NAG Library Routine Document

d02bhf  (ivp_rkm_zero_simple)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input/Output

On entry: must be set to the initial value of the independent variable $x$.

On exit: the point where $g\left(x,y\right)=0.0$ unless an error has occurred, when it contains the value of $x$ at the error. In particular, if $g\left(x,y\right)\ne 0.0$ anywhere on the range x to xend, it will contain xend on exit.

2:     $\mathbf{xend}$ – Real (Kind=nag_wp)Input

On entry: the final value of the independent variable $x$.

If $\mathbf{xend}<\mathbf{x}$ on entry, integration proceeds in a negative direction.

3:     $\mathbf{n}$ – IntegerInput

On entry: $\mathit{n}$, the number of differential equations.

Constraint: $\mathbf{n}>0$.

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

On entry: the initial values of the solution $y_1,y_2,\dots ,y_{\mathit{n}}$.

On exit: the computed values of the solution at the final point $x=\mathbf{x}$.

5:     $\mathbf{tol}$ – Real (Kind=nag_wp)Input/Output

On entry: must be set to a positive tolerance for controlling the error in the integration and in the determination of the position where $g\left(x,y\right)=0.0$.

d02bhf has been designed so that, for most problems, a reduction in tol leads to an approximately proportional reduction in the error in the solution obtained in the integration. The relation between changes in tol and the error in the determination of the position where $g\left(x,y\right)=0.0$ is less clear, but for tol small enough the error should be approximately proportional to tol. However, the actual relation between tol and the accuracy cannot be guaranteed. You are strongly recommended to call d02bhf with more than one value for tol and to compare the results obtained to estimate their accuracy. In the absence of any prior knowledge you might compare results obtained by calling d02bhf with $\mathbf{tol}={10.0}^{-p}$ and $\mathbf{tol}={10.0}^{-p-1}$ if $p$ correct decimal digits in the solution are required.

Constraint: $\mathbf{tol}>0.0$.

On exit: normally unchanged. However if the range from $x=\mathbf{x}$ to the position where $g\left(x,y\right)=0.0$ (or to the final value of $x$ if an error occurs) is so short that a small change in tol is unlikely to make any change in the computed solution, tol is returned with its sign changed. To check results returned with $\mathbf{tol}<0.0$, d02bhf should be called again with a positive value of tol whose magnitude is considerably smaller than that of the previous call.

6:     $\mathbf{irelab}$ – IntegerInput

On entry: determines the type of error control. At each step in the numerical solution an estimate of the local error, $\mathit{est}$, is made. For the current step to be accepted the following condition must be satisfied:

$\mathbf{irelab}=0$

$\mathit{est}\le \mathbf{tol}\times \max \left\{1.0,\left|y_1\right|,\left|y_2\right|,\dots ,\left|y_{\mathit{n}}\right|\right\}$;

$\mathbf{irelab}=1$

$\mathit{est}\le \mathbf{tol}$;

$\mathbf{irelab}=2$

$\mathit{est}\le \mathbf{tol}\times \max \left\{\epsilon ,\left|y_1\right|,\left|y_2\right|,\dots ,\left|y_{\mathit{n}}\right|\right\}$, where $\epsilon$ is machine precision.

If the appropriate condition is not satisfied, the step size is reduced and the solution recomputed on the current step.

If you wish to measure the error in the computed solution in terms of the number of correct decimal places, set $\mathbf{irelab}=1$ on entry, whereas if the error requirement is in terms of the number of correct significant digits, set $\mathbf{irelab}=2$. Where there is no preference in the choice of error test, $\mathbf{irelab}=0$ will result in a mixed error test. It should be borne in mind that the computed solution will be used in evaluating $g\left(x,y\right)$.

Constraint: $\mathbf{irelab}=0$, $1$ or $2$.

7:     $\mathbf{hmax}$ – Real (Kind=nag_wp)Input

On entry: if $\mathbf{hmax}=0.0$, no special action is taken.

