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NAG Toolbox: nag_ode_ivp_rkm_zero_simple (d02bh)
Purpose
nag_ode_ivp_rkm_zero_simple (d02bh) integrates a system of firstorder ordinary differential equations over an interval with suitable initial conditions, using a Runge–Kutta–Merson method, until a userspecified function of the solution is zero.
Syntax
[
x,
y,
tol,
ifail] = d02bh(
x,
xend,
y,
tol,
irelab,
hmax,
fcn,
g, 'n',
n)
[
x,
y,
tol,
ifail] = nag_ode_ivp_rkm_zero_simple(
x,
xend,
y,
tol,
irelab,
hmax,
fcn,
g, 'n',
n)
Description
nag_ode_ivp_rkm_zero_simple (d02bh) advances the solution of a system of ordinary differential equations
from
$x={\mathbf{x}}$ towards
$x={\mathbf{xend}}$ using a Merson form of the Runge–Kutta method. The system is defined by
fcn, which evaluates
${f}_{i}$ in terms of
$x$ and
${y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}}$ (see
Arguments), and the values of
${y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}}$ must be given at
$x={\mathbf{x}}$.
As the integration proceeds, a check is made on the function $g\left(x,y\right)$ specified by you, to determine an interval where it changes sign. The position of this sign change is then determined accurately by interpolating for the solution and its derivative. It is assumed that $g\left(x,y\right)$ is a continuous function of the variables, so that a solution of $g\left(x,y\right)=0$ can be determined by searching for a change in sign in $g\left(x,y\right)$.
The accuracy of the integration and, indirectly, of the determination of the position where
$g\left(x,y\right)=0$, is controlled by
tol.
For a description of Runge–Kutta methods and their practical implementation see
Hall and Watt (1976).
References
Hall G and Watt J M (ed.) (1976) Modern Numerical Methods for Ordinary Differential Equations Clarendon Press, Oxford
Parameters
Compulsory Input Parameters
 1:
$\mathrm{x}$ – double scalar

Must be set to the initial value of the independent variable $x$.
 2:
$\mathrm{xend}$ – double scalar

The final value of the independent variable
$x$.
If ${\mathbf{xend}}<{\mathbf{x}}$ on entry, integration proceeds in a negative direction.
 3:
$\mathrm{y}\left({\mathbf{n}}\right)$ – double array

The initial values of the solution ${y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}}$.
 4:
$\mathrm{tol}$ – double scalar

Must be set to a
positive tolerance for controlling the error in the integration and in the determination of the position where
$g\left(x,y\right)=0.0$.
nag_ode_ivp_rkm_zero_simple (d02bh) has been designed so that, for most problems, a reduction in
tol leads to an approximately proportional reduction in the error in the solution obtained in the integration. The relation between changes in
tol and the error in the determination of the position where
$g\left(x,y\right)=0.0$ is less clear, but for
tol small enough the error should be approximately proportional to
tol. However, the actual relation between
tol and the accuracy cannot be guaranteed. You are strongly recommended to call
nag_ode_ivp_rkm_zero_simple (d02bh) with more than one value for
tol and to compare the results obtained to estimate their accuracy. In the absence of any prior knowledge you might compare results obtained by calling
nag_ode_ivp_rkm_zero_simple (d02bh) with
${\mathbf{tol}}={10.0}^{p}$ and
${\mathbf{tol}}={10.0}^{p1}$ if
$p$ correct decimal digits in the solution are required.
Constraint:
${\mathbf{tol}}>0.0$.
 5:
$\mathrm{irelab}$ – int64int32nag_int scalar

