NAG Toolbox: nag_ode_ivp_rk_zero_simple (d02bj)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{x}$ – double scalar

The initial value of the independent variable $x$.

2:     $\mathrm{xend}$ – double scalar

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

The initial values of the solution $y_1,y_2,\dots ,y_{\mathit{n}}$ at $x=\mathbf{x}$.

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

fcn must evaluate the functions $f_i$ (i.e., the derivatives $y_i^{\prime }$) for given values of its arguments $x,y_1,\dots ,y_{\mathit{n}}$.

[f] = fcn(x, y)

Input Parameters

1:     $\mathrm{x}$ – double scalar

$x$, the value of the independent variable.

2:     $\mathrm{y}\left(\mathit{n}\right)$ – double array

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

Output Parameters

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

The value of $f_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.

5:     $\mathrm{tol}$ – double scalar

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.

nag_ode_ivp_rk_zero_simple (d02bj) 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 nag_ode_ivp_rk_zero_simple (d02bj) 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 nag_ode_ivp_rk_zero_simple (d02bj) 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: $10.0\times \mathit{machine precision}<\mathbf{tol}<0.01$.

6:     $\mathrm{relabs}$ – string (length ≥ 1)

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, then relabs should be set to 'A' on entry, whereas if the error requirement is in terms of the number of correct significant digits, then relabs should be set to 'R'. If you prefer a mixed error test, then 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 $\text{'D'}$.

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

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 nag_ode_ivp_rk_zero_simple (d02bj) 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, nag_ode_ivp_rk_zero_simple (d02bj) will integrate to xend with no further calls to output; if a call to output is required at the point $\mathbf{xsol}=\mathbf{xend}$, then xsol must be given precisely the value xend.

[xsol] = output(xsol, y)

Input Parameters

1:     $\mathrm{xsol}$ – double scalar

The output value of the independent variable $x$.

2:     $\mathrm{y}\left(\mathit{n}\right)$ – double array

The computed solution at the point xsol.

Output Parameters

1:     $\mathrm{xsol}$ – double scalar

You must set xsol to the next value of $x$ at which output is to be called.

If you do not wish to access intermediate output, the actual argument output must be the string nag_ode_ivp_rk_zero_simple_dummy_output (d02bjx). (nag_ode_ivp_rk_zero_simple_dummy_output (d02bjx) is included in the NAG Toolbox.)

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

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.

[result] = g(x, y)

Input Parameters

1:     $\mathrm{x}$ – double scalar

$x$, the value of the independent variable.

2:     $\mathrm{y}\left(\mathit{n}\right)$ – double array

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

Output Parameters

1:     $\mathrm{result}$ – double scalar

The value of $g\left(x,y\right)$ at the specified point.

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

Optional Input Parameters

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

Default: the dimension of the array y.

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

Constraint: $\mathbf{n}>0$.

Output Parameters

1:     $\mathrm{x}$ – double scalar

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

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

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

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

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry,	$\mathbf{tol}\ge 0.01$,
or	tol is too small
or	$\mathbf{n}\le 0$,
or	$\mathbf{relabs}\ne \text{'M'}$, $\text{'A'}$, $\text{'R'}$ or $\text{'D'}$,
or	$\mathbf{x}=\mathbf{xend}$.

   $\mathbf{ifail}=2$

With the given value of tol, no further progress can be made across the integration range from the current point $x=\mathbf{x}$. (See Further Comments 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 $g$, then no point at which $g\left(x,y\right)$ changes sign has been located up to the point $x=\mathbf{x}$.

   $\mathbf{ifail}=3$

tol is too small for nag_ode_ivp_rk_zero_simple (d02bj) 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.

   $\mathbf{ifail}=4$

xsol has not been reset or xsol lies behind x in the direction of integration, after the initial call to output, if the output option was selected.

   $\mathbf{ifail}=5$

A value of xsol returned by the output has not been reset or lies behind the last value of xsol in the direction of integration, if the output option was selected.

