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 $ifail = 1$ , $2$ or $5$ , in which case it contains no useful information.

eps: Type: System..::..Double
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 $α$ . Indeed, for very small positive values of eps, it is likely that the final approximation will satisfy $|x - α| < eps$ . You are advised to call the method with more than one value for eps to check the accuracy obtained.

Constraint: $eps > 0.0$ .

eta: Type: System..::..Double
On entry: a value such that if $|f (x)| < eta$ , $x$ is accepted as the zero. eta may be specified as $0.0$ (see [Accuracy]).

f: Type: NagLibrary..::..C05..::..C05AW_F
f must evaluate the function $f$ whose zero is to be determined.
A delegate of type C05AW_F.

nfmax: Type: System..::..Int32
On 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 $> 1000$ ).

Constraint: $nfmax > 0$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ 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

α

of the function

f (x)

given an initial approximation

x

α

. 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

α

is determined so that at least one of the following criteria is satisfied:

(i)	$\|x - α\| \sim eps$ ,
(ii)	$\|f (x)\| < eta$ .

References

None.

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$: On entry, $eps = 〈value〉$ .
Constraint: $eps > 0.0$ .

On entry, $nfmax = 〈value〉$ .
Constraint: $nfmax > 0$ .

$ifail = 2$: Internal scale factor invalid for this problem. Consider using c05ax instead and setting scal.

$ifail = 3$: Either f has no zero near x or too much accuracy has been requested. Check the coding of f or increase eps.

$ifail = 4$: 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.

$ifail = 5$: A serious error occurred in an internal call to an auxiliary method.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

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

ifail = 3

4

, although this may involve many more iterations than a lesser accuracy. You are recommended to set

eta = 0.0

and to use eps to control the accuracy, unless you have considerable knowledge of the size of

f (x)

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.

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 method c05ax is recommended instead of c05aw.

Example

This example calculates the zero of

f (x) = e^{- x} - x

from a starting value

x = 1.0

. Two calculations are made with

eps = 1.0E−3

and

1.0E−4

for comparison purposes, with

eta = 0.0

in both cases.

Example program (C#): c05awe.cs

Example program results: c05awe.r