Determines the type of error control. At each step in the numerical solution an estimate of the local error,
$\mathit{est}$, is made. For the current step to be accepted the following condition must be satisfied:
 ${\mathbf{irelab}}=0$
 $\mathit{est}\le {\mathbf{tol}}\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{1.0,\left{y}_{1}\right,\left{y}_{2}\right,\dots ,\left{y}_{\mathit{n}}\right\right\}$;
 ${\mathbf{irelab}}=1$
 $\mathit{est}\le {\mathbf{tol}}$;
 ${\mathbf{irelab}}=2$
 $\mathit{est}\le {\mathbf{tol}}\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{\epsilon ,\left{y}_{1}\right,\left{y}_{2}\right,\dots ,\left{y}_{\mathit{n}}\right\right\}$, where $\epsilon $ is machine precision.
If the appropriate condition is not satisfied, the step size is reduced and the solution recomputed on the current step.
If you wish to measure the error in the computed solution in terms of the number of correct decimal places, then set ${\mathbf{irelab}}=1$ on entry, whereas if the error requirement is in terms of the number of correct significant digits, then set ${\mathbf{irelab}}=2$. Where there is no preference in the choice of error test, ${\mathbf{irelab}}=0$ will result in a mixed error test. It should be borne in mind that the computed solution will be used in evaluating $g\left(x,y\right)$.
Constraint:
${\mathbf{irelab}}=0$, $1$ or $2$.
 6:
$\mathrm{hmax}$ – double scalar

If
${\mathbf{hmax}}=0.0$, no special action is taken.
If
${\mathbf{hmax}}\ne 0.0$, a check is made for a change in sign of
$g\left(x,y\right)$ at steps not greater than
$\left{\mathbf{hmax}}\right$. This facility should be used if there is any chance of ‘missing’ the change in sign by checking too infrequently. For example, if two changes of sign of
$g\left(x,y\right)$ are expected within a distance
$h$, say, of each other, then a suitable value for
hmax might be
${\mathbf{hmax}}=h/2$. If only one change of sign in
$g\left(x,y\right)$ is expected on the range
x to
xend, then the choice
${\mathbf{hmax}}=0.0$ is most appropriate.
 7:
$\mathrm{fcn}$ – function handle or string containing name of mfile

fcn must evaluate the functions
${f}_{i}$ (i.e., the derivatives
${y}_{i}^{\prime}$) for given values of its arguments
$x,{y}_{1},\dots ,{y}_{\mathit{n}}$.
[f] = fcn(x, y)
Input Parameters
 1:
$\mathrm{x}$ – double scalar

$x$, the value of the argument.
 2:
$\mathrm{y}\left(:\right)$ – double array

${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the argument.
Output Parameters
 1:
$\mathrm{f}\left(:\right)$ – double array

The value of
${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
 8:
$\mathrm{g}$ – function handle or string containing name of mfile

g must evaluate the function
$g\left(x,y\right)$ at a specified point.
[result] = g(x, y)
Input Parameters
 1:
$\mathrm{x}$ – double scalar

$x$, the value of the independent variable.
 2:
$\mathrm{y}\left(:\right)$ – double array

The value of
${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
Output Parameters
 1:
$\mathrm{result}$ – double scalar

The value of $g\left(x,y\right)$ at the specified point.
Optional Input Parameters
 1:
$\mathrm{n}$ – int64int32nag_int scalar

Default:
the dimension of the array
y.
$\mathit{n}$, the number of differential equations.
Constraint:
${\mathbf{n}}>0$.
Output Parameters
 1:
$\mathrm{x}$ – double scalar

The point where
$g\left(x,y\right)=0.0$ unless an error has occurred, when it contains the value of
$x$ at the error. In particular, if
$g\left(x,y\right)\ne 0.0$ anywhere on the range
x to
xend, it will contain
xend on exit.
 2:
$\mathrm{y}\left({\mathbf{n}}\right)$ – double array

The computed values of the solution at the final point $x={\mathbf{x}}$.
 3:
$\mathrm{tol}$ – double scalar

Normally unchanged. However if the range from
$x={\mathbf{x}}$ to the position where
$g\left(x,y\right)=0.0$ (or to the final value of
$x$ if an error occurs) is so short that a small change in
tol is unlikely to make any change in the computed solution, then
tol is returned with its sign changed. To check results returned with
${\mathbf{tol}}<0.0$,
nag_ode_ivp_rkm_zero_simple (d02bh) should be called again with a positive value of
tol whose magnitude is considerably smaller than that of the previous call.
 4:
$\mathrm{ifail}$ – 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:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{tol}}\le 0.0$, 
or  ${\mathbf{n}}\le 0$, 
or  ${\mathbf{irelab}}\ne 0$, $1$ or $2$. 
 ${\mathbf{ifail}}=2$

