c05ag Method
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c05ag locates a simple zero of a continuous function from a given starting value, using a binary search to locate an interval containing a zero of the function, then a combination of the methods of linear interpolation, extrapolation and bisection to locate the zero precisely.

Syntax

C#
public static void c05ag(
	ref double x,
	double h,
	double eps,
	double eta,
	C05..::.C05AG_F f,
	out double a,
	out double b,
	out int ifail
)
Visual Basic (Declaration)
Public Shared Sub c05ag ( _
	ByRef x As Double, _
	h As Double, _
	eps As Double, _
	eta As Double, _
	f As C05..::.C05AG_F, _
	<OutAttribute> ByRef a As Double, _
	<OutAttribute> ByRef b As Double, _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void c05ag(
	double% x, 
	double h, 
	double eps, 
	double eta, 
	C05..::.C05AG_F^ f, 
	[OutAttribute] double% a, 
	[OutAttribute] double% b, 
	[OutAttribute] int% ifail
)
F#
static member c05ag : 
        x:float byref * 
        h:float * 
        eps:float * 
        eta:float * 
        f:C05..::.C05AG_F * 
        a:float byref * 
        b:float byref * 
        ifail:int byref -> unit

Parameters

x
Type: System..::.Double %
On entry: an initial approximation to the zero.
On exit: if ifail=0 or 4, x is the final approximation to the zero. If ifail=3, x is likely to be a pole of fx. Otherwise, x contains no useful information.
h
Type: System..::.Double
On entry: a step length for use in the binary search for an interval containing the zero. The maximum interval searched is x-256.0×h,x+256.0×h .
Constraint: h  must be sufficiently large that x+h≠x  on the computer.
eps
Type: System..::.Double
On entry: the termination tolerance on x (see [Description]).
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..::.C05AG_F
f must evaluate the function f whose zero is to be determined.

A delegate of type C05AG_F.

a
Type: System..::.Double %
On exit: a and b are the lower and upper bounds respectively of the interval resulting from the binary search. If the zero is determined exactly such that fx=0.0  or is determined so that fx≤eta  at any stage in the calculation, then on exit a=b=x .
b
Type: System..::.Double %
On exit: a and b are the lower and upper bounds respectively of the interval resulting from the binary search. If the zero is determined exactly such that fx=0.0  or is determined so that fx≤eta  at any stage in the calculation, then on exit a=b=x .
ifail
Type: System..::.Int32 %
On exit: ifail=0 unless the method detects an error (see [Error Indicators and Warnings]).

Description

c05ag attempts to locate an interval a,b  containing a simple zero of the function fx  by a binary search starting from the initial point x=x  and using repeated calls to c05av. If this search succeeds, then the zero is determined to a user-specified accuracy by a call to (c05ay not in this release). The specifications of methods c05av should be consulted for details of the methods used.
The approximation x to the zero α is determined so that at least one of the following criteria is satisfied:
(i) x-α ≤ 4×eps× maxx,1 ,
(ii) fx≤eta .

References

Error Indicators and Warnings

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=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 c05ag depends primarily on the time spent evaluating f (see [Parameters]). The accuracy of the initial approximation x and the value of h will have a somewhat unpredictable effect on the timing.
If it is important to determine an interval of relative length less than 2×eps containing the zero, or if f is expensive to evaluate and the number of calls to f is to be restricted, then use of c05av followed by c05az is recommended. Use of this combination is also recommended when the structure of the problem to be solved does not permit a simple f to be written: the reverse communication facilities of these methods are more flexible than the direct communication of f required by c05ag.
If the iteration terminates with successful exit and a=b=x  there is no guarantee that the value returned in x corresponds to a simple zero 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′x=0.0 , then x must correspond to a multiple zero of f rather than a simple zero.

Example

See Also