NAG Library Routine Document

d02gaf  (bvp_fd_nonlin_fixedbc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{u}\left(\mathbf{n},2\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{u}\left(\mathit{i},1\right)$ must be set to the known or estimated value of $y_{\mathit{i}}$ at $a$ and $\mathbf{u}\left(\mathit{i},2\right)$ must be set to the known or estimated value of $y_{\mathit{i}}$ at $b$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.

2:     $\mathbf{v}\left(\mathbf{n},2\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{v}\left(\mathit{i},\mathit{j}\right)$ must be set to $0.0$ if $\mathbf{u}\left(\mathit{i},\mathit{j}\right)$ is a known value and to $1.0$ if $\mathbf{u}\left(\mathit{i},\mathit{j}\right)$ is an estimated value, for $\mathit{i}=1,2,\dots ,\mathit{n}$ and $\mathit{j}=1,2$.

Constraint: precisely $\mathit{n}$ of the $\mathbf{v}\left(i,j\right)$ must be set to $0.0$, i.e., precisely $\mathit{n}$ of the $\mathbf{u}\left(i,j\right)$ must be known values, and these must not be all at $a$ or all at $b$.

3:     $\mathbf{n}$ – IntegerInput

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

Constraint: $\mathbf{n}\ge 2$.

4:     $\mathbf{a}$ – Real (Kind=nag_wp)Input

On entry: $a$, the left-hand boundary point.

5:     $\mathbf{b}$ – Real (Kind=nag_wp)Input

On entry: $b$, the right-hand boundary point.

Constraint: $\mathbf{b}>\mathbf{a}$.

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

On entry: a positive absolute error tolerance. If

$$a=x_1<x_2<\cdots <x_{\mathbf{np}}=b$$

is the final mesh, $z_j\left(x_i\right)$ is the $j$th component of the approximate solution at $x_i$, and $y_j\left(x\right)$ is the $j$th component of the true solution of equation (1) (see Section 3) and the boundary conditions, then, except in extreme cases, it is expected that

$$\left|z_j\left(x_i\right)-y_j\left(x_i\right)\right|\le \mathbf{tol}\text{, \hspace{1em}}i=1,2,\dots ,\mathbf{np}\text{ and }j=1,2,\dots ,\mathit{n}\text{.}$$

(2)

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

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

fcn must evaluate the functions $f_{\mathit{i}}$ (i.e., the derivatives $y_{\mathit{i}}^{\prime }$), for $\mathit{i}=1,2,\dots ,\mathit{n}$, at a general point $x$.

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 d02gaf below, $\mathit{n}$ denotes the actual value of n in the call of d02gaf.

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 values 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 d02gaf 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 d02gaf. If your code inadvertently does return any NaNs or infinities, d02gaf is likely to produce unexpected results.

8:     $\mathbf{mnp}$ – IntegerInput

On entry: the maximum permitted number of mesh points.

Constraint: $\mathbf{mnp}\ge 32$.

9:     $\mathbf{x}\left(\mathbf{mnp}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: if $\mathbf{np}\ge 4$ (see np), the first np elements must define an initial mesh. Otherwise the elements of x need not be set.

Constraint:

$$\mathbf{a}=\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{np}\right)=\mathbf{b}\text{, \hspace{1em}}\mathbf{np}\ge 4\text{.}$$

(3)

On exit: $\mathbf{x}\left(1\right),\mathbf{x}\left(2\right),\dots ,\mathbf{x}\left(\mathbf{np}\right)$ define the final mesh (with the returned value of np) satisfying the relation (3).

10:   $\mathbf{y}\left(\mathbf{n},\mathbf{mnp}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the approximate solution $z_j\left(x_i\right)$ satisfying (2), on the final mesh, that is

$$\mathbf{y}\left(j,i\right)=z_j\left(x_i\right)\text{, \hspace{1em}}i=1,2,\dots ,\mathbf{np}\text{ and }j=1,2,\dots ,\mathit{n}\text{,}$$

where np is the number of points in the final mesh.

The remaining columns of y are not used.

11:   $\mathbf{np}$ – IntegerInput/Output

On entry: determines whether a default or user-supplied mesh is used.

$\mathbf{np}=0$

A default value of $4$ for np and a corresponding equispaced mesh $\mathbf{x}\left(1\right),\mathbf{x}\left(2\right),\dots ,\mathbf{x}\left(\mathbf{np}\right)$ are used.

$\mathbf{np}\ge 4$

You must define an initial mesh using the array x as described.

Constraint: $\mathbf{np}=0$ or $4\le \mathbf{np}\le \mathbf{mnp}$.

On exit: the number of points in the final (returned) mesh.

12:   $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) arrayWorkspace

13:   $\mathbf{lw}$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which d02gaf is called.

Constraint: $\mathbf{lw}\ge \mathbf{mnp}\times \left(3{\mathbf{n}}^2+6\mathbf{n}+2\right)+4{\mathbf{n}}^2+4\mathbf{n}$.

14:   $\mathbf{iw}\left(\mathbf{liw}\right)$ – Integer arrayWorkspace

15:   $\mathbf{liw}$ – IntegerInput

On entry: the dimension of the array iw as declared in the (sub)program from which d02gaf is called.

Constraint: $\mathbf{liw}\ge \mathbf{mnp}\times \left(2\mathbf{n}+1\right)+{\mathbf{n}}^2+4\mathbf{n}+2$.

16:   $\mathbf{ifail}$ – IntegerInput/Output

For this routine, the normal use of ifail is extended to control the printing of error and warning messages as well as specifying hard or soft failure (see Section 3.4 in How to Use the NAG Library and its Documentation).

On entry: ifail must be set to a value with the decimal expansion $\mathit{cba}$, where each of the decimal digits $c$, $b$ and $a$ must have a value of $0$ or $1$.

$a=0$	specifies hard failure, otherwise soft failure;
$b=0$	suppresses error messages, otherwise error messages will be printed (see Section 6);
$c=0$	suppresses warning messages, otherwise warning messages will be printed (see Section 6).

The recommended value for inexperienced users is $110$ (i.e., hard failure with all messages printed).

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

$\mathbf{ifail}=1$

One or more of the arguments n, tol, np, mnp, lw or liw has been incorrectly set, or $\mathbf{b}\le \mathbf{a}$, or the condition (3) on x is not satisfied, or the number of known boundary values (specified by v) is not n.

$\mathbf{ifail}=2$

The Newton iteration has failed to converge. This could be due to there being too few points in the initial mesh or to the initial approximate solution being too inaccurate. If this latter reason is suspected you should use d02raf instead. If the warning ‘Jacobian matrix is singular’ is printed this could be due to specifying zero estimated boundary values and these should be varied. This warning could also be printed in the unlikely event of the Jacobian matrix being calculated inaccurately. If you cannot make changes to prevent the warning then d02raf should be used.

$\mathbf{ifail}=3$

The Newton iteration has reached round-off level. It could be, however, that the answer returned is satisfactory. This error might occur if too much accuracy is requested.

$\mathbf{ifail}=4$

A finer mesh is required for the accuracy requested; that is mnp is not large enough.

$\mathbf{ifail}=5$

A serious error has occurred in a call to d02gaf. Check all array subscripts and subroutine argument lists in calls to d02gaf. 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

10

Example

10.1

Program Text

Program Text (d02gafe.f90)

10.2

Program Data

Program Data (d02gafe.d)

10.3

Program Results

Program Results (d02gafe.r)