# NAG Library Routine Document

## 1Purpose

c05azf locates a simple zero of a continuous function in a given interval by using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection. It uses reverse communication for evaluating the function.

## 2Specification

Fortran Interface
 Subroutine c05azf ( x, y, fx, tolx, ir, c, ind,
 Integer, Intent (In) :: ir Integer, Intent (Inout) :: ind, ifail Real (Kind=nag_wp), Intent (In) :: fx, tolx Real (Kind=nag_wp), Intent (Inout) :: x, y, c(17)
#include <nagmk26.h>
 void c05azf_ (double *x, double *y, const double *fx, const double *tolx, const Integer *ir, double c[], Integer *ind, Integer *ifail)

## 3Description

You must supply x and y to define an initial interval $\left[a,b\right]$ containing a simple zero of the function $f\left(x\right)$ (the choice of x and y must be such that $f\left({\mathbf{x}}\right)×f\left({\mathbf{y}}\right)\le 0.0$). The routine combines the methods of bisection, nonlinear interpolation and linear extrapolation (see Dahlquist and Björck (1974)), to find a sequence of sub-intervals of the initial interval such that the final interval $\left[{\mathbf{x}},{\mathbf{y}}\right]$ contains the zero and $\left|{\mathbf{x}}-{\mathbf{y}}\right|$ is less than some tolerance specified by tolx and ir (see Section 5). In fact, since the intermediate intervals $\left[{\mathbf{x}},{\mathbf{y}}\right]$ are determined only so that $f\left({\mathbf{x}}\right)×f\left({\mathbf{y}}\right)\le 0.0$, it is possible that the final interval may contain a discontinuity or a pole of $f$ (violating the requirement that $f$ be continuous). c05azf checks if the sign change is likely to correspond to a pole of $f$ and gives an error return in this case.
A feature of the algorithm used by this routine is that unlike some other methods it guarantees convergence within about ${\left({\mathrm{log}}_{2}\left[\left(b-a\right)/\delta \right]\right)}^{2}$ function evaluations, where $\delta$ is related to the argument tolx. See Brent (1973) for more details.
c05azf returns to the calling program for each evaluation of $f\left(x\right)$. On each return you should set ${\mathbf{fx}}=f\left({\mathbf{x}}\right)$ and call c05azf again.
The routine is a modified version of procedure ‘zeroin’ given by Brent (1973).

## 4References

Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
Bus J C P and Dekker T J (1975) Two efficient algorithms with guaranteed convergence for finding a zero of a function ACM Trans. Math. Software 1 330–345
Dahlquist G and Björck Å (1974) Numerical Methods Prentice–Hall

