# NAG Library Routine Document

## 1Purpose

c05ayf 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.

## 2Specification

Fortran Interface
 Subroutine c05ayf ( a, b, eps, eta, f, x,
 Integer, Intent (Inout) :: iuser(*), ifail Real (Kind=nag_wp), External :: f Real (Kind=nag_wp), Intent (In) :: a, b, eps, eta Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: x
#include nagmk26.h
 void c05ayf_ (const double *a, const double *b, const double *eps, const double *eta, double (NAG_CALL *f)(const double *x, Integer iuser[], double ruser[]),double *x, Integer iuser[], double ruser[], Integer *ifail)

## 3Description

c05ayf attempts to obtain an approximation to a simple zero of the function $f\left(x\right)$ given an initial interval $\left[a,b\right]$ such that $f\left(a\right)×f\left(b\right)\le 0$. The same core algorithm is used by c05azf whose specification should be consulted for details of the method used.
The approximation $x$ to the zero $\alpha$ is determined so that at least one of the following criteria is satisfied:
 (i) $\left|x-\alpha \right|\le {\mathbf{eps}}$, (ii) $\left|f\left(x\right)\right|\le {\mathbf{eta}}$.

## 4References

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

## 5Arguments

1:     $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: $a$, the lower bound of the interval.
2:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: $b$, the upper bound of the interval.
Constraint: ${\mathbf{b}}\ne {\mathbf{a}}$.
3:     $\mathbf{eps}$ – Real (Kind=nag_wp)Input
On entry: the termination tolerance on $x$ (see Section 3).
Constraint: ${\mathbf{eps}}>0.0$.
4:     $\mathbf{eta}$ – Real (Kind=nag_wp)Input
On entry: a value such that if $\left|f\left(x\right)\right|\le {\mathbf{eta}}$, $x$ is accepted as the zero. eta may be specified as $0.0$ (see Section 7).
5:     $\mathbf{f}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
f must evaluate the function $f$ whose zero is to be determined.
The specification of f is:
Fortran Interface
 Function f ( x,
 Real (Kind=nag_wp) :: f Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: x Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
#include nagmk26.h
 double f (const double *x, Integer iuser[], double ruser[])
1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the point at which the function must be evaluated.
2:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
3:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
f is called with the arguments iuser and ruser as supplied to c05ayf. You should use the arrays iuser and ruser to supply information to f.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05ayf is called. Arguments denoted as Input must not be changed by this procedure.
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05ayf. If your code inadvertently does return any NaNs or infinities, c05ayf is likely to produce unexpected results.
6:     $\mathbf{x}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{2}}$, x is the final approximation to the zero. If ${\mathbf{ifail}}={\mathbf{3}}$, x is likely to be a pole of $f\left(x\right)$. Otherwise, x contains no useful information.
7:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
8:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by c05ayf, but are passed directly to f and may be used to pass information to this routine.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$ and ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}\ne {\mathbf{b}}$.
On entry, ${\mathbf{eps}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{eps}}>0.0$.
On entry, ${\mathbf{f}}\left({\mathbf{a}}\right)$ and ${\mathbf{f}}\left({\mathbf{b}}\right)$ have the same sign with neither equalling $0.0$: ${\mathbf{f}}\left({\mathbf{a}}\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{f}}\left({\mathbf{b}}\right)=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=2$
No further improvement in the solution is possible. eps is too small: ${\mathbf{eps}}=〈\mathit{\text{value}}〉$. The final value of x returned is an accurate approximation to the zero.
${\mathbf{ifail}}=3$
The function values in the interval $\left[{\mathbf{a}},{\mathbf{b}}\right]$ might contain a pole rather than a zero. Reducing eps may help in distinguishing between a pole and a zero.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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 ${\mathbf{eta}}=0.0$ and to use eps to control the accuracy, unless you have considerable knowledge of the size of $f\left(x\right)$ for values of $x$ near the zero.

## 8Parallelism and Performance

c05ayf is not threaded in any implementation.

The time taken by c05ayf depends primarily on the time spent evaluating f (see Section 5).
If it is important to determine an interval of relative length less than $2×{\mathbf{eps}}$ containing the zero, or if f is expensive to evaluate and the number of calls to f is to be restricted, then use of c05azf is recommended. Use of c05azf 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 c05azf are more flexible than the direct communication of f required by c05ayf.

## 10Example

This example calculates an approximation to the zero of ${e}^{-x}-x$ within the interval $\left[0,1\right]$ using a tolerance of ${\mathbf{eps}}=\text{1.0E−5}$.

### 10.1Program Text

Program Text (c05ayfe.f90)

None.

### 10.3Program Results

Program Results (c05ayfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017