NAG Library Routine Document

d02bjf (ivp_rk_zero_simple)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

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

On exit: if $g$ is supplied by you, it contains the point where $g\left(x,y\right)=0$, unless $g\left(x,y\right)\ne 0$ anywhere on the range x to xend, in which case, x will contain xend (and the error indicator $\mathbf{ifail}=\mathbf{6}$ is set); if $g$ is not supplied by you it contains xend. However, if an error has occurred, it contains the value of $x$ at which the error occurred.

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

On entry: the final value of the independent variable. If $\mathbf{xend}<\mathbf{x}$, integration will proceed in the negative direction.

Constraint: $\mathbf{xend}\ne \mathbf{x}$.

3:     $\mathbf{n}$ – IntegerInput

On entry: $\mathit{n}$, the number of 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 $y_1,y_2,\dots ,y_{\mathit{n}}$ at $x=\mathbf{x}$.

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

5:     $\mathbf{fcn}$ – 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 (Inout) :: f(*)

C Header Interface

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

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

Note: the dimension, $\mathit{n}$, of y is n as in the call of d02bjf.

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

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

Note: the dimension, $\mathit{n}$, of f is n as in the call of d02bjf.

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 d02bjf 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 d02bjf. If your code inadvertently does return any NaNs or infinities, d02bjf is likely to produce unexpected results.

6:     $\mathbf{tol}$ – Real (Kind=nag_wp)Input

On entry: a positive tolerance for controlling the error in the integration. Hence tol affects the determination of the position where $g\left(x,y\right)=0$, if $g$ is supplied.

d02bjf has been designed so that, for most problems, a reduction in tol leads to an approximately proportional reduction in the error in the solution. However, the actual relation between tol and the accuracy achieved cannot be guaranteed. You are strongly recommended to call d02bjf 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 the results obtained by calling d02bjf with $\mathbf{relabs}=\text{'D'}$ and with each of $\mathbf{tol}={10.0}^{-p}$ and $\mathbf{tol}={10.0}^{-p-1}$ where $p$ correct significant digits are required in the solution, $y$. The accuracy of the value $x$ such that $g\left(x,y\right)=0$ is indirectly controlled by varying tol. You should experiment to determine this accuracy.

Constraint: $10.0\times \mathit{machine precision}<\mathbf{tol}<0.01$.

7:     $\mathbf{relabs}$ – Character(1)Input

On entry: 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:

$$\mathit{est}=\max \left(e_i/\left({\tau }_r\times \max \left(\left|y_i\right|,{\tau }_a\right)\right)\right)\le 1.0$$

where ${\tau }_r$ and ${\tau }_a$ are defined by

relabs	${\tau }_r$	${\tau }_a$
'M'	tol	1.0
'A'	${\epsilon }_r$	$\mathbf{tol}/{\epsilon }_r$
'R'	tol	${\epsilon }_a$
'D'	tol	${\epsilon }_a$

where ${\epsilon }_r$ and ${\epsilon }_a$ are small machine-dependent numbers and $e_i$ is an estimate of the local error at $y_i$, computed internally. If the condition is not satisfied, the step size is reduced and the solution is 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, relabs should be set to 'A' on entry, whereas if the error requirement is in terms of the number of correct significant digits, relabs should be set to 'R'. If you prefer a mixed error test, relabs should be set to 'M', otherwise if you have no preference, relabs should be set to the default 'D'. Note that in this case 'D' is taken to be 'R'.

Constraint: $\mathbf{relabs}=\text{'M'}$, $\text{'A'}$, $\text{'R'}$ or $\text{'D'}$.

8:     $\mathbf{output}$ – Subroutine, supplied by the NAG Library or the user.External Procedure

output permits access to intermediate values of the computed solution (for example to print or plot them), at successive user-specified points. It is initially called by d02bjf with $\mathbf{xsol}=\mathbf{x}$ (the initial value of $x$). You must reset xsol to the next point (between the current xsol and xend) where output is to be called, and so on at each call to output. If, after a call to output, the reset point xsol is beyond xend, d02bjf will integrate to xend with no further calls to output; if a call to output is required at the point $\mathbf{xsol}=\mathbf{xend}$, xsol must be given precisely the value xend.

