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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_roots_withdraw_contfn_brent_start (c05ag)

## Purpose

nag_roots_withdraw_contfn_brent_start (c05ag) locates a simple zero of a continuous function from a given starting value, using a binary search to locate an interval containing a zero of the function, then a combination of the methods of nonlinear interpolation, linear extrapolation and bisection to locate the zero precisely.
Note: this function is scheduled to be withdrawn, please see c05ag in Advice on Replacement Calls for Withdrawn/Superseded Routines..

## Syntax

[x, a, b, ifail] = c05ag(x, h, eps, eta, f)
[x, a, b, ifail] = nag_roots_withdraw_contfn_brent_start(x, h, eps, eta, f)

## Description

nag_roots_withdraw_contfn_brent_start (c05ag) 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] = 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.

None.

None.

### 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:     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${\mathbf{ifail}}=1$
On entry, either eps0.0 ${\mathbf{eps}}\le 0.0$, or x + h = x ${\mathbf{x}}+{\mathbf{h}}={\mathbf{x}}$ to machine accuracy (meaning that the search for an interval containing the zero cannot commence).
ifail = 2${\mathbf{ifail}}=2$
An interval containing the zero could not be found. Increasing h and calling nag_roots_withdraw_contfn_brent_start (c05ag) again will increase the range searched for the zero. Decreasing h and calling nag_roots_withdraw_contfn_brent_start (c05ag) again will refine the mesh used in the search for the zero.
W ifail = 3${\mathbf{ifail}}=3$
A change in sign of f(x) $f\left(x\right)$ has been determined as occurring near the point defined by the final value of x. However, there is some evidence that this sign-change corresponds to a pole of f(x) $f\left(x\right)$.
W ifail = 4${\mathbf{ifail}}=4$
Too much accuracy has been requested in the computation; that is, the zero has been located to relative accuracy at least ε$\epsilon$, where ε$\epsilon$ is the machine precision, but the exit conditions described in Section [Description] are not satisfied. It is unsafe for nag_roots_withdraw_contfn_brent_start (c05ag) to continue beyond this point, but the final value of x returned is an accurate approximation to the 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 ${\mathbf{ifail}}={\mathbf{4}}$, 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_brent_start (c05ag) 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_withdraw_contfn_brent_start (c05ag).
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_withdraw_contfn_brent_start_example
x = 1;
h = 0.1;
epsilon = 1e-05;
eta = 0;
f = @(x) exp(-x)-x;
[xOut, a, b, ifail] = nag_roots_withdraw_contfn_brent_start(x, h, epsilon, eta, f)
```
```

xOut =

0.5671

a =

0.5000

b =

0.9000

ifail =

0

```
```function c05ag_example
x = 1;
h = 0.1;
epsilon = 1e-05;
eta = 0;
f = @(x) exp(-x)-x;
[xOut, a, b, ifail] = c05ag(x, h, epsilon, eta, f)
```
```

xOut =

0.5671

a =

0.5000

b =

0.9000

ifail =

0

```