With the given value of
tol, no further progress can be made across the integration range from the current point
$x={\mathbf{x}}$, or dependence of the error on
tol would be lost if further progress across the integration range were attempted (see
Further Comments for a discussion of this error exit). The components
${\mathbf{y}}\left(1\right),{\mathbf{y}}\left(2\right),\dots ,{\mathbf{y}}\left(\mathit{n}\right)$ contain the computed values of the solution at the current point
$x={\mathbf{x}}$. No point at which
$g\left(x,y\right)$ changes sign has been located up to the point
$x={\mathbf{x}}$.
 ${\mathbf{ifail}}=3$

tol is too small for
nag_ode_ivp_rkm_zero_simple (d02bh) to take an initial step (see
Further Comments).
x and
${\mathbf{y}}\left(1\right),{\mathbf{y}}\left(2\right),\dots ,{\mathbf{y}}\left(\mathit{n}\right)$ retain their initial values.
 ${\mathbf{ifail}}=4$

At no point in the range
x to
xend did the function
$g\left(x,y\right)$ change sign. It is assumed that
$g\left(x,y\right)=0.0$ has no solution.
 ${\mathbf{ifail}}=5$ (nag_roots_contfn_brent_rcomm (c05az))

A serious error has occurred in an internal call to the specified function. Check all function calls and array dimensions. Seek expert help.
 ${\mathbf{ifail}}=6$

A serious error has occurred in an internal call to an integration function. Check all function calls and array dimensions. Seek expert help.
 ${\mathbf{ifail}}=7$

