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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_ode_ivp_rk_onestep (d02pd)

## Purpose

nag_ode_ivp_rk_onestep (d02pd) is a one-step function for solving an initial value problem for a first-order system of ordinary differential equations using Runge–Kutta methods.
Note: this function is scheduled to be withdrawn, please see d02pd in Advice on Replacement Calls for Withdrawn/Superseded Routines..

## Syntax

[tnow, ynow, ypnow, work, ifail] = d02pd(f, neq, work)
[tnow, ynow, ypnow, work, ifail] = nag_ode_ivp_rk_onestep(f, neq, work)

## Description

nag_ode_ivp_rk_onestep (d02pd) and its associated functions (nag_ode_ivp_rk_setup (d02pv), nag_ode_ivp_rk_reset_tend (d02pw), nag_ode_ivp_rk_interp (d02px), nag_ode_ivp_rk_diag (d02py) and nag_ode_ivp_rk_errass (d02pz)) solve an initial value problem for a first-order system of ordinary differential equations. The functions, based on Runge–Kutta methods and derived from RKSUITE (see Brankin et al. (1991)), integrate
 y′ = f(t,y)  given  y(t0) = y0 $y′=f(t,y) given y(t0)=y0$
where y$y$ is the vector of n$\mathit{n}$ solution components and t$t$ is the independent variable.
nag_ode_ivp_rk_onestep (d02pd) is designed to be used in complicated tasks when solving systems of ordinary differential equations. You must first call nag_ode_ivp_rk_setup (d02pv) to specify the problem and how it is to be solved. Thereafter you (repeatedly) call nag_ode_ivp_rk_onestep (d02pd) to take one integration step at a time from tstart in the direction of tend (as specified in nag_ode_ivp_rk_setup (d02pv)). In this manner nag_ode_ivp_rk_onestep (d02pd) returns an approximation to the solution ynow and its derivative ypnow at successive points tnow. If nag_ode_ivp_rk_onestep (d02pd) encounters some difficulty in taking a step, the integration is not advanced and the function returns with the same values of tnow, ynow and ypnow as returned on the previous successful step. nag_ode_ivp_rk_onestep (d02pd) tries to advance the integration as far as possible subject to passing the test on the local error and not going past tend.
In the call to nag_ode_ivp_rk_setup (d02pv) you can specify either the first step size for nag_ode_ivp_rk_onestep (d02pd) to attempt or that it compute automatically an appropriate value. Thereafter nag_ode_ivp_rk_onestep (d02pd) estimates an appropriate step size for its next step. This value and other details of the integration can be obtained after any call to nag_ode_ivp_rk_onestep (d02pd) by a call to nag_ode_ivp_rk_diag (d02py). The local error is controlled at every step as specified in nag_ode_ivp_rk_setup (d02pv). If you wish to assess the true error, you must set errass = true${\mathbf{errass}}=\mathbf{true}$ in the call to nag_ode_ivp_rk_setup (d02pv). This assessment can be obtained after any call to nag_ode_ivp_rk_onestep (d02pd) by a call to nag_ode_ivp_rk_errass (d02pz).
If you want answers at specific points there are two ways to proceed:
 (i) The more efficient way is to step past the point where a solution is desired, and then call nag_ode_ivp_rk_interp (d02px) to get an answer there. Within the span of the current step, you can get all the answers you want at very little cost by repeated calls to nag_ode_ivp_rk_interp (d02px). This is very valuable when you want to find where something happens, e.g., where a particular solution component vanishes. You cannot proceed in this way with method = 3${\mathbf{method}}=3$. (ii) The other way to get an answer at a specific point is to set tend to this value and integrate to tend. nag_ode_ivp_rk_onestep (d02pd) will not step past tend, so when a step would carry it past, it will reduce the step size so as to produce an answer at tend exactly. After getting an answer there (${\mathbf{tnow}}={\mathbf{tend}}$), you can reset tend to the next point where you want an answer, and repeat. tend could be reset by a call to nag_ode_ivp_rk_setup (d02pv), but you should not do this. You should use nag_ode_ivp_rk_reset_tend (d02pw) instead because it is both easier to use and much more efficient. This way of getting answers at specific points can be used with any of the available methods, but it is the only way with method = 3${\mathbf{method}}=3$. It can be inefficient. Should this be the case, the code will bring the matter to your attention.