## 5Arguments

Note: this routine uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument ind. Between intermediate exits and re-entries, all arguments other than fx must remain unchanged.
1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input/Output
2:     $\mathbf{y}$ – Real (Kind=nag_wp)Input/Output
On initial entry: x and y must define an initial interval $\left[a,b\right]$ containing the zero, such that $f\left({\mathbf{x}}\right)×f\left({\mathbf{y}}\right)\le 0.0$. It is not necessary that ${\mathbf{x}}<{\mathbf{y}}$.
On intermediate exit: x contains the point at which $f$ must be evaluated before re-entry to the routine.
On final exit: x and y define a smaller interval containing the zero, such that $f\left({\mathbf{x}}\right)×f\left({\mathbf{y}}\right)\le 0.0$, and $\left|{\mathbf{x}}-{\mathbf{y}}\right|$ satisfies the accuracy specified by tolx and ir, unless an error has occurred. If ${\mathbf{ifail}}={\mathbf{4}}$, x and y generally contain very good approximations to a pole; if ${\mathbf{ifail}}={\mathbf{5}}$, x and y generally contain very good approximations to the zero (see Section 6). If a point x is found such that $f\left({\mathbf{x}}\right)=0.0$, on final exit ${\mathbf{x}}={\mathbf{y}}$ (in this case there is no guarantee that x is a simple zero). In all cases, the value returned in x is the better approximation to the zero.
3:     $\mathbf{fx}$ – Real (Kind=nag_wp)Input
On initial entry: if ${\mathbf{ind}}=1$, fx need not be set.
If ${\mathbf{ind}}=-1$, fx must contain $f\left({\mathbf{x}}\right)$ for the initial value of x.
On intermediate re-entry: must contain $f\left({\mathbf{x}}\right)$ for the current value of x.
4:     $\mathbf{tolx}$ – Real (Kind=nag_wp)Input
On initial entry: the accuracy to which the zero is required. The type of error test is specified by ir.
Constraint: ${\mathbf{tolx}}>0.0$.
5:     $\mathbf{ir}$ – IntegerInput
On initial entry: indicates the type of error test.
${\mathbf{ir}}=0$
The test is: $\left|{\mathbf{x}}-{\mathbf{y}}\right|\le 2.0×{\mathbf{tolx}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,\left|{\mathbf{x}}\right|\right)$.
${\mathbf{ir}}=1$
The test is: $\left|{\mathbf{x}}-{\mathbf{y}}\right|\le 2.0×{\mathbf{tolx}}$.
${\mathbf{ir}}=2$
The test is: $\left|{\mathbf{x}}-{\mathbf{y}}\right|\le 2.0×{\mathbf{tolx}}×\left|{\mathbf{x}}\right|$.
Suggested value: ${\mathbf{ir}}=0$.
Constraint: ${\mathbf{ir}}=0$, $1$ or $2$.
6:     $\mathbf{c}\left(17\right)$ – Real (Kind=nag_wp) arrayInput/Output
On initial entry: if ${\mathbf{ind}}=1$, no elements of c need be set.
If ${\mathbf{ind}}=-1$, ${\mathbf{c}}\left(1\right)$ must contain $f\left({\mathbf{y}}\right)$, other elements of c need not be set.
On final exit: is undefined.
7:     $\mathbf{ind}$ – IntegerInput/Output
On initial entry: must be set to $1$ or $-1$.
${\mathbf{ind}}=1$
fx and ${\mathbf{c}}\left(1\right)$ need not be set.
${\mathbf{ind}}=-1$
fx and ${\mathbf{c}}\left(1\right)$ must contain $f\left({\mathbf{x}}\right)$ and $f\left({\mathbf{y}}\right)$ respectively.
On intermediate exit: contains $2$, $3$ or $4$. The calling program must evaluate $f$ at x, storing the result in fx, and re-enter c05azf with all other arguments unchanged.
On final exit: contains $0$.
Constraint: on entry ${\mathbf{ind}}=-1$, $1$, $2$, $3$ or $4$.
Note: any values you return to c05azf as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by c05azf. If your code does inadvertently return any NaNs or infinities, c05azf is likely to produce unexpected results.
8:     $\mathbf{ifail}$ – IntegerInput/Output
On initial entry: ifail must be set to $0$, . 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  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value  is used it is essential to test the value of ifail on exit.
On final 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, $f\left({\mathbf{x}}\right)$ and $f\left({\mathbf{y}}\right)$ have the same sign with neither equalling $0.0$: $f\left({\mathbf{x}}\right)=〈\mathit{\text{value}}〉$ and $f\left({\mathbf{y}}\right)=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ind}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ind}}=-1$, $1$, $2$, $3$ or $4$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ir}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ir}}=0$, $1$ or $2$.
On entry, ${\mathbf{tolx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tolx}}>0.0$.
${\mathbf{ifail}}=4$
The final interval may contain a pole rather than a zero. Note that this error exit is not completely reliable: it may be taken in extreme cases when $\left[{\mathbf{x}},{\mathbf{y}}\right]$ contains a zero, or it may not be taken when $\left[{\mathbf{x}},{\mathbf{y}}\right]$ contains a pole. Both these cases occur most frequently when tolx is large.
${\mathbf{ifail}}=5$
The tolerance tolx has been set too small for the problem being solved. However, the values x and y returned may well be good approximations to the zero. ${\mathbf{tolx}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
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 accuracy of the final value x as an approximation of the zero is determined by tolx and ir (see Section 5). A relative accuracy criterion (${\mathbf{ir}}=2$) should not be used when the initial values x and y are of different orders of magnitude. In this case a change of origin of the independent variable may be appropriate. For example, if the initial interval $\left[{\mathbf{x}},{\mathbf{y}}\right]$ is transformed linearly to the interval $\left[1,2\right]$, then the zero can be determined to a precise number of figures using an absolute (${\mathbf{ir}}=1$) or relative (${\mathbf{ir}}=2$) error test and the effect of the transformation back to the original interval can also be determined. Except for the accuracy check, such a transformation has no effect on the calculation of the zero.

## 8Parallelism and Performance

c05azf is not threaded in any implementation.

For most problems, the time taken on each call to c05azf will be negligible compared with the time spent evaluating $f\left(x\right)$ between calls to c05azf.
If the calculation terminates because $f\left({\mathbf{x}}\right)=0.0$, then on return y is set to x. (In fact, ${\mathbf{y}}={\mathbf{x}}$ on return only in this case and, possibly, when ${\mathbf{ifail}}={\mathbf{5}}$.) There is no guarantee that the value returned in x corresponds to a simple root and you should check whether it does. One way to check this is to compute the derivative of $f$ at the point x, preferably analytically, or, if this is not possible, numerically, perhaps by using a central difference estimate. If ${f}^{\prime }\left({\mathbf{x}}\right)=0.0$, then x must correspond to a multiple zero of $f$ rather than a simple zero.

## 10Example

This example calculates a zero of ${e}^{-x}-x$ with an initial interval $\left[0,1\right]$, ${\mathbf{tolx}}=\text{1.0E−5}$ and a mixed error test.

### 10.1Program Text

Program Text (c05azfe.f90)

None.

### 10.3Program Results

Program Results (c05azfe.r)