c05 Chapter Contents
c05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zero_cont_func_brent_rcomm (c05azc)

## 1  Purpose

nag_zero_cont_func_brent_rcomm (c05azc) 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.

## 2  Specification

 #include #include
 void nag_zero_cont_func_brent_rcomm (double *x, double *y, double fx, double tolx, Nag_ErrorControl ir, double c[], Integer *ind, NagError *fail)

## 3  Description

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 function 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). nag_zero_cont_func_brent_rcomm (c05azc) 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 function 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.
nag_zero_cont_func_brent_rcomm (c05azc) 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 nag_zero_cont_func_brent_rcomm (c05azc) again.
The function is a modified version of procedure ‘zeroin’ given by Brent (1973).

## 4  References

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

## 5  Arguments

Note: this function 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:    $\mathbf{x}$double *Input/Output
2:    $\mathbf{y}$double *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 function.
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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_PROBABLE_POLE, x and y generally contain very good approximations to a pole; if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_TOO_MUCH_ACC_REQUESTED, x and y generally contain very good approximations to the zero (see Section 6). If a point x is found such that $f\left({\mathbf{x}}\right)=0.0$, then 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:    $\mathbf{fx}$doubleInput
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:    $\mathbf{tolx}$doubleInput
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:    $\mathbf{ir}$Nag_ErrorControlInput
On initial entry: indicates the type of error test.
${\mathbf{ir}}=\mathrm{Nag_Mixed}$
The test is: $\left|{\mathbf{x}}-{\mathbf{y}}\right|\le 2.0×{\mathbf{tolx}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,\left|{\mathbf{x}}\right|\right)$.
${\mathbf{ir}}=\mathrm{Nag_Absolute}$
The test is: $\left|{\mathbf{x}}-{\mathbf{y}}\right|\le 2.0×{\mathbf{tolx}}$.
${\mathbf{ir}}=\mathrm{Nag_Relative}$
The test is: $\left|{\mathbf{x}}-{\mathbf{y}}\right|\le 2.0×{\mathbf{tolx}}×\left|{\mathbf{x}}\right|$.
Suggested value: ${\mathbf{ir}}=\mathrm{Nag_Mixed}$.
Constraint: ${\mathbf{ir}}=\mathrm{Nag_Mixed}$, $\mathrm{Nag_Absolute}$ or $\mathrm{Nag_Relative}$.
6:    $\mathbf{c}\left[17\right]$doubleInput/Output
On initial entry: if ${\mathbf{ind}}=1$, no elements of c need be set.
If ${\mathbf{ind}}=-1$, ${\mathbf{c}}\left[0\right]$ must contain $f\left({\mathbf{y}}\right)$, other elements of c need not be set.
On final exit: is undefined.
7:    $\mathbf{ind}$Integer *Input/Output
On initial entry: must be set to $1$ or $-1$.
${\mathbf{ind}}=1$
fx and ${\mathbf{c}}\left[0\right]$ need not be set.
${\mathbf{ind}}=-1$
fx and ${\mathbf{c}}\left[0\right]$ 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 nag_zero_cont_func_brent_rcomm (c05azc) with all other arguments unchanged.
On final exit: contains $0$.
Constraint: on entry ${\mathbf{ind}}=-1$, $1$, $2$, $3$ or $4$.
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{ind}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ind}}=-1$, $1$, $2$, $3$ or $4$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_SIGN_CHANGE
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 $f\left({\mathbf{y}}\right)=〈\mathit{\text{value}}〉$.
NE_PROBABLE_POLE
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.
NE_REAL
On entry, ${\mathbf{tolx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tolx}}>0.0$.
NW_TOO_MUCH_ACC_REQUESTED
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. ${\mathbf{tolx}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

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.

## 8  Parallelism and Performance

nag_zero_cont_func_brent_rcomm (c05azc) is not threaded in any implementation.

## 9  Further Comments

For most problems, the time taken on each call to nag_zero_cont_func_brent_rcomm (c05azc) will be negligible compared with the time spent evaluating $f\left(x\right)$ between calls to nag_zero_cont_func_brent_rcomm (c05azc).
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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_TOO_MUCH_ACC_REQUESTED.) 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.

## 10  Example

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.

### 10.1  Program Text

Program Text (c05azce.c)

None.

### 10.3  Program Results

Program Results (c05azce.r)