NAG Toolbox: nag_ode_bvp_fd_lin_gen (d02gb)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}$ – double scalar

$a$, the left-hand boundary point.

2:     $\mathrm{b}$ – double scalar

$b$, the right-hand boundary point.

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

3:     $\mathrm{tol}$ – double scalar

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 nag_ode_bvp_fd_lin_gen (d02gb) 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: $\mathbf{tol}>0.0$.

4:     $\mathrm{fcnf}$ – function handle or string containing name of m-file

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

[f] = fcnf(x)

Input Parameters

1:     $\mathrm{x}$ – double scalar

$x$, the value of the independent variable.

Output Parameters

1:     $\mathrm{f}\left(:\right)$ – double array

$\mathbf{f}\left(\mathit{n}\times \left(\mathit{i}-1\right)+\mathit{j}\right)$ 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 Example for an example.)

5:     $\mathrm{fcng}$ – function handle or string containing name of m-file

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

[g] = fcng(x)

Input Parameters

1:     $\mathrm{x}$ – double scalar

$x$, the value of the independent variable.

Output Parameters

1:     $\mathrm{g}\left(:\right)$ – double array

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

6:     $\mathrm{c}\left(\mathbf{n},\mathbf{n}\right)$ – double array

7:     $\mathrm{d}\left(\mathbf{n},\mathbf{n}\right)$ – double array

8:     $\mathrm{gam}\left(\mathbf{n}\right)$ – double array

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).

9:     $\mathrm{x}\left(\mathbf{mnp}\right)$ – double array

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)

10:   $\mathrm{np}$ – int64int32nag_int scalar

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).

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the dimension of the array gam and the first dimension of the arrays c, d and the second dimension of the arrays c, d. (An error is raised if these dimensions are not equal.)

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

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

2:     $\mathrm{mnp}$ – int64int32nag_int scalar

Default: the dimension of the array x.

The maximum permitted number of mesh points.

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

Output Parameters

1:     $\mathrm{c}\left(\mathbf{n},\mathbf{n}\right)$ – double array

2:     $\mathrm{d}\left(\mathbf{n},\mathbf{n}\right)$ – double array

3:     $\mathrm{gam}\left(\mathbf{n}\right)$ – double array

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 function 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.

4:     $\mathrm{x}\left(\mathbf{mnp}\right)$ – double array

$\mathbf{x}\left(1\right),\mathbf{x}\left(2\right),\dots ,\mathbf{x}\left(\mathbf{np}\right)$ define the final mesh (with the returned value of np) satisfying the relation (4).

5:     $\mathrm{y}\left(\mathbf{n},\mathbf{mnp}\right)$ – double array

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.

6:     $\mathrm{np}$ – int64int32nag_int scalar

The number of points in the final (returned) mesh.

7:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=1$

One or more of the arguments 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, $\mathit{iw}\left(1\right)=0$ 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 $\mathit{iw}\left(1\right)$ 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 $\mathit{iw}\left(1\right)$ contains minus the number of non-identically zero columns.

W  $\mathbf{ifail}=3$

The function 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 nag_ode_bvp_fd_lin_gen (d02gb). 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 nag_ode_bvp_fd_lin_gen (d02gb). Check all array subscripts and function argument lists in calls to nag_ode_bvp_fd_lin_gen (d02gb). 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 $\mathit{iw}\left(1\right)$ 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 $\mathit{iw}\left(1\right)$ 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 function.

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.

   $\mathbf{ifail}=-399$

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

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: d02gb_example

function d02gb_example


fprintf('d02gb example results\n\n');

% For communication with f.
global eps;

% Set up initial values before calling NAG routine for the first time.
mnp = int64(70);
a = 0;
b = 1;
tol = 0.001;
c = [1, 0; 0, 0];
d = [0, 0; 1, 0];
gam = [0; 1];
x = zeros(1,mnp);
np = int64(0);
n = int64(2);
xkeep = zeros(1,2);
npkeep = zeros(2);
ykeep = zeros(1,2);
espkeep = zeros(2);

for i = 1:2
    eps = 10^(-i);
    [c, d, gam, x, y, np, ifail] = ...
        d02gb(a, b, tol, @f, @g, c, d, gam, x, np, 'n', n, 'mnp', mnp);
    % save results.
    for j = 1:np 
        xkeep(j,i) = x(j);
        ykeep(j,i) = y(1,j);
    end
    % Store the number of points and epsilon for use in plot.
    npkeep(i) = np;
    espkeep(i) = eps;
    fprintf(' for eps = %7.4f, number of points required, np = %d\n', eps, np);
end

% Plot results.
fig1 = figure;
plot(xkeep(1:npkeep(1),1), ykeep(1:npkeep(1),1), '-+', ...
     xkeep(1:npkeep(2),2), ykeep(1:npkeep(2),2), ':x');
set(gca, 'YLim', [0 1.1]);
set(gca, 'XLim', [-0.01 1.01]);
title('Solution to Simple Second-order Problem');
xlabel('x');
ylabel('Solution');
legend(['\epsilon = ', num2str(espkeep(1))], ...
       ['\epsilon = ', num2str(espkeep(2))], 'Location', 'East');


function f = f(x)
% For communication with main routine.
global eps;

% Evaluate matrix.
f=zeros(2,2);
f(1,2) = 1;
f(2,2) = -1/eps;

function g = g(x)
% Evaluate vector.
g=zeros(2,1);

d02gb example results

 for eps =  0.1000, number of points required, np = 15
 for eps =  0.0100, number of points required, np = 49