NAG Library Routine Document

d02gbf (bvp_fd_lin_gen)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

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

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

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

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

3:     $\mathbf{n}$ – IntegerInput

On entry: the number of equations; that is $\mathit{n}$ is the order of system (1).

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

4:     $\mathbf{tol}$ – 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, $z\left(x\right)$ is the approximate solution from d02gbf and $y\left(x\right)$ is the true solution of equations (1) and (2) then, except in extreme cases, it is expected that

$$‖z-y‖\le \mathbf{tol}$$

(3)

where

$$‖u‖=\underset{1\le i\le \mathbf{n}}{\max }\underset{1\le j\le \mathbf{np}}{\max }\left|u_i\left(x_j\right)\right|\text{.}$$

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

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

fcnf must evaluate the matrix $F\left(x\right)$ in (1) at a general point $x$.

The specification of fcnf is:

Fortran Interface

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

C Header Interface

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

In the description of the arguments of d02gbf below, $\mathit{n}$ denotes the actual value of n in the call of d02gbf.

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

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

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

On exit: $\mathbf{f}\left(\mathit{n}\times \left(\mathit{i}-1\right)+\mathit{j}\right)$ must contain the $\left(\mathit{i},\mathit{j}\right)$th element of the matrix $F\left(x\right)$, for $\mathit{i}=1,2,\dots ,\mathit{n}$ and $\mathit{j}=1,2,\dots ,\mathit{n}$. (See Section 10 for an example.)

fcnf must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02gbf is called. Arguments denoted as Input must not be changed by this procedure.

Note: fcnf should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02gbf. If your code inadvertently does return any NaNs or infinities, d02gbf is likely to produce unexpected results.

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

fcng must evaluate the vector $g\left(x\right)$ in (1) at a general point $x$.

The specification of fcng is:

Fortran Interface

Subroutine fcng ( x, g) Real (Kind=nag_wp), Intent (In) :: x Real (Kind=nag_wp), Intent (Out) :: g(*)

C Header Interface

#include <nagmk26.h> void  fcng (const double *x, double g[])

In the description of the arguments of d02gbf below, $\mathit{n}$ denotes the actual value of n in the call of d02gbf.

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

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

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

On exit: the $\mathit{i}$th element of the vector $g\left(x\right)$, for $\mathit{i}=1,2,\dots ,\mathit{n}$. (See Section 10 for an example.)

fcng must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02gbf is called. Arguments denoted as Input must not be changed by this procedure.

Note: fcng should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02gbf. If your code inadvertently does return any NaNs or infinities, d02gbf is likely to produce unexpected results.

7:     $\mathbf{c}\left(\mathbf{n},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

8:     $\mathbf{d}\left(\mathbf{n},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

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

On entry: the arrays c and d must be set to the matrices $C$ and $D$ in (2)). gam must be set to the vector $\gamma$ in (2).

On exit: the rows of c and d and the components of gam are reordered so that the boundary conditions are in the order:

(i)	conditions on $y\left(a\right)$ only;
(ii)	condition involving $y\left(a\right)$ and $y\left(b\right)$; and
(iii)	conditions on $y\left(b\right)$ only.

The routine will be slightly more efficient if the arrays c, d and gam are ordered in this way before entry, and in this event they will be unchanged on exit.

Note that the problems (1) and (2) must be of boundary value type, that is neither $C$ nor $D$ may be identically zero. Note also that the rank of the matrix $\left[C,D\right]$ must be $\mathit{n}$ for the problem to be properly posed. Any violation of these conditions will lead to an error exit.

10:   $\mathbf{mnp}$ – IntegerInput

On entry: the maximum permitted number of mesh points.

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

11:   $\mathbf{x}\left(\mathbf{mnp}\right)$ – 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:

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

(4)

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 (4).

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

On exit: the approximate solution $z\left(x\right)$ satisfying (3), on the final mesh, that is

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

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

The remaining columns of y are not used.

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

On entry: determines whether a default mesh 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 x as in (4).

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

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

15:   $\mathbf{lw}$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which d02gbf is called.

Constraint: $\mathbf{lw}\ge \mathbf{mnp}\times \left(3{\mathbf{n}}^2+5\mathbf{n}+2\right)+3{\mathbf{n}}^2+5\mathbf{n}$.

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

17:   $\mathbf{liw}$ – IntegerInput

On entry: the dimension of the array iw as declared in the (sub)program from which d02gbf is called.

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

18:   $\mathbf{ifail}$ – 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

$\mathbf{ifail}=1$

On entry, $\mathbf{a}=〈\mathit{\text{value}}〉$ and $\mathbf{b}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{b}>\mathbf{a}$.

On entry, $\mathbf{liw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{liw}\ge \mathbf{mnp}\times \left(2\times \mathbf{n}+1\right)+\mathbf{n}$; that is, $〈\mathit{\text{value}}〉$.

