NAG FL Interface
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: $\mathbf{n}$ – Integer Input

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

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

4: $\mathbf{y}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Input/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: $\mathbf{irelab}$ – Integer Input

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

void  fcn_ (const double *x, const double y[], double f[])

C++ Header Interface

#include <nag.h> extern "C" { 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) array Input

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) array Output

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

double  g_ (const double *x, const double y[])

C++ Header Interface

#include <nag.h> extern "C" { 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) array Input

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) array Workspace

11: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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, $\mathbf{irelab}=〈\mathit{\text{value}}〉$.
Constraint: $0\le \mathbf{irelab}\le 2$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}>0$.

On entry, $\mathbf{tol}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{tol}>0.0$.

$\mathbf{ifail}=2$

The value of tol, $〈\mathit{\text{value}}〉$, is too small for the function to make any further progress across the integration range. Current value of $x=〈\mathit{\text{value}}〉$.

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 $x=\mathbf{x}$.

$\mathbf{ifail}=3$

The value of tol, $〈\mathit{\text{value}}〉$, is too small for the function to take an initial step.

$\mathbf{ifail}=4$

No change in sign of the function $g\left(x,y\right)$ was detected in the integration range.

$\mathbf{ifail}=5$

An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.

$\mathbf{ifail}=6$

A serious error occurred in a call to the internal integrator.
The error code internally was $〈\mathit{\text{value}}〉$. Please contact NAG.

$\mathbf{ifail}=7$

Unexpected internal error in call to interpolation routine.
The interpolation routine returned error flag $〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=-99$

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

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

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

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface 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

10.2 Program Data

10.3 Program Results