A serious error has occurred in an internal call to an interpolation function. Check all (sub)program calls and array dimensions. Seek expert help.
 ${\mathbf{ifail}}=99$
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 depends on
tol, on the mathematical properties of the differential system, on the position where
$g\left(x,y\right)=0.0$ and on the method. It can be controlled by varying
tol but the approximate proportionality of the error to
tol holds only for a restricted range of values of
tol. For
tol too large, the underlying theory may break down and the result of varying
tol may be unpredictable. For
tol too small, rounding error may affect the solution significantly and an error exit with
${\mathbf{ifail}}={\mathbf{2}}$ or
${\mathbf{3}}$ is possible.
The accuracy may also be restricted by the properties of
$g\left(x,y\right)$. You should try to code
g without introducing any unnecessary cancellation errors.
Further Comments
The time taken by
nag_ode_ivp_rkm_zero_simple (d02bh) depends on the complexity and mathematical properties of the system of differential equations defined by
fcn, the complexity of
g, on the range, the position of the solution and the tolerance. There is also an overhead of the form
$a+b\times \mathit{n}$ where
$a$ and
$b$ are machinedependent computing times.
For some problems it is possible that
nag_ode_ivp_rkm_zero_simple (d02bh) will return
${\mathbf{ifail}}={\mathbf{4}}$ because of inaccuracy of the computed values
y, leading to inaccuracy in the computed values of
$g\left(x,y\right)$ used in the search for the solution of
$g\left(x,y\right)=0.0$. This difficulty can be overcome by reducing
tol sufficiently, and if necessary, by choosing
hmax sufficiently small. If possible, you should choose
xend well beyond the expected point where
$g\left(x,y\right)=0.0$; for example make
$\left{\mathbf{xend}}{\mathbf{x}}\right$ about
$50\%$ larger than the expected range. As a simple check, if, with
xend fixed, a change in
tol does not lead to a significant change in
y at
xend, then inaccuracy is not a likely source of error.
If
nag_ode_ivp_rkm_zero_simple (d02bh) fails with
${\mathbf{ifail}}={\mathbf{3}}$, then it could be called again with a larger value of
tol if this has not already been tried. If the accuracy requested is really needed and cannot be obtained with this function, the system may be very stiff (see below) or so badly scaled that it cannot be solved to the required accuracy.
If
nag_ode_ivp_rkm_zero_simple (d02bh) fails with
${\mathbf{ifail}}={\mathbf{2}}$, it is likely that it has been called with a value of
tol which is so small that a solution cannot be obtained on the range
x to
xend. This can happen for wellbehaved systems and very small values of
tol. You should, however, consider whether there is a more fundamental difficulty. For example:
(a) 
in the region of a singularity (infinite value) of the solution, the function will usually stop with ${\mathbf{ifail}}={\mathbf{2}}$, unless overflow occurs first. If overflow occurs using nag_ode_ivp_rkm_zero_simple (d02bh), nag_ode_ivp_rkts_onestep (d02pf) can be used instead to detect the increasing solution, before overflow occurs. In any case, numerical integration cannot be continued through a singularity, and analytical treatment should be considered; 
(b) 
for ‘stiff’ equations, where the solution contains rapidly decaying components, the function will compute in very small steps in $x$ (internally to nag_ode_ivp_rkm_zero_simple (d02bh)) to preserve stability. This will usually exhibit itself by making the computing time excessively long, or occasionally by an exit with ${\mathbf{ifail}}={\mathbf{2}}$. Merson's method is not efficient in such cases, and you should try nag_ode_ivp_bdf_zero_simple (d02ej) which uses a Backward Differentiation Formula method. To determine whether a problem is stiff, nag_ode_ivp_rkts_range (d02pe) may be used. 
For wellbehaved systems with no difficulties such as stiffness or singularities, the Merson method should work well for low accuracy calculations (three or four figures). For high accuracy calculations or where
fcn is costly to evaluate, Merson's method may not be appropriate and a computationally less expensive method may be
nag_ode_ivp_adams_zero_simple (d02cj) which uses an Adams' method.
For problems for which
nag_ode_ivp_rkm_zero_simple (d02bh) is not sufficiently general, you should consider
nag_ode_ivp_rkts_onestep (d02pf).
nag_ode_ivp_rkts_onestep (d02pf) is a more general function with many facilities including a more general error control criterion.
nag_ode_ivp_rkts_onestep (d02pf) can be combined with the rootfinder
nag_roots_contfn_brent_rcomm (c05az) and the interpolation function
nag_ode_ivp_rkts_interp (d02ps) to solve equations involving
${y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}}$ and their derivatives.
nag_ode_ivp_rkm_zero_simple (d02bh) can also be used to solve an equation involving
$x$,
${y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}}$ and the derivatives of
${y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}}$. For example in
Example,
nag_ode_ivp_rkm_zero_simple (d02bh) is used to find a value of
${\mathbf{x}}>0.0$ where
${\mathbf{y}}\left(1\right)=0.0$. It could instead be used to find a turningpoint of
${y}_{1}$ by replacing the function
$g\left(x,y\right)$ in the program by:
function result = g(x,y)
f = d02bh_f(x,y);
result = f(1);
This function is only intended to locate the
first zero of
$g\left(x,y\right)$. If later zeros are required, you are strongly advised to construct your own more general rootfinding functions as discussed above.
Example
This example finds the value
${\mathbf{x}}>0.0$ at which
$y=0.0$, where
$y$,
$v$,
$\varphi $ are defined by
and where at
${\mathbf{x}}=0.0$ we are given
$y=0.5$,
$v=0.5$ and
$\varphi =\pi /5$. We write
$y={\mathbf{y}}\left(1\right)$,
$v={\mathbf{y}}\left(2\right)$ and
$\varphi ={\mathbf{y}}\left(3\right)$ and we set
${\mathbf{tol}}=\text{1.0e\u22124}$ and
${\mathbf{tol}}=\text{1.0e\u22125}$ in turn so that we can compare the solutions. We expect the solution
${\mathbf{x}}\simeq 7.3$ and so we set
${\mathbf{xend}}=10.0$ to avoid determining the solution of
$y=0.0$ too near the end of the range of integration. The initial values and range are read from a data file.
Open in the MATLAB editor:
d02bh_example
function d02bh_example
fprintf('d02bh example results\n\n');
x_init = 0;
y_init = [0.5 0.5 pi/5];
xend = 10;
tol = 0.00001;
irelab = int64(0);
hmax = 0;
m = int64(1);
val = 0;
[xroot, yroot, tol, ifail] = ...
d02bh(...
x_init, xend, y_init, tol, irelab, hmax, @fcn, @g);
fprintf('Root of Y(1) at %8.4f\n',xroot);
fprintf('Solution is %9.5f %9.5f %9.5f\n',yroot);
function f = fcn(x,y)
f = zeros(3,1);
f(1) = tan(y(3));
f(2) = 0.032*tan(y(3))/y(2)  0.02*y(2)/cos(y(3));
f(3) = 0.032/y(2)^2;
function result = g(x,y)
result = y(1);
d02bh example results
Root of Y(1) at 7.2883
Solution is 0.00000 0.47486 0.76011
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