On entry, $\mathbf{lw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lw}\ge \mathbf{mnp}\times \left(3\times {\mathbf{n}}^2+5\times \mathbf{n}+2\right)+3\times {\mathbf{n}}^2+5\times \mathbf{n}$; that is, $〈\mathit{\text{value}}〉$.

On entry, $\mathbf{mnp}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{mnp}\ge 32$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 2$.

On entry, $\mathbf{np}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{np}=0$ or $\mathbf{np}\ge 4$.

On entry, $\mathbf{np}=〈\mathit{\text{value}}〉$ and $\mathbf{mnp}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{np}\le \mathbf{mnp}$.

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

On entry: $\mathbf{a}=〈\mathit{\text{value}}〉$ and $\mathbf{x}\left(1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{a}=\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{np}\right)=\mathbf{b}\text{, \hspace{1em}}\mathbf{np}\ge 4$.

On entry: $\mathbf{b}=〈\mathit{\text{value}}〉$ and $\mathbf{x}\left(\mathbf{np}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{a}=\mathbf{x}\left(1\right)<\mathbf{x}\left(2\right)<\cdots <\mathbf{x}\left(\mathbf{np}\right)=\mathbf{b}\text{, \hspace{1em}}\mathbf{np}\ge 4$.

The sequence x is not strictly increasing. For $i=〈\mathit{\text{value}}〉$, $\mathbf{x}\left(i\right)=〈\mathit{\text{value}}〉$ and $\mathbf{x}\left(i+1\right)=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=2$

c is identically zero; n conditions are set in d.
At least one condition must be on the left. $\mathbf{n}=〈\mathit{\text{value}}〉$.

d is identically zero; n conditions are set in c.
At least one condition must be on the right. $\mathbf{n}=〈\mathit{\text{value}}〉$.

More than n columns of the n by $2\times \mathbf{n}$ matrix $\left[C,D\right]$ are identically zero, i.e., the boundary conditions are rank deficient. The number of non-identically zero columns is $〈\mathit{\text{value}}〉$.

Row $〈\mathit{\text{value}}〉$ of the array c and the corresponding row of array d are identically zero, i.e., the boundary conditions are rank deficient.

$\mathbf{ifail}=3$

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

Newton iteration has reached round-off level.
If desired accuracy has not been reached, tol is too small for this problem and this machine precision.

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 or you cannot make changes to prevent this error, you should use the routine with a continuation facility instead.

$\mathbf{ifail}=4$

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

$\mathbf{ifail}=5$

At least one row of the n by $2\times \mathbf{n}$ matrix $\left[C,D\right]$ is a linear combination of the other rows, i.e., the boundary conditions are rank deficient. The index of the first such row is $〈\mathit{\text{value}}〉$.

At least one row of the n by $2\times \mathbf{n}$ matrix $\left[C,D\right]$ is a linear combination of the other rows determined up to a numerical tolerance, i.e., the boundary conditions are rank deficient. The index of first such row is $〈\mathit{\text{value}}〉$.

There are two possible reasons for this error exit which occurs when checking the rank of the boundary conditions by reduction to a row echelon form:

(i)	at least one row of the $\mathit{n}$ by $2\mathit{n}$ matrix $\left[C,D\right]$ is a linear combination of the other rows and hence the boundary conditions are rank deficient. The index of the first such row encountered is given by $\mathbf{iw}\left(1\right)$ on exit; and
(ii)	as (i) but the rank deficiency implied by this error exit has only been determined up to a numerical tolerance. Minus the index of the first such row encountered is given by $\mathbf{iw}\left(1\right)$ on exit.

In case (ii) there is some doubt as to the rank deficiency of the boundary conditions. However even if the boundary conditions are not rank deficient they are not posed in a suitable form for use with this routine.

For example, if

$$C=\left(\begin{array}{ll}1& 0\\ 1& \epsilon \end{array}\right)\text{, \hspace{1em}}D=\left(\begin{array}{ll}1& 0\\ 1& 0\end{array}\right)\text{, \hspace{1em}}\gamma =\left(\begin{array}{l}{\gamma }_1\\ {\gamma }_2\end{array}\right)$$

and $\epsilon$ is small enough, this error exit is likely to be taken. A better form for the boundary conditions in this case would be

$$C=\left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right)\text{, \hspace{1em}}D=\left(\begin{array}{ll}1& 0\\ 0& 0\end{array}\right)\text{, \hspace{1em}}\gamma =\left(\begin{array}{l}{\gamma }_1\\ {\epsilon }^{-1}\left({\gamma }_2-{\gamma }_1\right)\end{array}\right)\text{.}$$

$\mathbf{ifail}=-99$

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 (d02gbfe.f90)

10.2

Program Data

Program Data (d02gbfe.d)

10.3

Program Results

Program Results (d02gbfe.r)