NAG Toolbox: nag_ode_ivp_rkts_range (d02pe)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

f must evaluate the functions $f_i$ (that is the first derivatives $y_i^{\prime }$) for given values of the arguments $t$, $y_i$.

[yp, user] = f(t, n, y, user)

Input Parameters

1:     $\mathrm{t}$ – double scalar

$t$, the current value of the independent variable.

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

$n$, the number of ordinary differential equations in the system to be solved.

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

The current values of the dependent variables, $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.

4:     $\mathrm{user}$ – Any MATLAB object

f is called from nag_ode_ivp_rkts_range (d02pe) with the object supplied to nag_ode_ivp_rkts_range (d02pe).

Output Parameters

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

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

2:     $\mathrm{user}$ – Any MATLAB object

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

$t$, the next value of the independent variable where a solution is desired.

Constraint: twant must be closer to tend than the previous value of tgot (or tstart on the first call to nag_ode_ivp_rkts_range (d02pe)); see nag_ode_ivp_rkts_setup (d02pq) for a description of tstart and tend. twant must not lie beyond tend in the direction of integration.

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

On the first call to nag_ode_ivp_rkts_range (d02pe), ygot need not be set. On all subsequent calls ygot must remain unchanged.

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

On the first call to nag_ode_ivp_rkts_range (d02pe), ymax need not be set. On all subsequent calls ymax must remain unchanged.

5:     $\mathrm{iwsav}\left(130\right)$ – int64int32nag_int array

6:     $\mathrm{rwsav}\left(32\times \mathbf{n}+350\right)$ – double array

These must be the same arrays supplied in a previous call to nag_ode_ivp_rkts_setup (d02pq). They must remain unchanged between calls.

Optional Input Parameters

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

Default: the dimension of the arrays ygot, ymax. (An error is raised if these dimensions are not equal.)

$n$, the number of ordinary differential equations in the system to be solved.

Constraint: $\mathbf{n}\ge 1$.

2:     $\mathrm{user}$ – Any MATLAB object

user is not used by nag_ode_ivp_rkts_range (d02pe), but is passed to f. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

Output Parameters

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

$t$, the value of the independent variable at which a solution has been computed. On successful exit with $\mathbf{ifail}=\mathbf{0}$, tgot will equal twant. On exit with $\mathbf{ifail}>\mathbf{1}$, a solution has still been computed at the value of tgot but in general tgot will not equal twant.

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

An approximation to the true solution at the value of tgot. At each step of the integration to tgot, the local error has been controlled as specified in nag_ode_ivp_rkts_setup (d02pq). The local error has still been controlled even when $\mathbf{tgot}\ne \mathbf{twant}$, that is after a return with $\mathbf{ifail}>\mathbf{1}$.

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

An approximation to the first derivative of the true solution at tgot.

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

$\mathbf{ymax}\left(i\right)$ contains the largest value of $\left|y_i\right|$ computed at any step in the integration so far.

5:     $\mathrm{user}$ – Any MATLAB object

6:     $\mathrm{iwsav}\left(130\right)$ – int64int32nag_int array

7:     $\mathrm{rwsav}\left(32\times \mathbf{n}+350\right)$ – double array

Information about the integration for use on subsequent calls to nag_ode_ivp_rkts_range (d02pe) or other associated functions.

8:     $\mathrm{ifail}$ – 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.

   $\mathbf{ifail}=1$

On entry, a previous call to the setup function has not been made or the communication arrays have become corrupted.

On entry, $\mathbf{n}=\_$, but the value passed to the setup function was .

On entry, the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere. You cannot continue integrating the problem.

tend (setup) had already been reached in a previous call.
To start a new problem, you will need to call the setup function.

twant does not lie in the direction of integration.

twant is too close to the last value of tgot (tstart on setup).

twant lies beyond tend (setup) in the direction of integration, but is very close to tend.

twant lies beyond tend (setup) in the direction of integration.

You cannot call this function after it has returned an error.
You must call the setup function to start another problem.

You cannot call this function when you have specified, in the setup function, that the step integrator will be used.

W  $\mathbf{ifail}=2$

This function is being used inefficiently because the step size has been reduced drastically many times to obtain answers at many points. Using the order $4$ and $5$ pair method at setup is more appropriate here. You can continue integrating this problem.