If $\mathbf{hmax}\ne 0.0$, a check is made for a change in sign of $g\left(x,y\right)$ at steps not greater than $\left|\mathbf{hmax}\right|$. This facility should be used if there is any chance of ‘missing’ the change in sign by checking too infrequently. For example, if two changes of sign of $g\left(x,y\right)$ are expected within a distance $h$, say, of each other, a suitable value for hmax might be $\mathbf{hmax}=h/2$. If only one change of sign in $g\left(x,y\right)$ is expected on the range x to xend, the choice $\mathbf{hmax}=0.0$ is most appropriate.

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

fcn must evaluate the functions $f_i$ (i.e., the derivatives $y_i^{\prime }$) for given values of its arguments $x,y_1,\dots ,y_{\mathit{n}}$.

The specification of fcn is:

Fortran Interface

Subroutine fcn ( x, y, f) Real (Kind=nag_wp), Intent (In) :: x, y(*) Real (Kind=nag_wp), Intent (Out) :: f(*)

C Header Interface

#include nagmk26.h void  fcn ( const double *x, const double y[], double f[])

In the description of the arguments of d02bhf below, $\mathit{n}$ denotes the value of n in the call of d02bhf.

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

On entry: $x$, the value of the argument.

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

On entry: $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the argument.

3:     $\mathbf{f}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

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

fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02bhf is called. Arguments denoted as Input must not be changed by this procedure.

Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02bhf. If your code inadvertently does return any NaNs or infinities, d02bhf is likely to produce unexpected results.

9:     $\mathbf{g}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

g must evaluate the function $g\left(x,y\right)$ at a specified point.

The specification of g is:

Fortran Interface

Function g ( x, y) Real (Kind=nag_wp) :: g Real (Kind=nag_wp), Intent (In) :: x, y(*)

C Header Interface

#include nagmk26.h double  g ( const double *x, const double y[])

In the description of the arguments of d02bhf below, $\mathit{n}$ denotes the value of n in the call of d02bhf.

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

On entry: $x$, the value of the independent variable.

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

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

g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02bhf is called. Arguments denoted as Input must not be changed by this procedure.

Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02bhf. If your code inadvertently does return any NaNs or infinities, d02bhf is likely to produce unexpected results.

10:   $\mathbf{w}\left(\mathbf{n},7\right)$ – Real (Kind=nag_wp) arrayWorkspace

11:   $\mathbf{ifail}$ – 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, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ 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,	$\mathbf{tol}\le 0.0$,
or	$\mathbf{n}\le 0$,
or	$\mathbf{irelab}\ne 0$, $1$ or $2$.

$\mathbf{ifail}=2$

With the given value of tol, no further progress can be made across the integration range from the current point $x=\mathbf{x}$, or dependence of the error on tol would be lost if further progress across the integration range were attempted (see Section 9 for a discussion of this error exit). The components $\mathbf{y}\left(1\right),\mathbf{y}\left(2\right),\dots ,\mathbf{y}\left(\mathit{n}\right)$ contain the computed values of the solution at the current point $x=\mathbf{x}$. No point at which $g\left(x,y\right)$ changes sign has been located up to the point $x=\mathbf{x}$.

$\mathbf{ifail}=3$

tol is too small for d02bhf to take an initial step (see Section 9). x and $\mathbf{y}\left(1\right),\mathbf{y}\left(2\right),\dots ,\mathbf{y}\left(\mathit{n}\right)$ retain their initial values.

$\mathbf{ifail}=4$

At no point in the range x to xend did the function $g\left(x,y\right)$ change sign. It is assumed that $g\left(x,y\right)=0.0$ has no solution.

$\mathbf{ifail}=5$ (c05azf)

A serious error has occurred in an internal call to the specified routine. Check all subroutine calls and array dimensions. Seek expert help.

$\mathbf{ifail}=6$

A serious error has occurred in an internal call to an integration routine. Check all subroutine calls and array dimensions. Seek expert help.

$\mathbf{ifail}=7$

A serious error has occurred in an internal call to an interpolation routine. Check all (sub)program calls and array dimensions. Seek expert help.

$\mathbf{ifail}=-99$

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

Real (kind=nag_wp) Function g(x,y)
Real (kind=nag_wp) x,y(3),f(3)
Call fcn(x,y,f)
g = f(1)
Return
End

10

Example

10.1

Program Text

Program Text (d02bhfe.f90)

10.2

Program Data

Program Data (d02bhfe.d)

10.3

Program Results

Program Results (d02bhfe.r)