## References

Brankin R W, Gladwell I and Shampine L F (1991) RKSUITE: A suite of Runge–Kutta codes for the initial value problems for ODEs SoftReport 91-S1 Southern Methodist University

## Parameters

### Compulsory Input Parameters

1:     f – function handle or string containing name of m-file
f must evaluate the functions fi${f}_{i}$ (that is the first derivatives yi${y}_{i}^{\prime }$) for given values of the arguments t$t$, yi${y}_{i}$.
[yp] = f(t, y)

Input Parameters

1:     t – double scalar
t$t$, the current value of the independent variable.
2:     y( : $:$) – double array
The current values of the dependent variables, yi${y}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,\mathit{n}$.

Output Parameters

1:     yp( : $:$) – double array
The values of fi${f}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,\mathit{n}$.
2:     neq – int64int32nag_int scalar
n$n$, the number of ordinary differential equations in the system to be solved by the integration function.
Constraint: neq1${\mathbf{neq}}\ge 1$.
3:     work( : $:$) – double array
Note: the dimension of the array work must be at least lenwrk${\mathbf{lenwrk}}$ (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.

None.

None.

### Output Parameters

1:     tnow – double scalar
t$t$, the value of the independent variable at which a solution has been computed.
2:     ynow( : $:$) – double array
Note: the dimension of the array ynow must be at least n$\mathit{n}$.
An approximation to the solution at tnow. The local error of the step to tnow was no greater than permitted by the specified tolerances (see nag_ode_ivp_rk_setup (d02pv)).
3:     ypnow( : $:$) – double array
Note: the dimension of the array ypnow must be at least n$\mathit{n}$.
An approximation to the derivative of the solution at tnow.
4:     work( : $:$) – double array
Note: the dimension of the array work must be at least lenwrk${\mathbf{lenwrk}}$ (see nag_ode_ivp_rk_setup (d02pv)).
Information about the integration for use on subsequent calls to nag_ode_ivp_rk_onestep (d02pd) or other associated functions.
5:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

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

ifail = 1${\mathbf{ifail}}=1$
On entry, an invalid call to nag_ode_ivp_rk_onestep (d02pd) 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 ifail = 2${\mathbf{ifail}}=2$
nag_ode_ivp_rk_onestep (d02pd) is being used inefficiently because the step size has been reduced drastically many times to obtain answers at many points tend. If you really need the solution at this many points, you should use nag_ode_ivp_rk_interp (d02px) to obtain the answers inexpensively. If you need to change from method = 3${\mathbf{method}}=3$ to do this, restart the integration from tnow, ynow by a call to nag_ode_ivp_rk_setup (d02pv). If you wish to continue as before, call nag_ode_ivp_rk_onestep (d02pd) again. The monitor of this kind of inefficiency will be reset automatically so that the integration can proceed.
W ifail = 3${\mathbf{ifail}}=3$
A considerable amount of work has been expended in the (primary) integration. This is measured by counting the number of calls to f. At least 5000$5000$ calls have been made since the last time this counter was reset. Calls to f in a secondary integration for global error assessment (when errass = true${\mathbf{errass}}=\mathbf{true}$ in the call to nag_ode_ivp_rk_setup (d02pv)) are not counted in this total. The integration was interrupted. If you wish to continue on towards tend, just call nag_ode_ivp_rk_onestep (d02pd) again. The counter measuring work will be reset to zero automatically.
W ifail = 4${\mathbf{ifail}}=4$
It appears that this problem is stiff. The methods implemented in nag_ode_ivp_rk_onestep (d02pd) 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. If you want to continue on towards tend, just call nag_ode_ivp_rk_onestep (d02pd) again. The stiffness monitor will be reset automatically.
W ifail = 5${\mathbf{ifail}}=5$
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 tnow, ynow by a call to nag_ode_ivp_rk_setup (d02pv).
W ifail = 6${\mathbf{ifail}}=6$
(This error exit can only occur if errass = true${\mathbf{errass}}=\mathbf{true}$ in the call to nag_ode_ivp_rk_setup (d02pv).) The global error assessment may not be reliable beyond the current integration point tnow. This may occur because either too little or too much accuracy has been requested or because f(t,y)$f\left(t,y\right)$ is not smooth enough for values of t$t$ just beyond tnow and current values of the solution y$y$. The integration cannot be continued. This return does not mean that you cannot integrate past tnow, rather that you cannot do it with errass = true${\mathbf{errass}}=\mathbf{true}$. However, it may also indicate problems with the primary integration.

