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NAG C Library Manual

NAG Library Function Documentnag_zero_cont_func_brent_bsrch (c05agc)

1  Purpose

nag_zero_cont_func_brent_bsrch (c05agc) 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 nonlinear interpolation, linear extrapolation and bisection to locate the zero precisely.

2  Specification

 #include #include
void  nag_zero_cont_func_brent_bsrch (double *x, double h, double xtol, double ftol,
 double (*f)(double xx, Nag_Comm *comm),
double *a, double *b, Nag_Comm *comm, NagError *fail)

3  Description

nag_zero_cont_func_brent_bsrch (c05agc) 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 nag_interval_zero_cont_func (c05avc). If this search succeeds, then the zero is determined to a user-specified accuracy by a call to nag_zero_cont_func_brent (c05ayc). The specifications of functions nag_interval_zero_cont_func (c05avc) and nag_zero_cont_func_brent (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) $\left|x-\alpha \right|\le {\mathbf{xtol}}$, (ii) $\left|f\left(x\right)\right|\le {\mathbf{ftol}}$.

4  References

Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall

5  Arguments

1:     xdouble *Input/Output
On entry: an initial approximation to the zero.
On exit: if NE_NOERROR or NW_TOO_MUCH_ACC_REQUESTED, x is the final approximation to the zero.
If NE_PROBABLE_POLE, x is likely to be a pole of $f\left(x\right)$.
Otherwise, x contains no useful information.
2:     hdoubleInput
On entry: a step length for use in the binary search for an interval containing the zero. The maximum interval searched is $\left[{\mathbf{x}}-256.0×{\mathbf{h}},{\mathbf{x}}+256.0×{\mathbf{h}}\right]$.
Constraint: ${\mathbf{h}}$ must be sufficiently large that ${\mathbf{x}}+{\mathbf{h}}\ne {\mathbf{x}}$ on the computer.
3:     xtoldoubleInput
On entry: the termination tolerance on $x$ (see Section 3).
Constraint: ${\mathbf{xtol}}>0.0$.
4:     ftoldoubleInput
On entry: a value such that if $\left|f\left(x\right)\right|\le {\mathbf{ftol}}$, $x$ is accepted as the zero. ftol may be specified as $0.0$ (see Section 7).
5:     ffunction, supplied by the userExternal Function
f must evaluate the function $f$ whose zero is to be determined.
The specification of f is:
 double f (double xx, Nag_Comm *comm)
1:     xxdoubleInput
On entry: the point at which the function must be evaluated.
2:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_zero_cont_func_brent_bsrch (c05agc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_zero_cont_func_brent_bsrch (c05agc) (see Section 3.2.1 in the Essential Introduction).
6:     adouble *Output
7:     bdouble *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 $\left|f\left(x\right)\right|\le {\mathbf{ftol}}$ at any stage in the calculation, then on exit ${\mathbf{a}}={\mathbf{b}}=x$.
8:     commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ 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.
NE_PROBABLE_POLE
Solution may be a pole rather than a zero.
NE_REAL
On entry, ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xtol}}>0.0$.
NE_REAL_2
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and ${\mathbf{h}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}+{\mathbf{h}}\ne {\mathbf{x}}$ (to machine accuracy).
NE_ZERO_NOT_FOUND
An interval containing the zero could not be found.
NW_TOO_MUCH_ACC_REQUESTED
The tolerance xtol has been set too small for the problem being solved. However, the value x returned is a good approximation to the zero. ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.

7  Accuracy

The levels of accuracy depend on the values of xtol and ftol. If full machine accuracy is required, they may be set very small, resulting in an exit with NW_TOO_MUCH_ACC_REQUESTED, although this may involve many more iterations than a lesser accuracy. You are recommended to set ${\mathbf{ftol}}=0.0$ and to use xtol 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  Further Comments

The time taken by nag_zero_cont_func_brent_bsrch (c05agc) 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 $2×{\mathbf{xtol}}$ containing the zero, or if f is expensive to evaluate and the number of calls to f is to be restricted, then use of nag_interval_zero_cont_func (c05avc) followed by nag_zero_cont_func_brent_rcomm (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 nag_zero_cont_func_brent_bsrch (c05agc).
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.

9  Example

This example calculates an approximation to the zero of $x-{e}^{-x}$ using a tolerance of ${\mathbf{xtol}}=\text{1.0e−5}$ starting from ${\mathbf{x}}=1.0$ and using an initial search step ${\mathbf{h}}=0.1$.

9.1  Program Text

Program Text (c05agce.c)

None.

9.3  Program Results

Program Results (c05agce.r)

nag_zero_cont_func_brent_bsrch (c05agc) (PDF version)
c05 Chapter Contents
c05 Chapter Introduction
NAG C Library Manual