NAG Library Routine Document

D02PCF

D02PCF and its associated routines (D02PVF, D02PYF and D02PZF) solve an initial value problem for a first-order system of ordinary differential equations. The routines, based on Runge–Kutta methods and derived from RKSUITE (see Brankin et al. (1991)), integrate

y^{'} = f (t, y) given y (t_{0}) = y_{0}

where

y

is the vector of

n

solution components and

t

is the independent variable.

D02PCF is designed for the usual task, namely to compute an approximate solution at a sequence of points. You must first call D02PVF to specify the problem and how it is to be solved. Thereafter you call D02PCF repeatedly with successive values of TWANT, the points at which you require the solution, in the range from TSTART to TEND (as specified in D02PVF). In this manner D02PCF returns the point at which it has computed a solution TGOT (usually TWANT), the solution there (YGOT) and its derivative (YPGOT). If D02PCF encounters some difficulty in taking a step toward TWANT, then it returns the point of difficulty (TGOT) and the solution and derivative computed there (YGOT and YPGOT, respectively).

In the call to D02PVF you can specify either the first step size for D02PCF to attempt or that it compute automatically an appropriate value. Thereafter D02PCF estimates an appropriate step size for its next step. This value and other details of the integration can be obtained after any call to D02PCF by a call to D02PYF. The local error is controlled at every step as specified in D02PVF. If you wish to assess the true error, you must set

ERRASS = .TRUE.

in the call to D02PVF. This assessment can be obtained after any call to D02PCF by a call to D02PZF.

For more complicated tasks, you are referred to routines D02PDF, D02PWF and D02PXF, all of which are used by D02PCF.

4 References

Brankin R W, Gladwell I and Shampine L F (1991) RKSUITE: A suite of Runge–Kutta codes for the initial value problems for ODEs SoftReport 91-S1 Southern Methodist University

5 Parameters

1: F – SUBROUTINE, supplied by the user.External Procedure

F must evaluate the functions

f_{i}

(that is the first derivatives

y_{i}^{'}

) for given values of the arguments

t

y_{i}

The specification of F is:

SUBROUTINE F (

T, Y, YP)

REAL (KIND=nag_wp)

T, Y(*), YP(*)

In the description of the parameters of D02PCF below,

n

denotes the value of NEQ in the call of D02PVF.

1: T – REAL (KIND=nag_wp)Input: On entry: $t$ , the current value of the independent variable.
2: Y( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: the current values of the dependent variables, $y_{i}$ , for $i = 1, 2, \dots, n$ .
3: YP( $*$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the values of $f_{i}$ , for $i = 1, 2, \dots, n$ .

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02PCF is called. Parameters denoted as Input must not be changed by this procedure.

2: 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 D02PCF); see D02PVF for a description of TSTART and TEND. TWANT must not lie beyond TEND in the direction of integration.

3: 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

IFAIL = 0

, TGOT will equal TWANT. On exit with

IFAIL > 1

, a solution has still been computed at the value of TGOT but in general TGOT will not equal TWANT.

4: YGOT( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array YGOT must be at least

n

On entry: on the first call to D02PCF, 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 D02PVF. The local error has still been controlled even when

TGOT \neq TWANT

, that is after a return with

IFAIL > 1

5: YPGOT( $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array YPGOT must be at least

n

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

6: YMAX( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array YMAX must be at least

n

On entry: on the first call to D02PCF, YMAX need not be set. On all subsequent calls YMAX must remain unchanged.

On exit:

YMAX (i)

contains the largest value of

|y_{i}|

computed at any step in the integration so far.

7: WORK( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array WORK must be at least

LENWRK

(see D02PVF).

On entry: this must be the same array as supplied to D02PVF. It must remain unchanged between calls.

On exit: information about the integration for use on subsequent calls to D02PCF or other associated routines.

8: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ 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 ​ 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

IFAIL \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, an invalid input value for TWANT was detected or an invalid call to D02PCF was made, for example without a previous call to the setup routine D02PVF. You cannot continue integrating the problem.

$IFAIL = 2$: This return is possible only when $METHOD = 3$ has been selected in the preceding call of D02PVF. D02PCF is being used inefficiently because the step size has been reduced drastically many times to get answers at many values of TWANT. If you really need the solution at this many points, you should change to $METHOD = 2$ because it is (much) more efficient in this situation. To change METHOD, restart the integration from TGOT, YGOT by a call to D02PVF. If you wish to continue with $METHOD = 3$ , just call D02PCF again without altering any of the arguments other than IFAIL. The monitor of this kind of inefficiency will be reset automatically so that the integration can proceed.

