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NAG Toolbox: nag_ode_ivp_rkts_range (d02pe)
Purpose
nag_ode_ivp_rkts_range (d02pe) solves an initial value problem for a firstorder system of ordinary differential equations using Runge–Kutta methods.
Syntax
[
tgot,
ygot,
ypgot,
ymax,
user,
iwsav,
rwsav,
ifail] = d02pe(
f,
twant,
ygot,
ymax,
iwsav,
rwsav, 'n',
n, 'user',
user)
[
tgot,
ygot,
ypgot,
ymax,
user,
iwsav,
rwsav,
ifail] = nag_ode_ivp_rkts_range(
f,
twant,
ygot,
ymax,
iwsav,
rwsav, 'n',
n, 'user',
user)
Description
nag_ode_ivp_rkts_range (d02pe) and its associated functions (
nag_ode_ivp_rkts_setup (d02pq),
nag_ode_ivp_rkts_diag (d02pt) and
nag_ode_ivp_rkts_errass (d02pu)) solve an initial value problem for a firstorder system of ordinary differential equations. The functions, based on Runge–Kutta methods and derived from RKSUITE (see
Brankin et al. (1991)), integrate
where
$y$ is the vector of
$\mathit{n}$ solution components and
$t$ is the independent variable.
nag_ode_ivp_rkts_range (d02pe) is designed for the usual task, namely to compute an approximate solution at a sequence of points. You must first call
nag_ode_ivp_rkts_setup (d02pq) to specify the problem and how it is to be solved. Thereafter you call
nag_ode_ivp_rkts_range (d02pe) repeatedly with successive values of
twant, the points at which you require the solution, in the range from
tstart to
tend (as specified in
nag_ode_ivp_rkts_setup (d02pq)). In this manner
nag_ode_ivp_rkts_range (d02pe) returns the point at which it has computed a solution
tgot (usually
twant), the solution there (
ygot) and its derivative (
ypgot). If
nag_ode_ivp_rkts_range (d02pe) encounters some difficulty in taking a step toward
twant, then it returns the point of difficulty (
tgot) and the solution and derivative computed there (
ygot and
ypgot, respectively).
In the call to
nag_ode_ivp_rkts_setup (d02pq) you can specify either the first step size for
nag_ode_ivp_rkts_range (d02pe) to attempt or that it computes automatically an appropriate value. Thereafter
nag_ode_ivp_rkts_range (d02pe) 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_rkts_range (d02pe) by a call to
nag_ode_ivp_rkts_diag (d02pt). The local error is controlled at every step as specified in
nag_ode_ivp_rkts_setup (d02pq). If you wish to assess the true error, you must set
method to a positive value
in the call to
nag_ode_ivp_rkts_setup (d02pq). This assessment can be obtained after any call to
nag_ode_ivp_rkts_range (d02pe) by a call to
nag_ode_ivp_rkts_errass (d02pu).
For more complicated tasks, you are referred to functions
nag_ode_ivp_rkts_onestep (d02pf),
nag_ode_ivp_rkts_reset_tend (d02pr) and
nag_ode_ivp_rkts_interp (d02ps), all of which are used by
nag_ode_ivp_rkts_range (d02pe).
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 91S1 Southern Methodist University
Parameters
Compulsory Input Parameters
 1:
$\mathrm{f}$ – function handle or string containing name of mfile

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 mfiles 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
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.
 ${\mathbf{ifail}}=1$

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
The accuracy of integration is determined by the arguments
tol and
thresh in a prior call to
nag_ode_ivp_rkts_setup (d02pq) (see the function document for
nag_ode_ivp_rkts_setup (d02pq) for further details and advice). Note that only the local error at each step is controlled by these arguments. 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.
Further Comments
If
nag_ode_ivp_rkts_range (d02pe) returns with
${\mathbf{ifail}}={\mathbf{5}}$ and the accuracy specified by
tol and
thresh 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
ygot and
ymax should be monitored (or
nag_ode_ivp_rkts_onestep (d02pf) should be used since this takes one integration step at a time) 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_rkts_range (d02pe) by a call to
nag_ode_ivp_rkts_diag (d02pt). If
${\mathbf{method}}>0$ in the call to
nag_ode_ivp_rkts_setup (d02pq), global error assessment is available after a return from
nag_ode_ivp_rkts_range (d02pe) with
${\mathbf{ifail}}={\mathbf{0}}$,
${\mathbf{2}}$,
${\mathbf{3}}$,
${\mathbf{4}}$,
${\mathbf{5}}$ or
${\mathbf{6}}$ by a call to
nag_ode_ivp_rkts_errass (d02pu).
After a failure with
${\mathbf{ifail}}={\mathbf{5}}$ or
${\mathbf{6}}$ each of the diagnostic functions
nag_ode_ivp_rkts_diag (d02pt) and
nag_ode_ivp_rkts_errass (d02pu) may be called only once.
If nag_ode_ivp_rkts_range (d02pe) 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_rkts_range (d02pe) 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
reposed as
over the range
$\left[0,2\pi \right]$ with initial conditions
${y}_{1}=0.0$ and
${y}_{2}=1.0$. Relative error control is used with threshold values of
$\text{1.0e\u22128}$ for each solution component and compute the solution at intervals of length
$\pi /4$ across the range. A loworder Runge–Kutta method (see
nag_ode_ivp_rkts_setup (d02pq)) is also used with tolerances
${\mathbf{tol}}=\text{1.0e\u22123}$ and
${\mathbf{tol}}=\text{1.0e\u22124}$ in turn so that the solutions can be compared.
See also
Example in
nag_ode_ivp_rkts_errass (d02pu).
Open in the MATLAB editor:
d02pe_example
function d02pe_example
fprintf('d02pe example results\n\n');
method = int64(1);
tstart = 0;
tend = 2*pi;
yinit = [0;1];
hstart = 0;
thresh = [1e08; 1e08];
npts = 40;
tol0 = 1.0E3;
ygot = zeros(npts+1, 2);
tgot = zeros(npts+1, 1);
err1 = zeros(npts+1, 2);
err2 = zeros(npts+1, 2);
ymax = zeros(1, 2);
tinc = (tendtstart)/npts;
tol = 10.0*tol0;
for i = 1:2
tol = tol*0.1;
[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
fig1 = figure;
title('Firstorder ODEs using RungeKutta Loworder 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');
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 1e7 0.01]);
ylabel('abs(Error)');
plot(ax2, tgot, abs(err1(:, 1)), '*b');
text(ceil(tgot(npts+1)), err1(npts+1, 1), 'yerror (tol=0.001)', 'Color', 'b');
plot(ax2, tgot, abs(err1(:, 2)), 'sm');
text(ceil(tgot(npts+1)), err1(npts+1, 2), 'yerror (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 = 1.0e03:
Number of evaluations of f = 421
Calculation with TOL = 1.0e04:
Number of evaluations of f = 871
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