NAG Library Routine Document

D02PSF

INTEGER	N, IDERIV, NWANT, LWCOMM, IUSER(*), IWSAV(130), IFAIL
REAL (KIND=nag_wp)	TWANT, YWANT(NWANT), YPWANT(NWANT), WCOMM(LWCOMM), RUSER(), RWSAV(32N+350)
EXTERNAL	F

3 Description

D02PSF and its associated routines (D02PFF, D02PQF, D02PRF, D02PTF and D02PUF) solve the 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.

D02PFF computes the solution at the end of an integration step. Using the information computed on that step D02PSF computes the solution by interpolation at any point on that step. It cannot be used if

METHOD = 3

- 3

was specified in the call to setup routine D02PQF.

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: $N$ – INTEGERInput

On entry:

n

, the number of ordinary differential equations in the system to be solved by the integration routine.

Constraint:

N \geq 1

2: $TWANT$ – REAL (KIND=nag_wp)Input

On entry:

t

, the value of the independent variable where a solution is desired.

3: $IDERIV$ – INTEGERInput

On entry: determines whether the solution and/or its first derivative are to be computed

$IDERIV = 0$: compute approximate solution.
$IDERIV = 1$: compute approximate first derivative.
$IDERIV = 2$: compute approximate solution and first derivative.

Constraint:

IDERIV = 0

1

2

4: $NWANT$ – INTEGERInput

On entry: the number of components of the solution to be computed. The first NWANT components are evaluated.

Constraint:

1 \leq NWANT \leq N

5: $YWANT (NWANT)$ – REAL (KIND=nag_wp) arrayOutput

On exit: an approximation to the first NWANT components of the solution at TWANT if

IDERIV = 0

2

. Otherwise YWANT is not defined.

6: $YPWANT (NWANT)$ – REAL (KIND=nag_wp) arrayOutput

On exit: an approximation to the first NWANT components of the first derivative at TWANT if

IDERIV = 1

2

. Otherwise YPWANT is not defined.

7: $F$ – 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}

. It must be the same procedure as supplied to D02PFF.

The specification of F is:

SUBROUTINE F (

T, N, Y, YP, IUSER, RUSER)

INTEGER	N, IUSER(*)
REAL (KIND=nag_wp)	T, Y(N), YP(N), RUSER(*)

1: $T$ – REAL (KIND=nag_wp)Input: On entry: $t$ , the current value of the independent variable.
2: $N$ – INTEGERInput: On entry: $n$ , the number of ordinary differential equations in the system to be solved.
3: $Y (N)$ – REAL (KIND=nag_wp) arrayInput: On entry: the current values of the dependent variables, $y_{i}$ , for $i = 1, 2, \dots, n$ .
4: $YP (N)$ – REAL (KIND=nag_wp) arrayOutput: On exit: the values of $f_{i}$ , for $i = 1, 2, \dots, n$ .
5: $IUSER (*)$ – INTEGER arrayUser Workspace
6: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace: F is called with the parameters IUSER and RUSER as supplied to D02PSF. You are free to use the arrays IUSER and RUSER to supply information to F as an alternative to using COMMON global variables.

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02PSF is called. Parameters denoted as Input must not be changed by this procedure.

8: $WCOMM (LWCOMM)$ – REAL (KIND=nag_wp) arrayCommunication Array

On entry: this array stores information that can be utilized on subsequent calls to D02PSF.

9: $LWCOMM$ – INTEGERInput

On entry: length of WCOMM.

If in a previous call to D02PQF:

$METHOD = 1$ or $- 1$ then LWCOMM must be at least $1$ .
$METHOD = 2$ or $- 2$ then LWCOMM must be at least $N + \max (N, 5 \times NWANT)$ .
$METHOD = 3$ or $- 3$ then WCOMM and LWCOMM are not referenced.

10: $IUSER (*)$ – INTEGER arrayUser Workspace

11: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by D02PSF, but are passed directly to F and may be used to pass information to this routine as an alternative to using COMMON global variables.

12: $IWSAV (130)$ – INTEGER arrayCommunication Array

13: $RWSAV (32 \times N + 350)$ – REAL (KIND=nag_wp) arrayCommunication Array

On entry: these must be the same arrays supplied in a previous call D02PFF. They must remain unchanged between calls.

On exit: information about the integration for use on subsequent calls to D02PFF, D02PSF or other associated routines.

14: $IFAIL$ – 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, if you are not familiar with this parameter, the recommended value is

0

. 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:

$IFAIL = 1$: $METHOD = - 3$ or $3$ in setup, but interpolation is not available for this method. Either use $METHOD = - 2$ or $2$ in setup or use reset routine to force the integrator to step to particular points.

On entry, a previous call to the setup routine has not been made or the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere.
You cannot continue integrating the problem.

On entry, $IDERIV = 〈value〉$ .
Constraint: $IDERIV = 0$ , $1$ or $2$ .

On entry, $LWCOMM = 〈value〉$ , $N = 〈value〉$ and $NWANT = 〈value〉$ .
Constraint: for $METHOD = - 2$ or $2$ , $LWCOMM \geq N + \max (N, 5 \times NWANT)$ .

On entry, $LWCOMM = 〈value〉$ .
Constraint: for $METHOD = - 1$ or $1$ , $LWCOMM \geq 1$ .

On entry, $N = 〈value〉$ , but the value passed to the setup routine was $N = 〈value〉$ .

On entry, $NWANT = 〈value〉$ and $N = 〈value〉$ .
Constraint: $1 \leq NWANT \leq N$ .

You cannot call this routine after the integrator has returned an error.

You cannot call this routine before you have called the step integrator.

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

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The computed values will be of a similar accuracy to that computed by D02PFF.

8 Parallelism and Performance

D02PSF is not threaded by NAG in any implementation.

D02PSF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 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. D02PFF is used to integrate the problem one step at a time and D02PSF is used to compute the first component of the solution and its derivative at intervals of length

π / 8

across the range whenever these points lie in one of those integration steps. A low order Runge–Kutta method (

METHOD = - 1

) is also used with tolerances

TOL = 1.0E−4

and

TOL = 1.0E−5

in turn so that solutions may be compared.

NAG Library Routine DocumentD02PSF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

D02PSF