NAG Toolbox: nag_ode_withdraw_ivp_rk_range (d02pc)

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] = f(t, y)

Input Parameters

1:     $\mathrm{t}$ – double scalar

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

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

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

Output Parameters

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

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

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

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

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

3:     $\mathrm{twant}$ – 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_rk_range (d02pc)); see nag_ode_ivp_rk_setup (d02pv) for a description of tstart and tend. twant must not lie beyond tend in the direction of integration.

4:     $\mathrm{ygot}\left(:\right)$ – double array

The dimension of the array ygot must be at least $\mathit{n}$

On the first call to nag_ode_ivp_rk_range (d02pc), ygot need not be set. On all subsequent calls ygot must remain unchanged.

5:     $\mathrm{ymax}\left(:\right)$ – double array

The dimension of the array ymax must be at least $\mathit{n}$

On the first call to nag_ode_ivp_rk_range (d02pc), ymax need not be set. On all subsequent calls ymax must remain unchanged.

6:     $\mathrm{work}\left(:\right)$ – double array

The dimension of the array work must be at least $\mathbf{lenwrk}$ (see nag_ode_ivp_rk_setup (d02pv))

This must be the same array as supplied to nag_ode_ivp_rk_setup (d02pv). It must remain unchanged between calls.

Optional Input Parameters

None.

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

The dimension of the array ygot will be $\mathit{n}$

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_rk_setup (d02pv). 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:     $\mathrm{ypgot}\left(:\right)$ – double array

The dimension of the array ypgot will be $\mathit{n}$

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

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

The dimension of the array ymax will be $\mathit{n}$

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

5:     $\mathrm{work}\left(:\right)$ – double array

The dimension of the array work will be $\mathbf{lenwrk}$ (see nag_ode_ivp_rk_setup (d02pv))

Information about the integration for use on subsequent calls to nag_ode_ivp_rk_range (d02pc) or other associated functions.

6:     $\mathrm{ifail}$ – 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, an invalid input value for twant was detected or an invalid call to nag_ode_ivp_rk_range (d02pc) was made, for example without a previous call to the setup function nag_ode_ivp_rk_setup (d02pv). You cannot continue integrating the problem.

W  $\mathbf{ifail}=2$

This return is possible only when $\mathbf{method}=3$ has been selected in the preceding call of nag_ode_ivp_rk_setup (d02pv). nag_ode_ivp_rk_range (d02pc) is being used inefficiently because the step size has been reduced drastically many times to get answers at many values of twant. If you really need the solution at this many points, you should change to $\mathbf{method}=2$ because it is (much) more efficient in this situation. To change method, restart the integration from tgot, ygot by a call to nag_ode_ivp_rk_setup (d02pv). If you wish to continue with $\mathbf{method}=3$, just call nag_ode_ivp_rk_range (d02pc) again without altering any of the arguments. The monitor of this kind of inefficiency will be reset automatically so that the integration can proceed.

W  $\mathbf{ifail}=3$

A considerable amount of work has been expended in the (primary) integration. This is measured by counting the number of calls to the supplied function f. At least $5000$ calls have been made since the last time this counter was reset. Calls to f in a secondary integration for global error assessment (when $\mathbf{errass}=\mathit{true}$ in the call to nag_ode_ivp_rk_setup (d02pv)) are not counted in this total. The integration was interrupted, so tgot is not equal to twant. If you wish to continue on towards twant, just call nag_ode_ivp_rk_range (d02pc) again without altering any of the arguments. The counter measuring work will be reset to zero automatically.

W  $\mathbf{ifail}=4$

It appears that this problem is stiff. The methods implemented in nag_ode_ivp_rk_range (d02pc) can solve such problems, but they are inefficient. You should change to another code based on methods appropriate for stiff problems. The integration was interrupted so tgot is not equal to twant. If you want to continue on towards twant, just call nag_ode_ivp_rk_range (d02pc) again without altering any of the arguments. The stiffness monitor will be reset automatically.

W  $\mathbf{ifail}=5$

It does not appear possible to achieve the accuracy specified by tol and thres in the call to nag_ode_ivp_rk_setup (d02pv) with the precision available on the computer being used and with this value of method. You cannot continue integrating this problem. A larger value for method, if possible, will permit greater accuracy with this precision. To increase method and/or continue with larger values of tol and/or thres, restart the integration from tgot, ygot by a call to nag_ode_ivp_rk_setup (d02pv).

W  $\mathbf{ifail}=6$

(This error exit can only occur if $\mathbf{errass}=\mathit{true}$ in the call to nag_ode_ivp_rk_setup (d02pv).) The global error assessment may not be reliable beyond the current integration point tgot. This may occur because either too little or too much accuracy has been requested or because $f\left(t,y\right)$ is not smooth enough for values of $t$ just past tgot and current values of the solution $y$. The integration cannot be continued. This return does not mean that you cannot integrate past tgot, rather that you cannot do it with $=\mathit{true}$. However, it may also indicate problems with the primary integration.

   $\mathbf{ifail}=-99$

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: d02pc_example

function d02pc_example


fprintf('d02pc example results\n\n');

% Set initial conditions and input
method = int64(1);
tstart = 0;
tend   = 2*pi;
n      = int64(2);
errass = false;
lenwrk = int64(32*n);
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
  [work, ifail] = d02pv(tstart, yinit, tend, tol, thresh, method, 'Usual', ...
                        errass, lenwrk);


  tgot(1) = tstart;
  ygot(1,:) = yinit;
  twant = tstart;
  for j=1:npts
    twant = twant + tinc;
    [tgot(j+1), ygot(j+1,:), ypgot, ymax, work, ifail] = d02pc(@f, n, ...
                                            twant, ygot(j,:), ymax, work);
    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);
  % d02py is a diagnostic routine.
  [fevals, stepcost, waste, stepsok, hnext, ifail] = d02py;
  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] = f(t, y)
% Evaluate derivative vector.
yp = zeros(2, 1);
yp(1) =  y(2);
yp(2) = -y(1);

d02pc example results


Calculation with TOL =  1.0e-03:

  Number of evaluations of f = 115

Calculation with TOL =  1.0e-04:

  Number of evaluations of f = 223