PDF version (NAG web site
, 64bit version, 64bit version)
NAG Toolbox: nag_ode_withdraw_ivp_rk_range (d02pc)
Purpose
nag_ode_ivp_rk_range (d02pc) solves an initial value problem for a firstorder system of ordinary differential equations using Runge–Kutta methods.
Note: this function is scheduled to be withdrawn, please see
d02pc in
Advice on Replacement Calls for Withdrawn/Superseded Routines..
Syntax
[
tgot,
ygot,
ypgot,
ymax,
work,
ifail] = d02pc(
f,
neq,
twant,
ygot,
ymax,
work)
[
tgot,
ygot,
ypgot,
ymax,
work,
ifail] = nag_ode_withdraw_ivp_rk_range(
f,
neq,
twant,
ygot,
ymax,
work)
Description
nag_ode_ivp_rk_range (d02pc) and its associated functions (
nag_ode_ivp_rk_setup (d02pv),
nag_ode_ivp_rk_diag (d02py) and
nag_ode_ivp_rk_errass (d02pz)) 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_rk_range (d02pc) is designed for the usual task, namely to compute an approximate solution at a sequence of points. You must first call
nag_ode_ivp_rk_setup (d02pv) to specify the problem and how it is to be solved. Thereafter you call
nag_ode_ivp_rk_range (d02pc) 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_rk_setup (d02pv)). In this manner
nag_ode_ivp_rk_range (d02pc) 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_rk_range (d02pc) 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_rk_setup (d02pv) you can specify either the first step size for
nag_ode_ivp_rk_range (d02pc) to attempt or that it compute automatically an appropriate value. Thereafter
nag_ode_ivp_rk_range (d02pc) 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_range (d02pc) 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
${\mathbf{errass}}=\mathit{true}$ in the call to
nag_ode_ivp_rk_setup (d02pv). This assessment can be obtained after any call to
nag_ode_ivp_rk_range (d02pc) by a call to
nag_ode_ivp_rk_errass (d02pz).
For more complicated tasks, you are referred to functions
nag_ode_ivp_rk_onestep (d02pd),
nag_ode_ivp_rk_reset_tend (d02pw) and
nag_ode_ivp_rk_interp (d02px), all of which are used by
nag_ode_ivp_rk_range (d02pc).
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] = 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
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, 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
The accuracy of integration is determined by the arguments
tol and
thres in a prior call to
nag_ode_ivp_rk_setup (d02pv) (see the function document for
nag_ode_ivp_rk_setup (d02pv) 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_rk_range (d02pc) 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
ygot and
ymax should be monitored (or
nag_ode_ivp_rk_onestep (d02pd) 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_rk_range (d02pc) by a call to
nag_ode_ivp_rk_diag (d02py). If
${\mathbf{errass}}=\mathit{true}$ in the call to
nag_ode_ivp_rk_setup (d02pv), global error assessment is available after any return from
nag_ode_ivp_rk_range (d02pc) (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
${\mathbf{6}}$ 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_range (d02pc) 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_range (d02pc) 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 (
${\mathbf{method}}=1$, see
nag_ode_ivp_rk_setup (d02pv)) 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. The value of
$\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).
See also
Example in
nag_ode_ivp_rk_errass (d02pz).
Open in the MATLAB editor:
d02pc_example
function d02pc_example
fprintf('d02pc example results\n\n');
method = int64(1);
tstart = 0;
tend = 2*pi;
n = int64(2);
errass = false;
lenwrk = int64(32*n);
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;
[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);
[fevals, stepcost, waste, stepsok, hnext, ifail] = d02py;
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] = f(t, y)
yp = zeros(2, 1);
yp(1) = y(2);
yp(2) = y(1);
d02pc example results
Calculation with TOL = 1.0e03:
Number of evaluations of f = 115
Calculation with TOL = 1.0e04:
Number of evaluations of f = 223
PDF version (NAG web site
, 64bit version, 64bit version)
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015