c05auf 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 routine may be called by the names c05auf or nagf_roots_contfn_brent_interval.
3Description
c05auf 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 c05avf. If this search succeeds, then the zero is determined to a user-specified accuracy by a call to c05ayf. The specifications of routines c05avfandc05ayf 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: $\mathbf{eps}$ – 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.
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 ${\mathbf{a}}={\mathbf{b}}=x$.
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 $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{eps}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{eps}}>0.0$.
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).
${\mathbf{ifail}}=2$
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. ${\mathbf{eps}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
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 ${\mathbf{ifail}}={\mathbf{4}}$, 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
c05auf is not threaded in any implementation.
9Further Comments
The time taken by c05auf 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 c05avf followed by c05azf 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 routines are more flexible than the direct communication of f required by c05auf.
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 ${\mathbf{h}}=0.1$.