The specification of output is:

Fortran Interface

Subroutine output ( xsol, y) Real (Kind=nag_wp), Intent (In) :: y(*) Real (Kind=nag_wp), Intent (Inout) :: xsol

C Header Interface

#include <nagmk26.h> void  output (double *xsol, const double y[])

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

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

On exit: you must set xsol to the next value of $x$ at which output is to be called.

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

Note: the dimension, $\mathit{n}$, of y is n as in the call of d02bjf.

On entry: the computed solution at the point xsol.

output must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02bjf is called. Arguments denoted as Input must not be changed by this procedure.

Note: output should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02bjf. If your code inadvertently does return any NaNs or infinities, d02bjf is likely to produce unexpected results.

If you do not wish to access intermediate output, the actual argument output must be the dummy routine d02bjx. (d02bjx is included in the NAG Library.)

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

g must evaluate the function $g\left(x,y\right)$ for specified values $x,y$. It specifies the function $g$ for which the first position $x$ where $g\left(x,y\right)=0$ is to be found.

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[])

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

Note: the dimension, $\mathit{n}$, of y is n as in the call of d02bjf.

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

g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02bjf 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 d02bjf. If your code inadvertently does return any NaNs or infinities, d02bjf is likely to produce unexpected results.

If you do not require the root-finding option, the actual argument g must be the dummy routine d02bjw. (d02bjw is included in the NAG Library.)

10:   $\mathbf{w}\left(20\times \mathbf{n}\right)$ – 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, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{relabs}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{relabs}=\text{'M'}$, $\text{'A'}$, $\text{'R'}$ or $\text{'D'}$.

On entry, $\mathbf{tol}=〈\mathit{\text{value}}〉$.
Constraint: $〈\mathit{\text{value}}〉\le \mathbf{tol}<0.01$.

On entry, $\mathbf{x}=\mathbf{xend}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{x}\ne \mathbf{xend}$.

$\mathbf{ifail}=2$

With the given value of tol, no further progress can be made across the integration range from the current point $x=\mathbf{x}$.
The solution is returned at this point, $\mathbf{x}=〈\mathit{\text{value}}〉$.

With the given value of tol, no further progress can be made across the integration range from the current point $x=\mathbf{x}$. (See Section 9 for a discussion of this error exit.) The components $\mathbf{y}\left(1\right),\mathbf{y}\left(2\right),\dots ,\mathbf{y}\left(\mathbf{n}\right)$ contain the computed values of the solution at the current point $x=\mathbf{x}$. If you have supplied $g$, no point at which $g\left(x,y\right)$ changes sign has been located up to the point $x=\mathbf{x}$.

$\mathbf{ifail}=3$

With the given value of tol, no initial progress can be made from the starting point of integration provided.

tol is too small for d02bjf to take an initial step. x and $\mathbf{y}\left(1\right),\mathbf{y}\left(2\right),\dots ,\mathbf{y}\left(\mathbf{n}\right)$ retain their initial values.

$\mathbf{ifail}=4$

On the first call to output, xsol remained unchanged.

On the first call to output, xsol was returned as $〈\mathit{\text{value}}〉$, which is inconsistent with $\left(\mathbf{x},\mathbf{xend}\right)$ — $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$.

$\mathbf{ifail}=5$

On a call to output, xsol remained unchanged.

On a call to output, xsol was returned as $〈\mathit{\text{value}}〉$, which is inconsistent with previous xsol and xend — $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$.

$\mathbf{ifail}=6$

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

$\mathbf{ifail}=7$

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

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

$\mathbf{ifail}=8$

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

$\mathbf{ifail}=-99$

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 (d02bjfe.f90)

10.2

Program Data

Program Data (d02bjfe.d)

10.3

Program Results

Program Results (d02bjfe.r)