# NAG Library Routine Document

## 1Purpose

c05awf attempts to locate a zero of a continuous function using a continuation method based on a secant iteration.

## 2Specification

Fortran Interface
 Subroutine c05awf ( x, eps, eta, f,
 Integer, Intent (In) :: nfmax Integer, Intent (Inout) :: iuser(*), ifail Real (Kind=nag_wp), External :: f Real (Kind=nag_wp), Intent (In) :: eps, eta Real (Kind=nag_wp), Intent (Inout) :: x, ruser(*)
#include nagmk26.h
 void c05awf_ (double *x, const double *eps, const double *eta, double (NAG_CALL *f)(const double *x, Integer iuser[], double ruser[]),const Integer *nfmax, Integer iuser[], double ruser[], Integer *ifail)

## 3Description

c05awf attempts to obtain an approximation to a simple zero $\alpha$ of the function $f\left(x\right)$ given an initial approximation $x$ to $\alpha$. The zero is found by a call to c05axf 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) $\left|x-\alpha \right|\sim {\mathbf{eps}}$, (ii) $\left|f\left(x\right)\right|<{\mathbf{eta}}$.
None.

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input/Output
On entry: an initial approximation to the zero.
On exit: the final approximation to the zero, unless ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$ or ${\mathbf{5}}$, in which case it contains no useful information.
2:     $\mathbf{eps}$ – Real (Kind=nag_wp)Input
On entry: an absolute tolerance to control the accuracy to which the zero is determined. In general, the smaller the value of eps the more accurate x will be as an approximation to $\alpha$. Indeed, for very small positive values of eps, it is likely that the final approximation will satisfy $\left|{\mathbf{x}}-\alpha \right|<{\mathbf{eps}}$. You are advised to call the routine with more than one value for eps to check the accuracy obtained.
Constraint: ${\mathbf{eps}}>0.0$.
3:     $\mathbf{eta}$ – Real (Kind=nag_wp)Input
On entry: a value such that if $\left|f\left(x\right)\right|<{\mathbf{eta}}$, $x$ is accepted as the zero. eta may be specified as $0.0$ (see Section 7).
4:     $\mathbf{f}$ – 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 c05awf. 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 c05awf 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 c05awf. If your code inadvertently does return any NaNs or infinities, c05awf is likely to produce unexpected results.
5:     $\mathbf{nfmax}$ – IntegerInput
On entry: the maximum permitted number of calls to f from c05awf. If f is inexpensive to evaluate, nfmax should be given a large value (say $\text{}>1000$).
Constraint: ${\mathbf{nfmax}}>0$.
6:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
7:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by c05awf, but are passed directly to f and may be used to pass information to this routine.
8:     $\mathbf{ifail}$ – 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, ${\mathbf{eps}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{eps}}>0.0$.
On entry, ${\mathbf{nfmax}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nfmax}}>0$.
${\mathbf{ifail}}=2$
Internal scale factor invalid for this problem. Consider using c05axf instead and setting scal.
${\mathbf{ifail}}=3$
Either f has no zero near x or too much accuracy has been requested. Check the coding of f or increase eps.
${\mathbf{ifail}}=4$
More than nfmax calls have been made to f.
nfmax may be too small for the problem (because x is too far away from the zero), or f has no zero near x, or too much accuracy has been requested in calculating the zero. Increase nfmax, check the coding of f or increase eps.
${\mathbf{ifail}}=5$
A serious error occurred in an internal call to an auxiliary routine.
${\mathbf{ifail}}=-99$
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 ${\mathbf{ifail}}={\mathbf{3}}$ or ${\mathbf{4}}$, 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

c05awf is not threaded in any implementation.

The time taken by c05awf depends primarily on the time spent evaluating the function $f$ (see Section 5) and on how close the initial value of x is to the zero.
If a more flexible way of specifying the function $f$ is required or if you wish to have closer control of the calculation, then the reverse communication routine c05axf is recommended instead of c05awf.

## 10Example

This example calculates the zero of $f\left(x\right)={e}^{-x}-x$ from a starting value ${\mathbf{x}}=1.0$. Two calculations are made with ${\mathbf{eps}}=\text{1.0E−3}$ and $\text{1.0E−4}$ for comparison purposes, with ${\mathbf{eta}}=0.0$ in both cases.

### 10.1Program Text

Program Text (c05awfe.f90)

None.

### 10.3Program Results

Program Results (c05awfe.r)

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