NAG Library Routine Document
S14BAF
1 Purpose
S14BAF computes values for the incomplete gamma functions and .
2 Specification
INTEGER |
IFAIL |
REAL (KIND=nag_wp) |
A, X, TOL, P, Q |
|
3 Description
S14BAF evaluates the incomplete gamma functions in the normalized form
with
and
, to a user-specified accuracy. With this normalization,
.
Several methods are used to evaluate the functions depending on the arguments
and
, the methods including Taylor expansion for
, Legendre's continued fraction for
, and power series for
. When both
and
are large, and
, the uniform asymptotic expansion of
Temme (1987) is employed for greater efficiency – specifically, this expansion is used when
and
.
Once either or is computed, the other is obtained by subtraction from . In order to avoid loss of relative precision in this subtraction, the smaller of and is computed first.
This routine is derived from the subroutine GAM in
Gautschi (1979b).
4 References
Gautschi W (1979a) A computational procedure for incomplete gamma functions ACM Trans. Math. Software 5 466–481
Gautschi W (1979b) Algorithm 542: Incomplete gamma functions ACM Trans. Math. Software 5 482–489
Temme N M (1987) On the computation of the incomplete gamma functions for large values of the parameters Algorithms for Approximation (eds J C Mason and M G Cox) Oxford University Press
5 Parameters
- 1: A – REAL (KIND=nag_wp)Input
On entry: the argument of the functions.
Constraint:
.
- 2: X – REAL (KIND=nag_wp)Input
On entry: the argument of the functions.
Constraint:
.
- 3: TOL – REAL (KIND=nag_wp)Input
On entry: the relative accuracy required by you in the results. If S14BAF is entered with
TOL greater than
or less than
machine precision, then the value of
machine precision is used instead.
- 4: P – REAL (KIND=nag_wp)Output
- 5: Q – REAL (KIND=nag_wp)Output
On exit: the values of the functions and respectively.
- 6: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL 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.
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
or
, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
Convergence of the Taylor series or Legendre continued fraction fails within
iterations. This error is extremely unlikely to occur; if it does, contact
NAG.
7 Accuracy
There are rare occasions when the relative accuracy attained is somewhat less than that specified by parameter
TOL. However, the error should never exceed more than one or two decimal places. Note also that there is a limit of
decimal places on the achievable accuracy, because constants in the routine are given to this precision.
The time taken for a call of S14BAF depends on the precision requested through
TOL, and also varies slightly with the input arguments
and
.
9 Example
This example reads values of the argument and from a file, evaluates the function and prints the results.
9.1 Program Text
Program Text (s14bafe.f90)
9.2 Program Data
Program Data (s14bafe.d)
9.3 Program Results
Program Results (s14bafe.r)