## Accuracy

The accuracy of integration is determined by the parameters tol and thres in a prior call to nag_ode_ivp_rk_setup (d02pv). Note that only the local error at each step is controlled by these parameters. The error estimates obtained are not strict bounds but are usually reliable over one step. Over a number of steps the overall error may accumulate in various ways, depending on the properties of the differential system.

If nag_ode_ivp_rk_onestep (d02pd) returns with ${\mathbf{ifail}}={\mathbf{5}}$ and the accuracy specified by tol and thres is really required then you should consider whether there is a more fundamental difficulty. For example, the solution may contain a singularity. In such a region the solution components will usually be large in magnitude. Successive output values of ynow should be monitored with the aim of trapping the solution before the singularity. In any case numerical integration cannot be continued through a singularity, and analytical treatment may be necessary.
Performance statistics are available after any return from nag_ode_ivp_rk_onestep (d02pd) (except when ${\mathbf{ifail}}={\mathbf{1}}$) by a call to nag_ode_ivp_rk_diag (d02py). If errass = true${\mathbf{errass}}=\mathbf{true}$ in the call to nag_ode_ivp_rk_setup (d02pv), global error assessment is available after any return from nag_ode_ivp_rk_onestep (d02pd) (except when ${\mathbf{ifail}}={\mathbf{1}}$) by a call to nag_ode_ivp_rk_errass (d02pz).
After a failure with ${\mathbf{ifail}}={\mathbf{5}}$ or 6${\mathbf{6}}$ the diagnostic functions nag_ode_ivp_rk_diag (d02py) and nag_ode_ivp_rk_errass (d02pz) may be called only once.
If nag_ode_ivp_rk_onestep (d02pd) returns with ${\mathbf{ifail}}={\mathbf{4}}$ then it is advisable to change to another code more suited to the solution of stiff problems. nag_ode_ivp_rk_onestep (d02pd) will not return with ${\mathbf{ifail}}={\mathbf{4}}$ if the problem is actually stiff but it is estimated that integration can be completed using less function evaluations than already computed.

## Example

This example solves the equation
 y ′ ′ = − y ,   y(0) = 0 ,   y′(0) = 1 $y′′ = -y , y(0) = 0 , y′(0) = 1$
reposed as
 y1 ′ = y2 $y1′ = y2$
 y2 ′ = − y1 $y2′ = -y1$
over the range [0,2π] $\left[0,2\pi \right]$ with initial conditions y1 = 0.0 ${y}_{1}=0.0$ and y2 = 1.0 ${y}_{2}=1.0$. We use relative error control with threshold values of 1.0e−8 $\text{1.0e−8}$ for each solution component and print the solution at each integration step across the range. We use a medium order Runge–Kutta method (method = 2${\mathbf{method}}=2$) with tolerances tol = 1.0e−4${\mathbf{tol}}=\text{1.0e−4}$ and tol = 1.0e−5${\mathbf{tol}}=\text{1.0e−5}$ in turn so that we may compare the solutions. The value of π$\pi$ is obtained by using nag_math_pi (x01aa).
Note that the length of work is large enough for any valid combination of input arguments to nag_ode_ivp_rk_setup (d02pv).
function nag_ode_ivp_rk_onestep_example
% Initialize variables and arrays.
tstart = 0;
hstart = 0;
ystart = [0; 1];
tend = 2*pi;
thres = [1e-08; 1e-08];
method = 2;
errass = false;
neq = 2;
lenwrk = 32*neq;

y1keep(1,:) = [ystart];
y2keep(1,:) = zeros(1,2);
y3keep(1,:) = zeros(1,2);
tkeep(1,:) = zeros(1,2);

fprintf('nag_ode_ivp_rk_onestep example program results\n\n');

% We run through the calculation twice with two tolerance values; we
% output and plot the results for both of them.
tol = [1e-04; 1e-05];
for itol = 1:2

