NAG Library Routine Document

d02pef (ivp_rkts_range)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{f}$ – Subroutine, supplied by the user.External Procedure

f must evaluate the functions $f_i$ (that is the first derivatives $y_i^{\prime }$) for given values of the arguments $t$, $y_i$.

The specification of f is:

Fortran Interface

Subroutine f ( t, n, y, yp, iuser, ruser) Integer, Intent (In) :: n Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: t, y(n) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: yp(n)

C Header Interface

#include <nagmk26.h> void  f (const double *t, const Integer *n, const double y[], double yp[], Integer iuser[], double ruser[])

1:     $\mathbf{t}$ – Real (Kind=nag_wp)Input

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

2:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of ordinary differential equations in the system to be solved.

3:     $\mathbf{y}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the current values of the dependent variables, $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.

4:     $\mathbf{yp}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the values of $f_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.

5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

f is called with the arguments iuser and ruser as supplied to d02pef. You should use the arrays iuser and ruser to supply information to f.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02pef is called. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02pef. If your code inadvertently does return any NaNs or infinities, d02pef is likely to produce unexpected results.

2:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of ordinary differential equations in the system to be solved.

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

3:     $\mathbf{twant}$ – Real (Kind=nag_wp)Input

On entry: $t$, the next value of the independent variable where a solution is desired.

Constraint: twant must be closer to tend than the previous value of tgot (or tstart on the first call to d02pef); see d02pqf for a description of tstart and tend. twant must not lie beyond tend in the direction of integration.

4:     $\mathbf{tgot}$ – Real (Kind=nag_wp)Output

On exit: $t$, the value of the independent variable at which a solution has been computed. On successful exit with $\mathbf{ifail}=\mathbf{0}$, tgot will equal twant. On exit with $\mathbf{ifail}>\mathbf{1}$, a solution has still been computed at the value of tgot but in general tgot will not equal twant.

5:     $\mathbf{ygot}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: on the first call to d02pef, ygot need not be set. On all subsequent calls ygot must remain unchanged.

On exit: an approximation to the true solution at the value of tgot. At each step of the integration to tgot, the local error has been controlled as specified in d02pqf. The local error has still been controlled even when $\mathbf{tgot}\ne \mathbf{twant}$, that is after a return with $\mathbf{ifail}>\mathbf{1}$.

6:     $\mathbf{ypgot}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: an approximation to the first derivative of the true solution at tgot.

7:     $\mathbf{ymax}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: on the first call to d02pef, ymax need not be set. On all subsequent calls ymax must remain unchanged.

On exit: $\mathbf{ymax}\left(i\right)$ contains the largest value of $\left|y_i\right|$ computed at any step in the integration so far.

8:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

9:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

iuser and ruser are not used by d02pef, but are passed directly to f and may be used to pass information to this routine.

10:   $\mathbf{iwsav}\left(130\right)$ – Integer arrayCommunication Array

11:   $\mathbf{rwsav}\left(32\times \mathbf{n}+350\right)$ – Real (Kind=nag_wp) arrayCommunication Array

On entry: these must be the same arrays supplied in a previous call to d02pqf. They must remain unchanged between calls.

On exit: information about the integration for use on subsequent calls to d02pef or other associated routines.

12:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 arguments 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, a previous call to the setup routine has not been made or the communication arrays have become corrupted.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$, but the value passed to the setup routine was $\mathbf{n}=〈\mathit{\text{value}}〉$.

On entry, the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere. You cannot continue integrating the problem.

tend (setup) had already been reached in a previous call.
To start a new problem, you will need to call the setup routine.

twant does not lie in the direction of integration. $\mathbf{twant}=〈\mathit{\text{value}}〉$.

twant is too close to the last value of tgot (tstart on setup).
When using the method of order $8$ at setup, these must differ by at least $〈\mathit{\text{value}}〉$. Their absolute difference is $〈\mathit{\text{value}}〉$.

twant lies beyond tend (setup) in the direction of integration, but is very close to tend.
You may have intended $\mathbf{twant}=\mathbf{tend}$.
$\left|\mathbf{twant}-\mathbf{tend}\right|=〈\mathit{\text{value}}〉$.

twant lies beyond tend (setup) in the direction of integration.
$\mathbf{twant}=〈\mathit{\text{value}}〉$ and $\mathbf{tend}=〈\mathit{\text{value}}〉$.

You cannot call this routine after it has returned an error.
You must call the setup routine to start another problem.

You cannot call this routine when you have specified, in the setup routine, that the step integrator will be used.

$\mathbf{ifail}=2$

This routine is being used inefficiently because the step size has been reduced drastically many times to obtain answers at many points. Using the order $4$ and $5$ pair method at setup is more appropriate here. You can continue integrating this problem.

$\mathbf{ifail}=3$

Approximately $〈\mathit{\text{value}}〉$ function evaluations have been used to compute the solution since the integration started or since this message was last printed. However, you can continue integrating the problem.

$\mathbf{ifail}=4$

Approximately $〈\mathit{\text{value}}〉$ function evaluations have been used to compute the solution since the integration started or since this message was last printed. Your problem has been diagnosed as stiff. If the situation persists, it will cost roughly $〈\mathit{\text{value}}〉$ times as much to reach tend (setup) as it has cost to reach the current time. You should probably call routines intended for stiff problems. However, you can continue integrating the problem.

$\mathbf{ifail}=5$

In order to satisfy your error requirements the solver has to use a step size of $〈\mathit{\text{value}}〉$ at the current time, $〈\mathit{\text{value}}〉$. This step size is too small for the machine precision, and is smaller than $〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=6$

The global error assessment algorithm failed at start of integration.
The integration is being terminated.

The global error assessment may not be reliable for times beyond $〈\mathit{\text{value}}〉$.
The integration is being terminated.

$\mathbf{ifail}=-99$

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

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

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

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d02pefe.f90)

10.2

Program Data

Program Data (d02pefe.d)

10.3

Program Results

Program Results (d02pefe.r)