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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_ode_ivp_rkts_range (d02pe)

## Purpose

nag_ode_ivp_rkts_range (d02pe) solves an initial value problem for a first-order 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 first-order system of ordinary differential equations. The functions, based on Runge–Kutta methods and derived from RKSUITE (see Brankin et al. (1991)), integrate
 $y′=ft,y given yt0=y0$
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 91-S1 Southern Methodist University

## 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:     $\mathrm{rwsav}\left(32×{\mathbf{n}}+350\right)$ – 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:     $\mathrm{rwsav}\left(32×{\mathbf{n}}+350\right)$ – 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.
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.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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.

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
 $y′′=-y, y0=0, y′0=1$
reposed as
 $y1′=y2$
 $y2′=-y1$
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 $\text{1.0e−8}$ for each solution component and compute the solution at intervals of length $\pi /4$ across the range. A low-order Runge–Kutta method (see nag_ode_ivp_rkts_setup (d02pq)) is also used with tolerances ${\mathbf{tol}}=\text{1.0e−3}$ and ${\mathbf{tol}}=\text{1.0e−4}$ in turn so that the solutions can be compared.
```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
``` 