nag_zero_cont_func_brent_rcomm (c05azc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_zero_cont_func_brent_rcomm (c05azc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zero_cont_func_brent_rcomm (c05azc) locates a simple zero of a continuous function in a given interval by using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection. It uses reverse communication for evaluating the function.

2  Specification

#include <nag.h>
#include <nagc05.h>
void  nag_zero_cont_func_brent_rcomm (double *x, double *y, double fx, double tolx, Nag_ErrorControl ir, double c[], Integer *ind, NagError *fail)

3  Description

You must supply x and y to define an initial interval a,b  containing a simple zero of the function fx  (the choice of x and y must be such that fx × fy 0.0 ). The function combines the methods of bisection, nonlinear interpolation and linear extrapolation (see Dahlquist and Björck (1974)), to find a sequence of sub-intervals of the initial interval such that the final interval x,y  contains the zero and x-y  is less than some tolerance specified by tolx and ir (see Section 5). In fact, since the intermediate intervals x,y  are determined only so that fx × fy 0.0 , it is possible that the final interval may contain a discontinuity or a pole of f (violating the requirement that f be continuous). nag_zero_cont_func_brent_rcomm (c05azc) checks if the sign change is likely to correspond to a pole of f and gives an error return in this case.
A feature of the algorithm used by this function is that unlike some other methods it guarantees convergence within about log2 b-a / δ 2  function evaluations, where δ is related to the argument tolx. See Brent (1973) for more details.
nag_zero_cont_func_brent_rcomm (c05azc) returns to the calling program for each evaluation of fx . On each return you should set fx = fx  and call nag_zero_cont_func_brent_rcomm (c05azc) again.
The function is a modified version of procedure ‘zeroin’ given by Brent (1973).

4  References

Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
Bus J C P and Dekker T J (1975) Two efficient algorithms with guaranteed convergence for finding a zero of a function ACM Trans. Math. Software 1 330–345
Dahlquist G and Björck Å (1974) Numerical Methods Prentice–Hall

5  Arguments

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument ind. Between intermediate exits and re-entries, all arguments other than fx must remain unchanged.
1:     xdouble *Input/Output
2:     ydouble *Input/Output
On initial entry: x and y must define an initial interval a,b  containing the zero, such that fx × fy0.0 . It is not necessary that x<y .
On intermediate exit: x contains the point at which f must be evaluated before re-entry to the function.
On final exit: x and y define a smaller interval containing the zero, such that fx×fy0.0 , and x-y  satisfies the accuracy specified by tolx and ir, unless an error has occurred. If fail.code= NE_PROBABLE_POLE, x and y generally contain very good approximations to a pole; if fail.code= NW_TOO_MUCH_ACC_REQUESTED, x and y generally contain very good approximations to the zero (see Section 6). If a point x is found such that fx=0.0 , then on final exit x=y  (in this case there is no guarantee that x is a simple zero). In all cases, the value returned in x is the better approximation to the zero.
3:     fxdoubleInput
On initial entry: if ind=1 , fx need not be set.
If ind=-1 , fx must contain fx  for the initial value of x.
On intermediate re-entry: must contain fx  for the current value of x.
4:     tolxdoubleInput
On initial entry: the accuracy to which the zero is required. The type of error test is specified by ir.
Constraint: tolx>0.0 .
5:     irNag_ErrorControlInput
On initial entry: indicates the type of error test.
The test is: x-y2.0×tolx×max1.0,x .
The test is: x-y2.0×tolx .
The test is: x-y2.0×tolx×x .
Suggested value: ir=Nag_Mixed.
Constraint: ir=Nag_Mixed, Nag_Absolute or Nag_Relative.
6:     c[17]doubleInput/Output
On initial entry: if ind=1 , no elements of c need be set.
If ind=-1 , c[0]  must contain fy , other elements of c need not be set.
On final exit: is undefined.
7:     indInteger *Input/Output
On initial entry: must be set to 1 or -1 .
fx and c[0]  need not be set.
fx and c[0]  must contain fx  and fy  respectively.
On intermediate exit: contains 2, 3 or 4. The calling program must evaluate f at x, storing the result in fx, and re-enter nag_zero_cont_func_brent_rcomm (c05azc) with all other arguments unchanged.
On final exit: contains 0.
Constraint: on entry ind=-1, 1, 2, 3 or 4.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
On entry, ind=value.
Constraint: ind=-1, 1, 2, 3 or 4.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
On entry, fx and fy have the same sign with neither equalling 0.0: fx=value and fy=value.
The final interval may contain a pole rather than a zero. Note that this error exit is not completely reliable: it may be taken in extreme cases when x,y  contains a zero, or it may not be taken when x,y  contains a pole. Both these cases occur most frequently when tolx is large.
On entry, tolx=value.
Constraint: tolx>0.0.
The tolerance tolx has been set too small for the problem being solved. However, the values x and y returned may well be good approximations to the zero. tolx=value.

7  Accuracy

The accuracy of the final value x as an approximation of the zero is determined by tolx and ir (see Section 5). A relative accuracy criterion ( ir=2 ) should not be used when the initial values x and y are of different orders of magnitude. In this case a change of origin of the independent variable may be appropriate. For example, if the initial interval x,y  is transformed linearly to the interval 1,2 , then the zero can be determined to a precise number of figures using an absolute ( ir=1 ) or relative ( ir=2 ) error test and the effect of the transformation back to the original interval can also be determined. Except for the accuracy check, such a transformation has no effect on the calculation of the zero.

8  Further Comments

For most problems, the time taken on each call to nag_zero_cont_func_brent_rcomm (c05azc) will be negligible compared with the time spent evaluating fx  between calls to nag_zero_cont_func_brent_rcomm (c05azc).
If the calculation terminates because fx=0.0 , then on return y is set to x. (In fact, y=x  on return only in this case and, possibly, when fail.code= NW_TOO_MUCH_ACC_REQUESTED.) There is no guarantee that the value returned in x corresponds to a simple root 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 fx=0.0 , then x must correspond to a multiple zero of f rather than a simple zero.

9  Example

This example calculates a zero of e-x - x  with an initial interval 0,1 , tolx=1.0e−5  and a mixed error test.

9.1  Program Text

Program Text (c05azce.c)

9.2  Program Data


9.3  Program Results

Program Results (c05azce.r)

nag_zero_cont_func_brent_rcomm (c05azc) (PDF version)
c05 Chapter Contents
c05 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012