NAG CL Interface
c05auc (contfn_brent_interval)
1
Purpose
c05auc locates a simple zero of a continuous function from a given starting value. It uses a binary search to locate an interval containing a zero of the function, then Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection, to locate the zero precisely.
2
Specification
void 
c05auc (double *x,
double h,
double eps,
double eta,
double 
(*f)(double x,
Nag_Comm *comm),


double *a,
double *b,
Nag_Comm *comm,
NagError *fail) 

The function may be called by the names: c05auc, nag_roots_contfn_brent_interval or nag_zero_cont_func_brent_binsrch.
3
Description
c05auc attempts to locate an interval
$\left[a,b\right]$ containing a simple zero of the function
$f\left(x\right)$ by a binary search starting from the initial point
$x={\mathbf{x}}$ and using repeated calls to
c05avc. If this search succeeds, then the zero is determined to a userspecified accuracy by a call to
c05ayc. The specifications of functions
c05avc and
c05ayc should be consulted for details of the methods used.
The approximation
$x$ to the zero
$\alpha $ is determined so that at least one of the following criteria is satisfied:

(i)$\leftx\alpha \right\le {\mathbf{eps}}$,

(ii)$\leftf\left(x\right)\right\le {\mathbf{eta}}$.
4
References
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
5
Arguments

1:
$\mathbf{x}$ – double *
Input/Output

On entry: an initial approximation to the zero.
On exit: if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or
NW_TOO_MUCH_ACC_REQUESTED,
x is the final approximation to the zero.
If
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_PROBABLE_POLE,
x is likely to be a pole of
$f\left(x\right)$.
Otherwise,
x contains no useful information.

2:
$\mathbf{h}$ – double
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:
$\mathbf{eps}$ – double
Input

On entry: the termination tolerance on
$x$ (see
Section 3).
Constraint:
${\mathbf{eps}}>0.0$.

4:
$\mathbf{eta}$ – double
Input

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

5:
$\mathbf{f}$ – function, supplied by the user
External Function

f must evaluate the function
$f$ whose zero is to be determined.
The specification of
f is:
double 
f (double x,
Nag_Comm *comm)



1:
$\mathbf{x}$ – double
Input

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

2:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
f.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
c05auc you may allocate memory and initialize these pointers with various quantities for use by
f when called from
c05auc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: f should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
c05auc. If your code inadvertently
does return any NaNs or infinities,
c05auc is likely to produce unexpected results.

6:
$\mathbf{a}$ – double *
Output

7:
$\mathbf{b}$ – double *
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 $\leftf\left(x\right)\right\le {\mathbf{eta}}$ at any stage in the calculation, on exit ${\mathbf{a}}={\mathbf{b}}=x$.

8:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

9:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INTERNAL_ERROR

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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_PROBABLE_POLE

Solution may be a pole rather than a zero.
 NE_REAL

On entry, ${\mathbf{eps}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{eps}}>0.0$.
 NE_REAL_2

On entry, ${\mathbf{x}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{h}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{x}}+{\mathbf{h}}\ne {\mathbf{x}}$ (to machine accuracy).
 NE_ZERO_NOT_FOUND

An interval containing the zero could not be found. Increasing
h and calling
c05auc again will increase the range searched for the zero. Decreasing
h and calling
c05auc again will refine the mesh used in the search for the zero.
 NW_TOO_MUCH_ACC_REQUESTED

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.
${\mathbf{eps}}=\u2329\mathit{\text{value}}\u232a$.
7
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
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_TOO_MUCH_ACC_REQUESTED, although this may involve many more iterations than a lesser accuracy. You are recommended to set
${\mathbf{eta}}=0.0$ and to use
eps to control the accuracy, unless you have considerable knowledge of the size of
$f\left(x\right)$ for values of
$x$ near the zero.
8
Parallelism and Performance
c05auc is not threaded in any implementation.
The time taken by
c05auc depends primarily on the time spent evaluating
f (see
Section 5). 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
$2\times {\mathbf{eps}}$ containing the zero, or if
f is expensive to evaluate and the number of calls to
f is to be restricted, then use of
c05avc followed by
c05azc 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 functions are more flexible than the direct communication of
f required by
c05auc.
If the iteration terminates with successful exit and
${\mathbf{a}}={\mathbf{b}}={\mathbf{x}}$ 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
${f}^{\prime}\left({\mathbf{x}}\right)=0.0$, then
x must correspond to a multiple zero of
$f$ rather than a simple zero.
10
Example
This example calculates an approximation to the zero of $x{e}^{x}$ using a tolerance of ${\mathbf{eps}}=\text{1.0e\u22125}$ starting from ${\mathbf{x}}=1.0$ and using an initial search step ${\mathbf{h}}=0.1$.
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results