   $\mathbf{ifail}=6$

At no point in the range x to xend did the function $g\left(x,y\right)$ change sign, if $g$ was supplied. It is assumed that $g\left(x,y\right)=0$ has no solution.

   $\mathbf{ifail}=7$

A serious error has occurred in an internal call to an interpolation function. Check all (sub)program calls and array dimensions. Seek expert help.

   $\mathbf{ifail}=-99$

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: d02bj_example

function d02bj_example


fprintf('d02bj example results\n\n');

% For communication with save.
global ykeep ncall xkeep;

% Initialize variables and arrays.
x = 0;
xend = 10;
y = [0.5; 0.5; pi/5];
relabs = 'Default';

xOut1 = [];
ncall = 0;
ykeep = zeros(1,length(y));
xkeep = zeros(1,1);

fprintf('Case 1: intermediate output, root-finding\n\n');
for j = 4:5
    tol = double(10)^(-j);
    disp(['Calculation with tol = ',num2str(tol)]);
    disp(' X     Y(1)       Y(2)       Y(3)');

    [xOut, yOut, ifail] = d02bj(x, xend, y, @fcn, tol, ...
        relabs, @output, @g);

    disp(' ');
    disp(['Root of Y(1) = 0.0 at ',num2str(xOut)]);
    disp('Solution is' );
    fprintf('  %8.4f   %8.4f   %8.4f\n\n', yOut);
end

fprintf('Case 2: no intermediate output, root-finding \n\n');
for j = 4:5
    tol = double(10)^(-j);
    disp(['Calculation with tol = ',num2str(tol)]);

    [xOut, yOut, ifail] = d02bj(x, xend, y, @fcn, tol, ...
        relabs, 'd02bjx',@g);

    disp(['Root of Y(1) = 0.0 at ',num2str(xOut)]);
    disp('Solution is' );
    fprintf('  %8.4f   %8.4f   %8.4f\n\n', yOut);

    % Store the x value for plotting.
    xOut1 = xOut;
end

fprintf('Case 3: intermediate output, no root-finding\n\n');
for j = 4:5
    tol = double(10)^(-j);
    disp(['Calculation with tol = ',num2str(tol)]);
    fprintf(' X     Y(1)       Y(2)       Y(3) \n');

    % save stores intermediate values in xkeep, ykeep, which are
    % plotted later (it also outputs them).
    [xOut, yOut, ifail] = d02bj(x, xend, y, @fcn, tol, ...
        relabs, @save,'d02bjw');
    fprintf('\n');
end

fprintf(['Case 4: no intermediate output, no root-finding ', ...
    '(integrate to xend)\n\n']);
for j = 4:5
    tol = double(10)^(-j);
    disp(['Calculation with tol = ',num2str(tol)]);
    disp(' X     Y(1)       Y(2)       Y(3)');
    fprintf('%2d   %8.4f   %8.4f   %8.4f\n', x, y);

    [xOut, yOut, ifail] = ...
        d02bj(x, xend, y, @fcn, tol, relabs, 'd02bjx','d02bjw');
    fprintf('%2d   %8.4f   %8.4f   %8.4f\n\n', xOut, yOut);
end

% Plot results.
nres = 0.5*length(xkeep);
xplot = xkeep(nres+1:2*nres);
yplot = ykeep(nres+1:2*nres, :);
fig1 = figure;
display_plot(xplot, yplot, xOut1)


function xsolOut = save(xsol, y)
% For communication with main routine.
global ykeep ncall xkeep;

