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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_roots_withdraw_contfn_cntin_start (c05aj)

Purpose

nag_roots_withdraw_contfn_cntin_start (c05aj) attempts to locate a zero of a continuous function using a continuation method based on a secant iteration.
Note: this function is scheduled to be withdrawn, please see c05aj in Advice on Replacement Calls for Withdrawn/Superseded Routines..

Syntax

[x, ifail] = c05aj(x, eps, eta, f, nfmax)
[x, ifail] = nag_roots_withdraw_contfn_cntin_start(x, eps, eta, f, nfmax)

Description

nag_roots_withdraw_contfn_cntin_start (c05aj) attempts to obtain an approximation to a simple zero α$\alpha$ of the function f(x) $f\left(x\right)$ given an initial approximation x$x$ to α$\alpha$. The zero is found by a call to nag_roots_contfn_cntin_rcomm (c05ax) whose specification should be consulted for details of the method used.
The approximation x$x$ to the zero α$\alpha$ is determined so that at least one of the following criteria is satisfied:
 (i) |x − α| ∼ eps $|x-\alpha |\sim {\mathbf{eps}}$, (ii) |f(x)| < eta $|f\left(x\right)|<{\mathbf{eta}}$.

None.

Parameters

Compulsory Input Parameters

1:     x – double scalar
An initial approximation to the zero.
2:     eps – double scalar
An absolute tolerance to control the accuracy to which the zero is determined. In general, the smaller the value of eps the more accurate x will be as an approximation to α$\alpha$. Indeed, for very small positive values of eps, it is likely that the final approximation will satisfy |xα| < eps $|{\mathbf{x}}-\alpha |<{\mathbf{eps}}$. You are advised to call the function with more than one value for eps to check the accuracy obtained.
Constraint: eps > 0.0 ${\mathbf{eps}}>0.0$.
3:     eta – double scalar
A value such that if |f(x)| < eta $|f\left(x\right)|<{\mathbf{eta}}$, x$x$ is accepted as the zero. eta may be specified as 0.0$0.0$ (see Section [Accuracy]).
4:     f – function handle or string containing name of m-file
f must evaluate the function f$f$ whose zero is to be determined.
[result] = f(xx)

Input Parameters

1:     xx – double scalar
The point at which the function must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
5:     nfmax – int64int32nag_int scalar
The maximum permitted number of calls to f from nag_roots_withdraw_contfn_cntin_start (c05aj). If f is inexpensive to evaluate, nfmax should be given a large value (say > 1000 $\text{}>1000$).
Constraint: nfmax > 0 ${\mathbf{nfmax}}>0$.

None.

None.

Output Parameters

1:     x – double scalar
The final approximation to the zero, unless ${\mathbf{ifail}}={\mathbf{1}}$, 2${\mathbf{2}}$ or 5${\mathbf{5}}$, in which case it contains no useful information.
2:     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:
ifail = 1${\mathbf{ifail}}=1$
 On entry, eps ≤ 0.0 ${\mathbf{eps}}\le 0.0$, or nfmax ≤ 0 ${\mathbf{nfmax}}\le 0$.
ifail = 2${\mathbf{ifail}}=2$
An internally calculated scale factor has the wrong order of magnitude for the problem. If this error exit occurs, you are advised to call nag_roots_contfn_cntin_rcomm (c05ax) instead where different scale values can be tried.
ifail = 3${\mathbf{ifail}}=3$
Either the function f(x) $f\left(x\right)$ given by f has no zero near x or too much accuracy has been requested in calculating the zero. The first is a more likely cause of this error exit and you should check the coding of f and make an independent investigation of its behaviour near x. The second can be alleviated by increasing eps.
ifail = 4${\mathbf{ifail}}=4$
More than nfmax calls have been made to f. This error exit can occur because nfmax is too small for the problem (essentially because x is too far away from the zero) or for either of the reasons given under ${\mathbf{ifail}}={\mathbf{3}}$ above. If nfmax is increased considerably and this error exit occurs again at approximately the same final value of x, then it is likely that one of the reasons given under ${\mathbf{ifail}}={\mathbf{3}}$ is the cause.
ifail = 5 ${\mathbf{ifail}}=5$ (nag_roots_contfn_cntin_rcomm (c05ax))
A serious error has occurred in the specified function. Check all function calls. Seek expert help.

Accuracy

The levels of accuracy depend on the values of eps and eta. If full machine accuracy is required, they may be set very small, resulting in an exit with ${\mathbf{ifail}}={\mathbf{3}}$ or 4${\mathbf{4}}$, although this may involve many more iterations than a lesser accuracy. You are recommended to set eta = 0.0 ${\mathbf{eta}}=0.0$ and to use eps to control the accuracy, unless you have considerable knowledge of the size of f(x) $f\left(x\right)$ for values of x$x$ near the zero.

The time taken by nag_roots_withdraw_contfn_cntin_start (c05aj) depends primarily on the time spent evaluating the function f$f$ (see Section [Parameters]) and on how close the initial value of x is to the zero.
If a more flexible way of specifying the function f$f$ is required or if you wish to have closer control of the calculation, then the reverse communication function nag_roots_contfn_cntin_rcomm (c05ax) is recommended instead of nag_roots_withdraw_contfn_cntin_start (c05aj).

Example

```function nag_roots_withdraw_contfn_cntin_start_example
x = 1;
eta = 0;
nfmax = int64(200);
f = @(x) exp(-x)-x;
for k=3:4
[xOut, ifail] = nag_roots_withdraw_contfn_cntin_start(x, 10^-k, eta, f, nfmax);

if ifail == 3 || ifail == 4
fprintf('With epsilon = %10.2e, final value = %14.5f\n', 10^-k, xOut);
else
fprintf('With epsilon = %10.2e, root = %14.5f\n', 10^-k, xOut);
end
end
```
```
With epsilon =   1.00e-03, root =        0.56715
With epsilon =   1.00e-04, root =        0.56715

```
```function c05aj_example
x = 1;
eta = 0;
nfmax = int64(200);
f = @(x) exp(-x)-x;
for k=3:4
[xOut, ifail] = c05aj(x, 10^-k, eta, f, nfmax);

if ifail == 3 || ifail == 4
fprintf('With epsilon = %10.2e, final value = %14.5f\n', 10^-k, xOut);
else
fprintf('With epsilon = %10.2e, root = %14.5f\n', 10^-k, xOut);
end
end
```
```
With epsilon =   1.00e-03, root =        0.56715
With epsilon =   1.00e-04, root =        0.56715

```