NAG Library Routine Document
C05AYF locates a simple zero of a continuous function in a given interval using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection.
||A, B, EPS, ETA, F, X, RUSER(*)
C05AYF attempts to obtain an approximation to a simple zero of the function
given an initial interval
The same core algorithm is used by C05AZF
whose specification should be consulted for details of the method used.
to the zero
is determined so that at least one of the following criteria is satisfied:
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
- 1: A – REAL (KIND=nag_wp)Input
On entry: , the lower bound of the interval.
- 2: B – REAL (KIND=nag_wp)Input
On entry: , the upper bound of the interval.
- 3: EPS – REAL (KIND=nag_wp)Input
: the termination tolerance on
(see Section 3
- 4: ETA – REAL (KIND=nag_wp)Input
: a value such that if
is accepted as the zero. ETA
may be specified as
(see Section 7
- 5: F – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
must evaluate the function
whose zero is to be determined.
The specification of F
- 1: X – REAL (KIND=nag_wp)Input
On entry: the point at which the function must be evaluated.
- 2: IUSER() – INTEGER arrayUser Workspace
- 3: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
is called with the parameters IUSER
as supplied to C05AYF. You are free to use the arrays IUSER
to supply information to F
as an alternative to using COMMON global variables.
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05AYF is called. Parameters denoted as Input
be changed by this procedure.
- 6: X – REAL (KIND=nag_wp)Output
is the final approximation to the zero. If
is likely to be a pole of
. Otherwise, X
contains no useful information.
- 7: IUSER() – INTEGER arrayUser Workspace
- 8: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
are not used by C05AYF, but are passed directly to F
and may be used to pass information to this routine as an alternative to using COMMON global variables.
- 9: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
Too much accuracy has been requested in the computation; that is, the zero has been located to relative accuracy at least
is the machine precision
, but the exit conditions described in Section 3
are not satisfied. It is unsafe for C05AYF to continue beyond this point, but the final value of X
returned is an accurate approximation to the zero.
A change in sign of
has been determined as occurring near the point defined by the final value of X
. However, there is some evidence that this sign-change corresponds to a pole of
The levels of accuracy depend on the values of EPS
. If full machine accuracy is required, they may be set very small, resulting in an exit with
, 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.
The time taken by C05AYF depends primarily on the time spent evaluating F
(see Section 5
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 C05AZF
is recommended. Use of C05AZF
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 C05AZF
are more flexible than the direct communication of F
required by C05AYF.
This example calculates an approximation to the zero of within the interval using a tolerance of .
9.1 Program Text
Program Text (c05ayfe.f90)
9.2 Program Data
9.3 Program Results
Program Results (c05ayfe.r)