% nag_ode_ivp_rk_setup is a setup routine to be called prior to nag_ode_ivp_rk_onestep.
[work, ifail] = nag_ode_ivp_rk_setup(tstart, ystart, tend, tol(itol), thres, ...
int64(method), task, errass, int64(lenwrk), 'neq', ...
int64(neq));
if ifail ~= 0
% Illegal input.  Print message and exit.
error('Warning: nag_ode_ivp_rk_setup returned with ifail = %1d ',ifail);
end

fprintf('Calculation with tol = %1.3e\n\n',tol(itol));
disp('  t           y1           y1''');
fprintf('%1.3f     %8.4f     %8.4f\n', tstart, ystart(1), ystart(2));

% Initialize the storage counter, and the current t value before
% looping over t.
ncall = 0;
tnow = tstart;
while (tnow < tend)
[tnow, ynow, ypnow, work, ifail] = nag_ode_ivp_rk_onestep(@f, ...
int64(neq), work);
if ifail ~= 0
% Illegal input.  Print message and exit.
error('Warning: nag_ode_ivp_rk_onestep returned with ifail = %1d \n\n',...
ifail);
end

% Output results, then store them for plotting.
fprintf('%1.3f     %8.4f     %8.4f\n', tnow, ynow(1), ynow(2));
ncall = ncall + 1;
tkeep(ncall,itol) = tnow;
if (itol == 2)
y1keep(ncall,:) = [ynow(1) ynow(2)];
y2keep(ncall,:) = [ynow(1)-sin(tnow) ynow(2)-cos(tnow)];
else
y3keep(ncall,:) = [ynow(1)-sin(tnow) ynow(2)-cos(tnow)];
end

end

% nag_ode_ivp_rk_diag is a diagnostic routine.
[totfcn, stpcst, waste, stpsok, hnext, ifail] = nag_ode_ivp_rk_diag;

fprintf(['\nCost of the integration in evaluations of F is ' ...
'%1.0f\n\n'], totfcn);
end
% Plot results.
fig = figure('Number', 'off');
display_plot(tkeep, y1keep, y2keep, y3keep)

function [yp] = f(t, y)
% Evaluate derivative vector.
yp = zeros(2, 1);
yp(1) =  y(2);
yp(2) = -y(1);
function display_plot(tkeep,y1keep,y2keep,y3keep)
% Formatting for title and axis labels.
titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14};
labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13};
set(gca, 'FontSize', 13); % for legend, axis tick labels, etc.
% Calculate spline through data values.
xx = tkeep(1,2):0.01:tkeep(length(tkeep(:,2)),2);
yy1 = spline(tkeep(:,2),y1keep(:,1),xx);
yy2 = spline(tkeep(:,2),y1keep(:,2),xx);
% Plot two of the curves and then add the other three.
[haxes, hline1, hline2] = plotyy(xx, yy2, tkeep(1:length(y3keep),1), ...
abs(y3keep(:,1)), 'plot', 'semilogy');
hold on
plot(xx,yy1,':');
hold on
plot(tkeep(:,2),y1keep(:,1),'+');
hold on
plot(tkeep(:,2),y1keep(:,2),'x');
% Select the required y axis, and plot the sixth curve.
axes(haxes(2));
hold on;
hline3 = plot(tkeep(:,2), abs(y2keep(:,1)));
% Set the axis limits and the tick specifications to beautify the plot.
set(haxes(1), 'YLim', [-1 1.2]);
set(haxes(1), 'YMinorTick', 'on');
set(haxes(1), 'YTick', [-1 -0.6 -0.2 0.2 0.6 1.0 1.2]);
set(haxes(2), 'YLim', [1e-9 1e-4]);
set(haxes(2), 'YMinorTick', 'on');
set(haxes(2), 'YTick', [1e-9 1e-8 1e-7 1e-6 1e-5 1e-4]);
for iaxis = 1:2
% These properties must be the same for both sets of axes.
set(haxes(iaxis), 'XLim', [0 7]);
set(haxes(iaxis), 'XTick', [0 1 2 3 4 5 6 7]);
set(haxes(iaxis), 'FontSize', 13);
end
set(gca, 'box', 'off'); % so ticks aren't shown on opposite axes.
title(['1st-order ODEs using Step-by-step RK ',...
'Medium-order Method with 2 Tolerances'], titleFmt{:});
% Label the x axis, and both y axes.
xlabel('t', labFmt{:});
ylabel(haxes(1),'Solution (y,y'')', labFmt{:});
ylabel(haxes(2), 'abs(Error)', labFmt{:});
legend('y-error (tol= 0.0001)','y-error (tol= 0.00001)','cos(x)',...
'sin(x)','y-solution','y''-solution','Location','Best');
% Set some features of the lines.
set(hline1, 'Linewidth', 0.5,'LineStyle','--');
set(hline2, 'Linewidth', 0.5, 'Marker', 's','LineStyle','-.');
set(hline3, 'Linewidth', 0.5, 'Marker', '*','LineStyle',':');
set(hline3,'Visible','off');

