NAG Library Function Document
nag_zero_cont_func_brent_rcomm (c05azc)
1
Purpose
nag_zero_cont_func_brent_rcomm (c05azc) locates a simple zero of a continuous function in a given interval by using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection. It uses reverse communication for evaluating the function.
2
Specification
#include <nag.h> |
#include <nagc05.h> |
void |
nag_zero_cont_func_brent_rcomm (double *x,
double *y,
double fx,
double tolx,
Nag_ErrorControl ir,
double c[],
Integer *ind,
NagError *fail) |
|
3
Description
You must supply
x and
y to define an initial interval
containing a simple zero of the function
(the choice of
x and
y
must be such that
). The function combines the methods of bisection, nonlinear interpolation and linear extrapolation (see
Dahlquist and Björck (1974)), to find a sequence of sub-intervals of the initial interval such that the final interval
contains the zero and
is less than some tolerance specified by
tolx and
ir (see
Section 5). In fact,
since the intermediate intervals
are determined only so that
, it is possible that the final interval may contain a discontinuity or a pole of
(violating the requirement that
be continuous).
nag_zero_cont_func_brent_rcomm (c05azc) checks if the sign change is likely to correspond to a pole of
and gives an error return in this case.
A feature of the algorithm used by this function is that unlike some other methods it guarantees convergence within about
function evaluations, where
is related to the argument
tolx. See
Brent (1973) for more details.
nag_zero_cont_func_brent_rcomm (c05azc) returns to the calling program for each evaluation of . On each return you should set and call nag_zero_cont_func_brent_rcomm (c05azc)
again.
The function is a modified version of procedure ‘zeroin’ given by
Brent (1973).
4
References
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
Bus J C P and Dekker T J (1975) Two efficient algorithms with guaranteed convergence for finding a zero of a function ACM Trans. Math. Software 1 330–345
Dahlquist G and Björck Å (1974) Numerical Methods Prentice–Hall
5
Arguments
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument
ind. Between intermediate exits and re-entries,
all arguments other than fx must remain unchanged.
- 1:
– double *Input/Output
- 2:
– double *Input/Output
-
On initial entry:
x and
y must define an initial interval
containing the zero, such that
. It is not necessary that
.
On intermediate exit:
x contains the point at which
must be evaluated before re-entry to the function.
On final exit:
x and
y define a smaller interval containing the zero, such that
, and
satisfies the accuracy specified by
tolx and
ir, unless an error has occurred. If
NE_PROBABLE_POLE,
x and
y generally contain very good approximations to a pole; if
NW_TOO_MUCH_ACC_REQUESTED,
x and
y generally contain very good approximations to the zero (see
Section 6). If a point
x is found such that
, on final exit
(in this case there is no guarantee that
x is a simple zero). In all cases, the value returned in
x is the better approximation to the zero.
- 3:
– doubleInput
-
On initial entry: if
,
fx need not be set.
If
,
fx must contain
for the initial value of
x.
On intermediate re-entry: must contain
for the current value of
x.
- 4:
– doubleInput
-
On initial entry: the accuracy to which the zero is required. The type of error test is specified by
ir.
Constraint:
.
- 5:
– Nag_ErrorControlInput
-
On initial entry: indicates the type of error test.
- The test is: .
- The test is: .
- The test is: .
Suggested value:
.
Constraint:
, or .
- 6:
– doubleInput/Output
-
On initial entry: if
, no elements of
c need be set.
If
,
must contain
, other elements of
c need not be set.
On final exit: is undefined.
- 7:
– Integer *Input/Output
-
On initial entry: must be set to
or
.
- fx and need not be set.
- fx and must contain and respectively.
On intermediate exit:
contains
,
or
. The calling program must evaluate
at
x, storing the result in
fx, and re-enter
nag_zero_cont_func_brent_rcomm (c05azc) with all other arguments unchanged.
On final exit: contains .
Constraint:
on entry , , , or .
Note: any values you return to nag_zero_cont_func_brent_rcomm (c05azc) as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_zero_cont_func_brent_rcomm (c05azc). If your code inadvertently does return any NaNs or infinities, nag_zero_cont_func_brent_rcomm (c05azc) is likely to produce unexpected results.
- 8:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: , , , or .
- 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_NOT_SIGN_CHANGE
-
On entry, and have the same sign with neither equalling : and .
- NE_PROBABLE_POLE
-
The final interval may contain a pole rather than a zero. Note that this error exit is not completely reliable: it may be taken in extreme cases when
contains a zero, or it may not be taken when
contains a pole. Both these cases occur most frequently when
tolx is large.
- NE_REAL
-
On entry, .
Constraint: .
- NW_TOO_MUCH_ACC_REQUESTED
-
The tolerance
tolx has been set too small for the problem being solved. However, the values
x and
y returned may well be good approximations to the zero.
.
7
Accuracy
The accuracy of the final value
x as an approximation of the zero is determined by
tolx and
ir (see
Section 5). A relative accuracy criterion (
) should not be used when the initial values
x and
y are of different orders of magnitude. In this case a change of origin of the independent variable may be appropriate. For example, if the initial interval
is transformed linearly to the interval
, then the zero can be determined to a precise number of figures using an absolute
(
) or relative (
) error test and the effect of the transformation back to the original interval can also be determined. Except for the accuracy check, such a transformation has no effect on the calculation of the zero.
8
Parallelism and Performance
nag_zero_cont_func_brent_rcomm (c05azc) is not threaded in any implementation.
For most problems, the time taken on each call to nag_zero_cont_func_brent_rcomm (c05azc) will be negligible compared with the time spent evaluating between calls to nag_zero_cont_func_brent_rcomm (c05azc).
If the calculation terminates because
,
then on return
y is set to
x. (In fact,
on return only in this case and, possibly, when
NW_TOO_MUCH_ACC_REQUESTED.) There is no guarantee that the value returned in
x corresponds to a
simple
root 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 a zero of with an initial interval , and a mixed error test.
10.1
Program Text
Program Text (c05azce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (c05azce.r)