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.
The function may be called by the names: c05auc, nag_roots_contfn_brent_interval or nag_zero_cont_func_brent_binsrch.
3Description
c05auc attempts to locate an interval $[a,b]$ 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 user-specified accuracy by a call to c05ayc. The specifications of functions c05avcandc05ayc 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:
On entry: a step length for use in the binary search for an interval containing the zero. The maximum interval searched is $[{\mathbf{x}}-256.0\times {\mathbf{h}},{\mathbf{x}}+256.0\times {\mathbf{h}}]$.
Constraint:
${\mathbf{h}}$ must be sufficiently large that ${\mathbf{x}}+{\mathbf{h}}\ne {\mathbf{x}}$ on the computer.
3: $\mathbf{eps}$ – doubleInput
On entry: the termination tolerance on $x$ (see Section 3).
Constraint:
${\mathbf{eps}}>0.0$.
4: $\mathbf{eta}$ – doubleInput
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: $\mathbf{f}$ – function, supplied by the userExternal Function
f must evaluate the function $f$ whose zero is to be determined.
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 floating-point 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 $\left|f\left(x\right)\right|\le {\mathbf{eta}}$ 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).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ 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, ${\mathbf{eps}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{eps}}>0.0$.
NE_REAL_2
On entry, ${\mathbf{x}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{h}}=\u27e8\mathit{\text{value}}\u27e9$.
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. ${\mathbf{eps}}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
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.
8Parallelism and Performance
c05auc is not threaded in any implementation.
9Further Comments
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.
10Example
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 ${\mathbf{h}}=0.1$.