c05aj Method
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c05aj attempts to locate a zero of a continuous function using a continuation method based on a secant iteration.

Syntax

C#
public static void c05aj(
	ref double x,
	double eps,
	double eta,
	C05..::.C05AJ_F f,
	int nfmax,
	out int ifail
)
Visual Basic (Declaration)
Public Shared Sub c05aj ( _
	ByRef x As Double, _
	eps As Double, _
	eta As Double, _
	f As C05..::.C05AJ_F, _
	nfmax As Integer, _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void c05aj(
	double% x, 
	double eps, 
	double eta, 
	C05..::.C05AJ_F^ f, 
	int nfmax, 
	[OutAttribute] int% ifail
)
F#
static member c05aj : 
        x:float byref * 
        eps:float * 
        eta:float * 
        f:C05..::.C05AJ_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 on exit, 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 fx<eta , x is accepted as the zero. eta may be specified as 0.0 (see [Accuracy]).
f
Type: NagLibrary..::.C05..::.C05AJ_F
f must evaluate the function f whose zero is to be determined.

A delegate of type C05AJ_F.

nfmax
Type: System..::.Int32
On entry: the maximum permitted number of calls to f from c05aj. 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 (see [Error Indicators and Warnings]).

Description

c05aj attempts to obtain an approximation to a simple zero α of the function fx  given an initial approximation x to 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∼eps ,
(ii) fx<eta .

References

Error Indicators and Warnings

Errors or warnings detected by the method:
ifail=1
On entry, eps≤0.0 ,
or nfmax≤0 .
ifail=2
An internally calculated scale factor has the wrong order of magnitude for the problem. If this error exit occurs, you are advised to call c05ax instead where different scale values can be tried.
ifail=3
Either the function fx  given by f has no zero near x or too much accuracy has been requested in calculating the zero. The first is a more likely cause of this error exit and you should check the coding of f and make an independent investigation of its behaviour near x. The second can be alleviated by increasing eps.
ifail=4
More than nfmax calls have been made to f. This error exit can occur because nfmax is too small for the problem (essentially because x is too far away from the zero) or for either of the reasons given under ifail=3 above. If nfmax is increased considerably and this error exit occurs again at approximately the same final value of x, then it is likely that one of the reasons given under ifail=3 is the cause.
ifail=5  (c05ax)
A serious error has occurred in the specified method. Check all method calls. Seek expert help.
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 or 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 fx  for values of x near the zero.

Further Comments

The time taken by c05aj 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 c05aj.

Example

See Also