NAG Library Routine Document

c05auf (contfn_brent_interval)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input/Output

On entry: an initial approximation to the zero.

On exit: if $\mathbf{ifail}=\mathbf{0}$ or $\mathbf{4}$, x is the final approximation to the zero.

If $\mathbf{ifail}=\mathbf{3}$, x is likely to be a pole of $f\left(x\right)$.

Otherwise, x contains no useful information.

2:     $\mathbf{h}$ – Real (Kind=nag_wp)Input

On entry: a step length for use in the binary search for an interval containing the zero. The maximum interval searched is $\left[\mathbf{x}-256.0\times \mathbf{h},\mathbf{x}+256.0\times \mathbf{h}\right]$.

Constraint: $\mathbf{h}$ must be sufficiently large that $\mathbf{x}+\mathbf{h}\ne \mathbf{x}$ on the computer.

3:     $\mathbf{eps}$ – Real (Kind=nag_wp)Input

On entry: the termination tolerance on $x$ (see Section 3).

Constraint: $\mathbf{eps}>0.0$.

4:     $\mathbf{eta}$ – Real (Kind=nag_wp)Input

On entry: a value such that if $\left|f\left(x\right)\right|\le \mathbf{eta}$, $x$ is accepted as the zero. eta may be specified as $0.0$ (see Section 7).

5:     $\mathbf{f}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

f must evaluate the function $f$ whose zero is to be determined.

The specification of f is:

Fortran Interface

Function f ( x, iuser, ruser) Real (Kind=nag_wp) :: f Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: x Real (Kind=nag_wp), Intent (Inout) :: ruser(*)

C Header Interface

#include <nagmk26.h> double  f (const double *x, Integer iuser[], double ruser[])

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the point at which the function must be evaluated.

2:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

3:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

f is called with the arguments iuser and ruser as supplied to c05auf. You should use the arrays iuser and ruser to supply information to f.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05auf is called. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05auf. If your code inadvertently does return any NaNs or infinities, c05auf is likely to produce unexpected results.

6:     $\mathbf{a}$ – Real (Kind=nag_wp)Output

7:     $\mathbf{b}$ – Real (Kind=nag_wp)Output

On exit: the lower and upper bounds respectively of the interval resulting from the binary search. If the zero is determined exactly such that $f\left(x\right)=0.0$ or is determined so that $\left|f\left(x\right)\right|\le \mathbf{eta}$ at any stage in the calculation, on exit $\mathbf{a}=\mathbf{b}=x$.

8:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

9:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

iuser and ruser are not used by c05auf, but are passed directly to f and may be used to pass information to this routine.

10:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{eps}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{eps}>0.0$.

On entry, $\mathbf{x}=〈\mathit{\text{value}}〉$ and $\mathbf{h}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{x}+\mathbf{h}\ne \mathbf{x}$ (to machine accuracy).

$\mathbf{ifail}=2$

An interval containing the zero could not be found. Increasing h and calling c05auf again will increase the range searched for the zero. Decreasing h and calling c05auf again will refine the mesh used in the search for the zero.

$\mathbf{ifail}=3$

Solution may be a pole rather than a zero.

$\mathbf{ifail}=4$

The tolerance eps has been set too small for the problem being solved. However, the value x returned is a good approximation to the zero. $\mathbf{eps}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (c05aufe.f90)

10.2

Program Data

10.3

Program Results

Program Results (c05aufe.r)