% This version of the intermediate output routine stores the values
% (so they can be plotted in the main routine).
ncall = ncall+1;
ykeep(ncall,:) = y;
xkeep(ncall,:) = xsol;
fprintf('%2d   %8.4f   %8.4f   %8.4f\n', xsol, y);
xsolOut = xsol + 2;

function xsolOut = output(xsol, y)
% Output intermediate values of solution.
fprintf('%2d   %8.4f   %8.4f   %8.4f\n', xsol, y);
xsolOut = xsol + 2;

function f = fcn(x,y)
% Evaluate the derivatives.
f = zeros(3,1);
f(1) = tan(y(3));
f(2) = -0.032*tan(y(3))/y(2) - 0.02*y(2)/cos(y(3));
f(3) = -0.032/y(2)^2;

function result = g(x,y)
% Evaluate g(x,y) when root-finding option is selected.
result = y(1);

function display_plot(xplot, yplot, xOut1)
% Formatting for title and axis labels.
% Plot the three curves.
plot(xplot, yplot(:,1), '-+', ...
     xplot, yplot(:,2), '--x', ...
     xplot, yplot(:,3), ':*');
% Mark the height=0 point.
do_stem(xplot, yplot, xOut1);
% Add title.
title('ODE Solution using Runge-Kutta with Root-finding');
% Label the axes.
xlabel('x');
ylabel('Solution');
% Add legend.
legend('height','velocity','angle','height = 0','Location','Best');

function do_stem(xplot, yplot, xOut1)
% Find the x bin that xOut1 lies in.
for i = 1:length(xplot)
    if xplot(i) > xOut1
        break
    end
end
% Use linear interpolation to find the corresponding y values on the
% two curves.
dx = xplot(i)-xplot(i-1);
ddx = xOut1-xplot(i-1);
d1 = ddx/dx;

f1 = yplot(i-1,2) + d1*(yplot(i,2)-yplot(i-1,2));
f2 = yplot(i-1,3) + d1*(yplot(i,3)-yplot(i-1,3));

% Plot the line from the x axis to the two y values.
hold on
stem([xOut1,xOut1],[f1,0],'k:s');
hold on
stem([xOut1,xOut1],[f2,0],'k:s');

d02bj example results

Case 1: intermediate output, root-finding

Calculation with tol = 0.0001
 X     Y(1)       Y(2)       Y(3)
 0     0.5000     0.5000     0.6283
 2     1.5493     0.4055     0.3066
 4     1.7423     0.3743    -0.1289
 6     1.0055     0.4173    -0.5507
 
Root of Y(1) = 0.0 at 7.2882
Solution is
   -0.0000     0.4749    -0.7601

Calculation with tol = 1e-05
 X     Y(1)       Y(2)       Y(3)
 0     0.5000     0.5000     0.6283
 2     1.5493     0.4055     0.3066
 4     1.7423     0.3743    -0.1289
 6     1.0055     0.4173    -0.5507
 
Root of Y(1) = 0.0 at 7.2883
Solution is
   -0.0000     0.4749    -0.7601

Case 2: no intermediate output, root-finding 

Calculation with tol = 0.0001
Root of Y(1) = 0.0 at 7.2882
Solution is
   -0.0000     0.4749    -0.7601

Calculation with tol = 1e-05
Root of Y(1) = 0.0 at 7.2883
Solution is
   -0.0000     0.4749    -0.7601

Case 3: intermediate output, no root-finding

Calculation with tol = 0.0001
 X     Y(1)       Y(2)       Y(3) 
 0     0.5000     0.5000     0.6283
 2     1.5493     0.4055     0.3066
 4     1.7423     0.3743    -0.1289
 6     1.0055     0.4173    -0.5507
 8    -0.7460     0.5130    -0.8537
10    -3.6283     0.6333    -1.0515

Calculation with tol = 1e-05
 X     Y(1)       Y(2)       Y(3) 
 0     0.5000     0.5000     0.6283
 2     1.5493     0.4055     0.3066
 4     1.7423     0.3743    -0.1289
 6     1.0055     0.4173    -0.5507
 8    -0.7459     0.5130    -0.8537
10    -3.6282     0.6333    -1.0515

Case 4: no intermediate output, no root-finding (integrate to xend)

Calculation with tol = 0.0001
 X     Y(1)       Y(2)       Y(3)
 0     0.5000     0.5000     0.6283
10    -3.6283     0.6333    -1.0515

Calculation with tol = 1e-05
 X     Y(1)       Y(2)       Y(3)
 0     0.5000     0.5000     0.6283
10    -3.6282     0.6333    -1.0515