W  $\mathbf{ifail}=3$

Approximately $\_$ function evaluations have been used to compute the solution since the integration started or since this message was last printed. However, you can continue integrating the problem.

W  $\mathbf{ifail}=4$

Approximately $\_$ function evaluations have been used to compute the solution since the integration started or since this message was last printed. Your problem has been diagnosed as stiff. If the situation persists, it will cost roughly $\_$ times as much to reach tend (setup) as it has cost to reach the current time. You should probably call functions intended for stiff problems. However, you can continue integrating the problem.

W  $\mathbf{ifail}=5$

In order to satisfy your error requirements the solver has to use a step size of $\_$ at the current time, $\_$. This step size is too small for the machine precision, and is smaller than $\_$.

W  $\mathbf{ifail}=6$

The global error assessment algorithm failed at start of integration.
The integration is being terminated.

The global error assessment may not be reliable for times beyond $\_$.
The integration is being terminated.

   $\mathbf{ifail}=-99$

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: d02pe_example

function d02pe_example


fprintf('d02pe example results\n\n');

% Set initial conditions and input
method = int64(1);
tstart = 0;
tend = 2*pi;
yinit = [0;1];
hstart = 0;
thresh = [1e-08; 1e-08];
npts = 40;
tol0 =  1.0E-3;
ygot = zeros(npts+1, 2);
tgot = zeros(npts+1, 1);
err1 = zeros(npts+1, 2);
err2 = zeros(npts+1, 2);
ymax = zeros(1, 2);

% Set control for output
tinc = (tend-tstart)/npts;
tol = 10.0*tol0;

% We run through the calculation twice with two tolerance values
for i = 1:2

  tol = tol*0.1;

  % Call setup function
  [iwsav, rwsav, ifail] = d02pq(tstart, tend, yinit, tol, thresh, method);

  tgot(1) = tstart;
  ygot(1,:) = yinit;
  twant = tstart;
  for j=1:npts
    twant = twant + tinc;
    [tgot(j+1), ygot(j+1,:), ypgot, ymax, user, iwsav, rwsav, ifail] = ...
      d02pe(@f, twant, ygot(j, :), ymax, iwsav, rwsav);
    err1(j+1, i) =  ygot(j+1, 1)-sin(tgot(j+1));
    err2(j+1, i) =  ygot(j+1, 2)-cos(tgot(j+1));
  end

  fprintf('\nCalculation with TOL = %8.1e:\n\n', tol);
  [fevals, stepcost, waste, stepsok, hnext, iwsav, ifail] = d02pt(iwsav, rwsav);
  fprintf('  Number of evaluations of f = %d\n', fevals);

end

% Plot results
fig1 = figure;
title('First-order ODEs using Runge-Kutta Low-order Method, Two Tolerances');
hold on;
axis([0 10 -1.2 1.2]);
xlabel('t');
ylabel('Solution (y, y'')');
plot(tgot, ygot(:, 1), '-xr');
text(ceil(tgot(npts+1)), ygot(npts+1, 1)-0.2, 'y', 'Color', 'r');
plot(tgot, ygot(:, 2), '-xg');
text(ceil(tgot(npts+1)), ygot(npts+1, 2), 'y''', 'Color', 'g');
% Plot errors with a different (log) scale
ax1 = gca;
ax2 = axes('Position',get(ax1,'Position'),...
           'XAxisLocation','bottom','YAxisLocation','right',...
           'YScale','log','Color','none','XColor','k','YColor','k');
hold on;
axis([0 10 1e-7 0.01]);
ylabel('abs(Error)');
plot(ax2, tgot, abs(err1(:, 1)), '-*b');
text(ceil(tgot(npts+1)), err1(npts+1, 1), 'y-error (tol=0.001)', 'Color', 'b');
plot(ax2, tgot, abs(err1(:, 2)), '-sm');
text(ceil(tgot(npts+1)), err1(npts+1, 2), 'y-error (tol=0.0001)', 'Color', 'm');
hold off



function [yp, user] = f(t, n, y, user)
  yp = [y(2); -y(1)];

d02pe example results


Calculation with TOL =  1.0e-03:

  Number of evaluations of f = 421

Calculation with TOL =  1.0e-04:

  Number of evaluations of f = 871