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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_roots_contfn_brent_interval (c05au)

Purpose

nag_roots_contfn_brent_interval (c05au) locates a simple zero of a continuous function from a given starting value. It uses a binary search to locate an interval containing a zero of the function, then Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection, to locate the zero precisely.

Syntax

[x, a, b, user, ifail] = c05au(x, h, eps, eta, f, 'user', user)
[x, a, b, user, ifail] = nag_roots_contfn_brent_interval(x, h, eps, eta, f, 'user', user)

Description

nag_roots_contfn_brent_interval (c05au) attempts to locate an interval [a,b] $\left[a,b\right]$ containing a simple zero of the function f(x) $f\left(x\right)$ by a binary search starting from the initial point x = x $x={\mathbf{x}}$ and using repeated calls to nag_roots_contfn_interval_rcomm (c05av). If this search succeeds, then the zero is determined to a user-specified accuracy by a call to nag_roots_contfn_brent (c05ay). The specifications of functions nag_roots_contfn_interval_rcomm (c05av) and nag_roots_contfn_brent (c05ay) should be consulted for details of the methods 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:     x – double scalar
An initial approximation to the zero.
2:     h – double scalar
A step length for use in the binary search for an interval containing the zero. The maximum interval searched is [x256.0 × h,x + 256.0 × h] $\left[{\mathbf{x}}-256.0×{\mathbf{h}},{\mathbf{x}}+256.0×{\mathbf{h}}\right]$.
Constraint: ${\mathbf{h}}$ must be sufficiently large that x + hx ${\mathbf{x}}+{\mathbf{h}}\ne {\mathbf{x}}$ on the computer.
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_interval (c05au) with the object supplied to nag_roots_contfn_brent_interval (c05au).

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_interval (c05au), 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 4${\mathbf{4}}$, 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:     a – double scalar
3:     b – double scalar
The lower and upper bounds respectively of the interval resulting from the binary search. If the zero is determined exactly such that f(x) = 0.0 $f\left(x\right)=0.0$ or is determined so that |f(x)|eta $|f\left(x\right)|\le {\mathbf{eta}}$ at any stage in the calculation, then on exit a = b = x ${\mathbf{a}}={\mathbf{b}}=x$.
4:     user – Any MATLAB object
5:     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: eps > 0.0${\mathbf{eps}}>0.0$.
Constraint: x + hx${\mathbf{x}}+{\mathbf{h}}\ne {\mathbf{x}}$ (to machine accuracy).
ifail = 2${\mathbf{ifail}}=2$
An interval containing the zero could not be found. Increasing h and calling nag_roots_contfn_brent_interval (c05au) again will increase the range searched for the zero. Decreasing h and calling nag_roots_contfn_brent_interval (c05au) again will refine the mesh used in the search for the zero.
W ifail = 3${\mathbf{ifail}}=3$
Solution may be a pole rather than a zero.
W ifail = 4${\mathbf{ifail}}=4$
The tolerance eps has been set too small for the problem being solved.

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 ${\mathbf{ifail}}={\mathbf{4}}$, 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_contfn_brent_interval (c05au) depends primarily on the time spent evaluating f (see Section [Parameters]). The accuracy of the initial approximation x and the value of h will have a somewhat unpredictable effect on the timing.
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_interval_rcomm (c05av) followed by nag_roots_contfn_brent_rcomm (c05az) is recommended. Use of this combination 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 these functions are more flexible than the direct communication of f required by nag_roots_contfn_brent_interval (c05au).
If the iteration terminates with successful exit and a = b = x ${\mathbf{a}}={\mathbf{b}}={\mathbf{x}}$ there is no guarantee that the value returned in x corresponds to a simple zero and you should check whether it does.
One way to check this is to compute the derivative of f$f$ at the point x, preferably analytically, or, if this is not possible, numerically, perhaps by using a central difference estimate. If f(x) = 0.0 ${f}^{\prime }\left({\mathbf{x}}\right)=0.0$, then x must correspond to a multiple zero of f$f$ rather than a simple zero.

Example

```function nag_roots_contfn_brent_interval_example
x = 1;
h = 0.1;
eps = 1e-5;
eta = 0;
[xOut, a, b, user, ifail] = nag_roots_contfn_brent_interval(x, h, eps, eta, @f)

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

xOut =

0.5671

a =

0.5000

b =

0.9000

user =

0

ifail =

0

```
```function c05au_example
x = 1;
h = 0.1;
eps = 1e-5;
eta = 0;
[xOut, a, b, user, ifail] = c05au(x, h, eps, eta, @f)

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

xOut =

0.5671

a =

0.5000

b =

0.9000

user =

0

ifail =

0

```