$IFAIL = 3$: A considerable amount of work has been expended in the (primary) integration. This is measured by counting the number of calls to the supplied routine F. At least $5000$ calls have been made since the last time this counter was reset. Calls to F in a secondary integration for global error assessment (when $ERRASS = .TRUE.$ in the call to D02PVF) are not counted in this total. The integration was interrupted, so TGOT is not equal to TWANT. If you wish to continue on towards TWANT, just call D02PCF again without altering any of the arguments other than IFAIL. The counter measuring work will be reset to zero automatically.

$IFAIL = 4$: It appears that this problem is stiff. The methods implemented in D02PCF can solve such problems, but they are inefficient. You should change to another code based on methods appropriate for stiff problems. The integration was interrupted so TGOT is not equal to TWANT. If you want to continue on towards TWANT, just call D02PCF again without altering any of the arguments other than IFAIL. The stiffness monitor will be reset automatically.

$IFAIL = 5$: It does not appear possible to achieve the accuracy specified by TOL and THRES in the call to D02PVF with the precision available on the computer being used and with this value of METHOD. You cannot continue integrating this problem. A larger value for METHOD, if possible, will permit greater accuracy with this precision. To increase METHOD and/or continue with larger values of TOL and/or THRES, restart the integration from TGOT, YGOT by a call to D02PVF.

$IFAIL = 6$: (This error exit can only occur if $ERRASS = .TRUE.$ in the call to D02PVF.) The global error assessment may not be reliable beyond the current integration point TGOT. This may occur because either too little or too much accuracy has been requested or because $f (t, y)$ is not smooth enough for values of $t$ just past TGOT and current values of the solution $y$ . The integration cannot be continued. This return does not mean that you cannot integrate past TGOT, rather that you cannot do it with $= .TRUE.$ . However, it may also indicate problems with the primary integration.

7 Accuracy

The accuracy of integration is determined by the parameters TOL and THRES in a prior call to D02PVF (see the routine document for D02PVF for further details and advice). Note that only the local error at each step is controlled by these parameters. The error estimates obtained are not strict bounds but are usually reliable over one step. Over a number of steps the overall error may accumulate in various ways, depending on the properties of the differential system.

8 Further Comments

If D02PCF returns with

IFAIL = 5

and the accuracy specified by TOL and THRES is really required then you should consider whether there is a more fundamental difficulty. For example, the solution may contain a singularity. In such a region the solution components will usually be large in magnitude. Successive output values of YGOT and YMAX should be monitored (or D02PDF should be used since this takes one integration step at a time) with the aim of trapping the solution before the singularity. In any case numerical integration cannot be continued through a singularity, and analytical treatment may be necessary.

Performance statistics are available after any return from D02PCF by a call to D02PYF. If

ERRASS = .TRUE.

in the call to D02PVF, global error assessment is available after any return from D02PCF (except when

IFAIL = 1

) by a call to D02PZF.

After a failure with

IFAIL = 5

6

the diagnostic routines D02PYF and D02PZF may be called only once.

If D02PCF returns with

IFAIL = 4

then it is advisable to change to another code more suited to the solution of stiff problems. D02PCF will not return with

IFAIL = 4

if the problem is actually stiff but it is estimated that integration can be completed using less function evaluations than already computed.

9 Example

This example solves the equation

y^{''} = - y, y (0) = 0, y^{'} (0) = 1

reposed as

y_{1}^{'} = y_{2}

y_{2}^{'} = - y_{1}

over the range

[0, 2 π]

with initial conditions

y_{1} = 0.0

and

y_{2} = 1.0

. Relative error control is used with threshold values of

1.0E−8

for each solution component and compute the solution at intervals of length

π / 4

across the range. A low-order Runge–Kutta method (

METHOD = 1

, see D02PVF) is also used with tolerances

TOL = 1.0E−3

and

TOL = 1.0E−4

in turn so that the solutions can be compared. The value of

π

is obtained by using X01AAF.

Note that the length of WORK is large enough for any valid combination of input arguments to D02PVF.

NAG Library Routine DocumentD02PCF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D02PCF