NAG Library Function Document
nag_zero_cont_func_brent_binsrch (c05auc)
1 Purpose
nag_zero_cont_func_brent_binsrch (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
| #include <nag.h> |
| #include <nagc05.h> |
| void |
nag_zero_cont_func_brent_binsrch (double *x,
double h,
double eps,
double eta,
| double |
(*f)(double x,
Nag_Comm *comm),
|
|
double *a,
double *b,
Nag_Comm *comm,
NagError *fail) |
|
3 Description
nag_zero_cont_func_brent_binsrch (c05auc) attempts to locate an interval
containing a simple zero of the function
by a binary search starting from the initial point
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
to the zero
is determined so that at least one of the following criteria is satisfied:
| (i) |
, |
| (ii) |
. |
4 References
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
5 Arguments
- 1:
x – double *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
.
Otherwise,
x contains no useful information.
- 2:
h – doubleInput
On entry: a step length for use in the binary search for an interval containing the zero. The maximum interval searched is .
Constraint:
must be sufficiently large that on the computer.
- 3:
eps – doubleInput
On entry: the termination tolerance on
(see
Section 3).
Constraint:
.
- 4:
eta – doubleInput
On entry: a value such that if
,
is accepted as the zero.
eta may be specified as
(see
Section 7).
- 5:
f – function, supplied by the userExternal Function
f must evaluate the function
whose zero is to be determined.
The specification of
f is:
| double |
f (double x,
Nag_Comm *comm)
|
|
- 1:
x – doubleInput
On entry: the point at which the function must be evaluated.
- 2:
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 nag_zero_cont_func_brent_binsrch (c05auc) you may allocate memory and initialize these pointers with various quantities for use by
f when called from nag_zero_cont_func_brent_binsrch (c05auc) (see
Section 3.2.1.1 in the Essential Introduction).
- 6:
a – double *Output
- 7:
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 or is determined so that at any stage in the calculation, then on exit .
- 8:
comm – Nag_Comm *Communication Structure
-
The NAG communication argument (see
Section 3.2.1.1 in the Essential Introduction).
- 9:
fail – NagError *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 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, .
Constraint: .
- NE_REAL_2
-
On entry, and .
Constraint: (to machine accuracy).
- NE_ZERO_NOT_FOUND
-
An interval containing the zero could not be found. Increasing
h and calling nag_zero_cont_func_brent_binsrch (c05auc) again will increase the range searched for the zero. Decreasing
h and calling nag_zero_cont_func_brent_binsrch (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.
.
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
NW_TOO_MUCH_ACC_REQUESTED, although this may involve many more iterations than a lesser accuracy. You are recommended to set
and to use
eps to control the accuracy, unless you have considerable knowledge of the size of
for values of
near the zero.
8 Parallelism and Performance
Not applicable.
The time taken by nag_zero_cont_func_brent_binsrch (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
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_binsrch (c05auc).
If the iteration terminates with successful exit and
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
at the point
x, preferably analytically, or, if this is not possible, numerically, perhaps by using a central difference estimate. If
, then
x must correspond to a multiple zero of
rather than a simple zero.
10 Example
This example calculates an approximation to the zero of using a tolerance of starting from and using an initial search step .
10.1 Program Text
Program Text (c05auce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (c05auce.r)