NAG Library Routine Document

D02GBF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     A – REAL (KIND=nag_wp)Input

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

2:     B – REAL (KIND=nag_wp)Input

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

Constraint: $\mathbf{B}>\mathbf{A}$.

3:     N – INTEGERInput

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

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

4:     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:     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:

SUBROUTINE FCNF ( X, F) REAL (KIND=nag_wp)  X, F(*)

In the description of the parameters of D02GBF below, $\mathit{n}$ denotes the actual value of N in the call of D02GBF.

1:     X – REAL (KIND=nag_wp)Input

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

2:     F($*$) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{F}\left(\mathit{j},\mathit{i}\right)$ 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 9 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. Parameters denoted as Input must not be changed by this procedure.

6:     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:

SUBROUTINE FCNG ( X, G) REAL (KIND=nag_wp)  X, G(*)

In the description of the parameters of D02GBF below, $\mathit{n}$ denotes the actual value of N in the call of D02GBF.

1:     X – REAL (KIND=nag_wp)Input

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

2:     G($*$) – 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 9 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. Parameters denoted as Input must not be changed by this procedure.

7:     C(N,N) – REAL (KIND=nag_wp) arrayInput/Output

8:     D(N,N) – REAL (KIND=nag_wp) arrayInput/Output

9:     GAM(N) – 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:   MNP – INTEGERInput

On entry: the maximum permitted number of mesh points.

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

11:   X(MNP) – 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:   Y(N,MNP) – 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:   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:   W(LW) – REAL (KIND=nag_wp) arrayWorkspace

15:   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:   IW(LIW) – INTEGER arrayWorkspace

17:   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:   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 3.3 in the Essential Introduction).

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 parameters N, TOL, NP, MNP, LW or LIW is incorrectly set, $\mathbf{B}\le \mathbf{A}$ or the condition (4) on X is not satisfied.

$\mathbf{IFAIL}=2$

There are three possible reasons for this error exit to be taken:

(i)	one of the matrices $C$ or $D$ is identically zero (that is, the problem is of initial value and not boundary value type). In this case, $\mathbf{IW}\left(1\right)=0$ on exit;
(ii)	a row of $C$ and the corresponding row of $D$ are identically zero (that is, the boundary conditions are rank deficient). In this case, on exit $\mathbf{IW}\left(1\right)$ contains the index of the first such row encountered; and
(iii)	more than $\mathit{n}$ of the columns of the $\mathit{n}$ by $2\mathit{n}$ matrix $\left[C,D\right]$ are identically zero (that is, the boundary conditions are rank deficient). In this case, on exit $\mathbf{IW}\left(1\right)$ contains minus the number of non-identically zero columns.

$\mathbf{IFAIL}=3$

The routine has failed to find a solution to the specified accuracy. There are a variety of possible reasons including:

(i)	the boundary conditions are rank deficient, which may be indicated by the message that the Jacobian is singular. However this is an unlikely explanation for the error exit as all rank deficient boundary conditions should lead instead to error exits with either $\mathbf{IFAIL}=\mathbf{2}$ or $\mathbf{5}$; see also (iv);
(ii)	not enough mesh points are permitted in order to attain the required accuracy. This is indicated by $\mathbf{NP}=\mathbf{MNP}$ on return from a call to D02GBF. This difficulty may be aggravated by a poor initial choice of mesh points;
(iii)	the accuracy requested cannot be attained on the computer being used; and
(iv)	an unlikely combination of values of $Fx$ has led to a singular Jacobian. The error should not persist if more mesh points are allowed.

$\mathbf{IFAIL}=4$

A serious error has occurred in a call to D02GBF. Check all array subscripts and subroutine parameter lists in calls to D02GBF. Seek expert help.

$\mathbf{IFAIL}=5$

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

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

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (d02gbfe.f90)

9.2  Program Data

Program Data (d02gbfe.d)

9.3  Program Results

Program Results (d02gbfe.r)