nag_ode_ivp_rk_onestep example program results

Calculation with tol = 1.000e-04

t           y1           y1'
0.000       0.0000       1.0000
0.785       0.7071       0.7071
1.519       0.9987       0.0513
2.282       0.7573      -0.6531
2.911       0.2285      -0.9735
3.706      -0.5348      -0.8450
4.364      -0.9399      -0.3414
5.320      -0.8209       0.5710
5.802      -0.4631       0.8863
6.283       0.0000       1.0000

Cost of the integration in evaluations of F is 78

Calculation with tol = 1.000e-05

t           y1           y1'
0.000       0.0000       1.0000
0.393       0.3827       0.9239
0.785       0.7071       0.7071
1.416       0.9881       0.1538
1.870       0.9557      -0.2943
2.204       0.8062      -0.5916
2.761       0.3711      -0.9286
3.230      -0.0880      -0.9961
3.587      -0.4304      -0.9026
4.022      -0.7710      -0.6368
4.641      -0.9974      -0.0717
5.152      -0.9049       0.4256
5.521      -0.6903       0.7235
5.902      -0.3718       0.9283
6.283       0.0000       1.0000

Cost of the integration in evaluations of F is 118

function d02pd_example
% Initialize variables and arrays.
tstart = 0;
hstart = 0;
ystart = [0; 1];
tend = 2*pi;
thres = [1e-08; 1e-08];
method = 2;
errass = false;
neq = 2;
lenwrk = 32*neq;

y1keep(1,:) = [ystart];
y2keep(1,:) = zeros(1,2);
y3keep(1,:) = zeros(1,2);
tkeep(1,:) = zeros(1,2);

fprintf('d02pd example program results\n\n');

% We run through the calculation twice with two tolerance values; we
% output and plot the results for both of them.
tol = [1e-04; 1e-05];
for itol = 1:2

% d02pv is a setup routine to be called prior to d02pd.
[work, ifail] = d02pv(tstart, ystart, tend, tol(itol), thres, ...
int64(method), task, errass, int64(lenwrk), 'neq', ...
int64(neq));
if ifail ~= 0
% Illegal input.  Print message and exit.
error('Warning: d02pv returned with ifail = %1d ',ifail);
end

fprintf('Calculation with tol = %1.3e\n\n',tol(itol));
disp('  t           y1           y1''');
fprintf('%1.3f     %8.4f     %8.4f\n', tstart, ystart(1), ystart(2));

% Initialize the storage counter, and the current t value before
% looping over t.
ncall = 0;
tnow = tstart;
while (tnow < tend)
[tnow, ynow, ypnow, work, ifail] = d02pd(@f, ...
int64(neq), work);
if ifail ~= 0
% Illegal input.  Print message and exit.
error('Warning: d02pd returned with ifail = %1d ',...
ifail);
end

% Output results, then store them for plotting.
fprintf('%1.3f     %8.4f     %8.4f\n', tnow, ynow(1), ynow(2));
ncall = ncall + 1;
tkeep(ncall,itol) = tnow;
if (itol == 2)
y1keep(ncall,:) = [ynow(1) ynow(2)];
y2keep(ncall,:) = [ynow(1)-sin(tnow) ynow(2)-cos(tnow)];
else
y3keep(ncall,:) = [ynow(1)-sin(tnow) ynow(2)-cos(tnow)];
end

end

% d02py is a diagnostic routine.
[totfcn, stpcst, waste, stpsok, hnext, ifail] = d02py;

fprintf(['\nCost of the integration in evaluations of F is ' ...
'%1.0f\n\n'], totfcn);
end
% Plot results.
fig = figure('Number', 'off');
display_plot(tkeep, y1keep, y2keep, y3keep)

