c05aw attempts to locate a zero of a continuous function using a continuation method based on a secant iteration.
Syntax
C# 

public static void c05aw( ref double x, double eps, double eta, C05..::..C05AW_F f, int nfmax, out int ifail ) 
Visual Basic 

Public Shared Sub c05aw ( _ ByRef x As Double, _ eps As Double, _ eta As Double, _ f As C05..::..C05AW_F, _ nfmax As Integer, _ <OutAttribute> ByRef ifail As Integer _ ) 
Visual C++ 

public: static void c05aw( double% x, double eps, double eta, C05..::..C05AW_F^ f, int nfmax, [OutAttribute] int% ifail ) 
F# 

static member c05aw : x : float byref * eps : float * eta : float * f : C05..::..C05AW_F * nfmax : int * ifail : int byref > unit 
Parameters
 x
 Type: System..::..Double%On entry: an initial approximation to the zero.On exit: the final approximation to the zero, unless ${\mathbf{ifail}}={1}$, ${2}$ or ${5}$, in which case it contains no useful information.
 eps
 Type: System..::..DoubleOn 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 method with more than one value for eps to check the accuracy obtained.Constraint: ${\mathbf{eps}}>0.0$.
 eta
 Type: System..::..DoubleOn entry: a value such that if $\leftf\left(x\right)\right<{\mathbf{eta}}$, $x$ is accepted as the zero. eta may be specified as $0.0$ (see [Accuracy]).
 f
 Type: NagLibrary..::..C05..::..C05AW_Ff must evaluate the function $f$ whose zero is to be determined.
A delegate of type C05AW_F.
 nfmax
 Type: System..::..Int32On entry: the maximum permitted number of calls to f from c05aw. If f is inexpensive to evaluate, nfmax should be given a large value (say $\text{}>1000$).Constraint: ${\mathbf{nfmax}}>0$.
 ifail
 Type: System..::..Int32%On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).
Description
c05aw 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 c05ax 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)  $\leftx\alpha \right\sim {\mathbf{eps}}$, 
(ii)  $\leftf\left(x\right)\right<{\mathbf{eta}}$. 
References
None.
Error Indicators and Warnings
Errors or warnings detected by the method:
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{eps}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{eps}}>0.0$.On entry, ${\mathbf{nfmax}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nfmax}}>0$.
 ${\mathbf{ifail}}=2$
 ${\mathbf{ifail}}=3$
 ${\mathbf{ifail}}=4$
 ${\mathbf{ifail}}=5$

A serious error occurred in an internal call to an auxiliary method.
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 ${\mathbf{ifail}}={3}$ or ${4}$, 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.
Parallelism and Performance
None.
Further Comments
The time taken by c05aw depends primarily on the time spent evaluating the function $f$ (see [Parameters]) and on how close the initial value of x is to the zero.
Example
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 ${\mathbf{eps}}=\text{1.0E\u22123}$ and $\text{1.0E\u22124}$ for comparison purposes, with ${\mathbf{eta}}=0.0$ in both cases.