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# NAG Toolbox: nag_roots_contfn_brent (c05ay)

## Purpose

nag_roots_contfn_brent (c05ay) locates a simple zero of a continuous function in a given interval using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection.

## Syntax

[x, user, ifail] = c05ay(a, b, eps, eta, f, 'user', user)
[x, user, ifail] = nag_roots_contfn_brent(a, b, eps, eta, f, 'user', user)

## Description

nag_roots_contfn_brent (c05ay) attempts to obtain an approximation to a simple zero of the function f(x) $f\left(x\right)$ given an initial interval [a,b] $\left[a,b\right]$ such that f(a) × f(b) 0 $f\left(a\right)×f\left(b\right)\le 0$. The same core algorithm is used by nag_roots_contfn_brent_rcomm (c05az) 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 $|x-\alpha |\le {\mathbf{eps}}$, (ii) |f(x)| ≤ eta $|f\left(x\right)|\le {\mathbf{eta}}$.

## References

Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall

## Parameters

### Compulsory Input Parameters

1:     a – double scalar
a$a$, the lower bound of the interval.
2:     b – double scalar
b$b$, the upper bound of the interval.
Constraint: ba ${\mathbf{b}}\ne {\mathbf{a}}$.
3:     eps – double scalar
The termination tolerance on x$x$ (see Section [Description]).
Constraint: eps > 0.0 ${\mathbf{eps}}>0.0$.
4:     eta – double scalar
A value such that if |f(x)|eta $|f\left(x\right)|\le {\mathbf{eta}}$, x$x$ is accepted as the zero. eta may be specified as 0.0$0.0$ (see Section [Accuracy]).
5:     f – function handle or string containing name of m-file
f must evaluate the function f$f$ whose zero is to be determined.
[result, user] = f(x, user)

Input Parameters

1:     x – double scalar
The point at which the function must be evaluated.
2:     user – Any MATLAB object
f is called from nag_roots_contfn_brent (c05ay) with the object supplied to nag_roots_contfn_brent (c05ay).

Output Parameters

1:     result – double scalar
The result of the function.
2:     user – Any MATLAB object

### Optional Input Parameters

1:     user – Any MATLAB object
user is not used by nag_roots_contfn_brent (c05ay), 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.

iuser ruser

### Output Parameters

1:     x – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or 2${\mathbf{2}}$, x is the final approximation to the zero. If ${\mathbf{ifail}}={\mathbf{3}}$, x is likely to be a pole of f(x)$f\left(x\right)$. Otherwise, x contains no useful information.
2:     user – Any MATLAB object
3:     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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail = 1${\mathbf{ifail}}=1$
Constraint: ab${\mathbf{a}}\ne {\mathbf{b}}$.
Constraint: eps > 0.0${\mathbf{eps}}>0.0$.
On entry, f(a)${\mathbf{f}}\left({\mathbf{a}}\right)$ and f(b)${\mathbf{f}}\left({\mathbf{b}}\right)$ have the same sign with neither equalling 0.0$0.0$.
W ifail = 2${\mathbf{ifail}}=2$
No further improvement in the solution is possible.
W ifail = 3${\mathbf{ifail}}=3$
The function values in the interval [a,b] $\left[{\mathbf{a}},{\mathbf{b}}\right]$ might contain a pole rather than a zero. Reducing eps may help in distinguishing between a pole and a zero.

## 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 ${\mathbf{ifail}}={\mathbf{2}}$, 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.

## Further Comments

The time taken by nag_roots_contfn_brent (c05ay) depends primarily on the time spent evaluating f (see Section [Parameters]).
If it is important to determine an interval of relative length less than 2 × eps$2×{\mathbf{eps}}$ containing the zero, or if f is expensive to evaluate and the number of calls to f is to be restricted, then use of nag_roots_contfn_brent_rcomm (c05az) is recommended. Use of nag_roots_contfn_brent_rcomm (c05az) is also recommended when the structure of the problem to be solved does not permit a simple f to be written: the reverse communication facilities of nag_roots_contfn_brent_rcomm (c05az) are more flexible than the direct communication of f required by nag_roots_contfn_brent (c05ay).

## Example

```function nag_roots_contfn_brent_example
a = 0;
b = 1;
eps = 1e-5;
eta = 0;
fprintf('\n');
[x, user, ifail] = nag_roots_contfn_brent(a, b, eps, eta, @f);
switch ifail
case {0}
fprintf('With eps = %10.2e, root = %14.5f\n', eps, x);
case {2, 3}
fprintf('With eps = %10.2e, final value = %14.5f\n', eps, x);
end

function [result, user] = f(x, user)
result = x - exp(-x);
```
```

With eps =   1.00e-05, root =        0.56714

```
```function c05ay_example
a = 0;
b = 1;
eps = 1e-5;
eta = 0;
fprintf('\n');
[x, user, ifail] = c05ay(a, b, eps, eta, @f);
switch ifail
case {0}
fprintf('With eps = %10.2e, root = %14.5f\n', eps, x);
case {2, 3}
fprintf('With eps = %10.2e, final value = %14.5f\n', eps, x);
end

function [result, user] = f(x, user)
result = x - exp(-x);
```
```

With eps =   1.00e-05, root =        0.56714

```

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