function [yp] = f(t, y)
% Evaluate derivative vector.
yp = zeros(2, 1);
yp(1) =  y(2);
yp(2) = -y(1);
function display_plot(tkeep,y1keep,y2keep,y3keep)
% Formatting for title and axis labels.
titleFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 14};
labFmt = {'FontName', 'Helvetica', 'FontWeight', 'Bold', 'FontSize', 13};
set(gca, 'FontSize', 13); % for legend, axis tick labels, etc.
% Calculate spline through data values.
xx = tkeep(1,2):0.01:tkeep(length(tkeep(:,2)),2);
yy1 = spline(tkeep(:,2),y1keep(:,1),xx);
yy2 = spline(tkeep(:,2),y1keep(:,2),xx);
% Plot two of the curves and then add the other three.
[haxes, hline1, hline2] = plotyy(xx, yy2, tkeep(1:length(y3keep),1), ...
abs(y3keep(:,1)), 'plot', 'semilogy');
hold on
plot(xx,yy1,':');
hold on
plot(tkeep(:,2),y1keep(:,1),'+');
hold on
plot(tkeep(:,2),y1keep(:,2),'x');
% Select the required y axis, and plot the sixth curve.
axes(haxes(2));
hold on;
hline3 = plot(tkeep(:,2), abs(y2keep(:,1)));
% Set the axis limits and the tick specifications to beautify the plot.
set(haxes(1), 'YLim', [-1 1.2]);
set(haxes(1), 'YMinorTick', 'on');
set(haxes(1), 'YTick', [-1 -0.6 -0.2 0.2 0.6 1.0 1.2]);
set(haxes(2), 'YLim', [1e-9 1e-4]);
set(haxes(2), 'YMinorTick', 'on');
set(haxes(2), 'YTick', [1e-9 1e-8 1e-7 1e-6 1e-5 1e-4]);
for iaxis = 1:2
% These properties must be the same for both sets of axes.
set(haxes(iaxis), 'XLim', [0 7]);
set(haxes(iaxis), 'XTick', [0 1 2 3 4 5 6 7]);
set(haxes(iaxis), 'FontSize', 13);
end
set(gca, 'box', 'off'); % so ticks aren't shown on opposite axes.
title(['1st-order ODEs using Step-by-step RK ',...
'Medium-order Method with 2 Tolerances'], titleFmt{:});
% Label the x axis, and both y axes.
xlabel('t', labFmt{:});
ylabel(haxes(1),'Solution (y,y'')', labFmt{:});
ylabel(haxes(2), 'abs(Error)', labFmt{:});
legend('y-error (tol= 0.0001)','y-error (tol= 0.00001)','cos(x)',...
'sin(x)','y-solution','y''-solution','Location','Best');
% Set some features of the lines.
set(hline1, 'Linewidth', 0.5,'LineStyle','--');
set(hline2, 'Linewidth', 0.5, 'Marker', 's','LineStyle','-.');
set(hline3, 'Linewidth', 0.5, 'Marker', '*','LineStyle',':');

d02pd example program results

Calculation with tol = 1.000e-04

t           y1           y1'
0.000       0.0000       1.0000
0.785       0.7071       0.7071
1.519       0.9987       0.0513
2.282       0.7573      -0.6531
2.911       0.2285      -0.9735
3.706      -0.5348      -0.8450
4.364      -0.9399      -0.3414
5.320      -0.8209       0.5710
5.802      -0.4631       0.8863
6.283       0.0000       1.0000

Cost of the integration in evaluations of F is 78

Calculation with tol = 1.000e-05

t           y1           y1'
0.000       0.0000       1.0000
0.393       0.3827       0.9239
0.785       0.7071       0.7071
1.416       0.9881       0.1538
1.870       0.9557      -0.2943
2.204       0.8062      -0.5916
2.761       0.3711      -0.9286
3.230      -0.0880      -0.9961
3.587      -0.4304      -0.9026
4.022      -0.7710      -0.6368
4.641      -0.9974      -0.0717
5.152      -0.9049       0.4256
5.521      -0.6903       0.7235
5.902      -0.3718       0.9283
6.283       0.0000       1.0000

Cost of the integration in evaluations of F is 118