integration has reached. This will be different from T only when interpolation has been performed ($\mathbf{ITASK}=3$).

13:   $\mathrm{LCOM}$ – INTEGERInput

On entry: the dimension of the array COM as declared in the (sub)program from which D02NEF is called.

Constraint: $\mathbf{LCOM}\ge 40+\left(\mathit{maxorder}+4\right)\times \mathbf{NEQ}+\mathbf{NEQ}\times p+q$ where $\mathit{maxorder}$ is the maximum order that can be used by the integration method (see MAXORD in D02MWF); $p=\mathbf{NEQ}$ when the Jacobian is full and $p=\left(2\times \mathbf{ML}+\mathbf{MU}+1\right)$ when the Jacobian is banded; and, $q=\left(\mathbf{NEQ}/\left(\mathbf{ML}+\mathbf{MU}+1\right)\right)+1$ when the Jacobian is to be evaluated numerically and $q=0$ otherwise.

14:   $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace

15:   $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by D02NEF, but are passed directly to RES and JAC and may be used to pass information to these routines as an alternative to using COMMON global variables.

16:   $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry, $\mathbf{NEQ}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{NEQ}\ge 1$.

$\mathbf{IFAIL}=3$

On entry, $\mathbf{T}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{TOUT}\ne \mathbf{T}$.

TOUT is behind T in the direction of $h$: $\mathbf{TOUT}-\mathbf{T}=〈\mathit{\text{value}}〉$, $h=〈\mathit{\text{value}}〉$.

TOUT is too close to T to start integration: $\mathbf{TOUT}-\mathbf{T}=〈\mathit{\text{value}}〉$: $\mathit{hmin}=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=6$

Some element of RTOL is less than zero.

$\mathbf{IFAIL}=7$

Some element of ATOL is less than zero.

$\mathbf{IFAIL}=8$

A previous call to this routine returned with $\mathbf{ITASK}=〈\mathit{\text{value}}〉$ and no appropriate action was taken.

$\mathbf{IFAIL}=11$

Either the initialization routine has not been called prior to the first call of this routine or a communication array has become corrupted.

$\mathbf{IFAIL}=12$

Either the initialization routine has not been called prior to the first call of this routine or a communication array has become corrupted.

$\mathbf{IFAIL}=13$

COM array is of insufficient length; length required $=〈\mathit{\text{value}}〉$; actual length $\mathbf{LCOM}=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=14$

All elements of RTOL and ATOL are zero.

$\mathbf{IFAIL}=15$

Maximum number of steps taken on this call before reaching TOUT: $\mathbf{T}=〈\mathit{\text{value}}〉$, maximum number of steps $=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=16$

Too much accuracy requested for precision of machine. RTOL and ATOL were increased by scale factor $R$. Try running again with these scaled tolerances. $\mathbf{T}=〈\mathit{\text{value}}〉$, $R=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=17$

A solution component has become zero when a purely relative tolerance (zero absolute tolerance) was selected for that component. $\mathbf{T}=〈\mathit{\text{value}}〉$, $\mathbf{Y}\left(\mathit{I}\right)=〈\mathit{\text{value}}〉$ for component $\mathit{I}=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=18$

The error test failed repeatedly with $\left|h\right|=\mathit{hmin}$. $\mathbf{T}=〈\mathit{\text{value}}〉$. Stepsize $h=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=19$

The corrector repeatedly failed to converge with $\left|h\right|=\mathit{hmin}$. $\mathbf{T}=〈\mathit{\text{value}}〉$. Stepsize $h=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=20$

The iteration matrix is singular. $\mathbf{T}=〈\mathit{\text{value}}〉$. Stepsize $h=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=21$

The corrector could not converge and the error test failed repeatedly. $\mathbf{T}=〈\mathit{\text{value}}〉$. Stepsize $h=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=22$

IRES was set to $-1$ during a call to RES and could not be resolved. $\mathbf{T}=〈\mathit{\text{value}}〉$. Stepsize $h=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=23$

IRES was set to $-2$ during a call to RES. $\mathbf{T}=〈\mathit{\text{value}}〉$. Stepsize $=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=24$

The initial YDOT could not be computed. $\mathbf{T}=〈\mathit{\text{value}}〉$. Stepsize $h=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=25$

Repeated occurrences of input constraint violations have been detected. This could result in a potential infinite loop: $\mathbf{ITASK}=〈\mathit{\text{value}}〉$. Current violation corresponds to exit with $\mathbf{IFAIL}=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-999$

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (d02nefe.f90)

10.2  Program Data

Program Data (d02nefe.d)

10.3  Program Results

Program Results (d02nefe.r)