NAG C Library Function Document
nag_incomplete_gamma (s14bac)
1
Purpose
nag_incomplete_gamma (s14bac) computes values for the incomplete gamma functions and .
2
Specification
| #include <nag.h> |
| #include <nags.h> |
| void |
nag_incomplete_gamma (double a,
double x,
double tol,
double *p,
double *q,
NagError *fail) |
|
3
Description
nag_incomplete_gamma (s14bac) 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 function is derived from the function 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
Arguments
- 1:
– doubleInput
-
On entry: the argument of the functions.
Constraint:
.
- 2:
– doubleInput
-
On entry: the argument of the functions.
Constraint:
.
- 3:
– doubleInput
-
On entry: the relative accuracy required by you in the results. If
nag_incomplete_gamma (s14bac) is entered with
tol greater than
or less than
machine precision, then the value of
machine precision is used instead.
- 4:
– double *Output
- 5:
– double *Output
-
On exit: the values of the functions and respectively.
- 6:
– 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_ALG_NOT_CONV
-
Algorithm fails to terminate in iterations.
- 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_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_REAL_ARG_LE
-
On entry, .
Constraint: .
- NE_REAL_ARG_LT
-
On entry, .
Constraint: .
7
Accuracy
There are rare occasions when the relative accuracy attained is somewhat less than that specified by argument
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 function are given to this precision.
8
Parallelism and Performance
nag_incomplete_gamma (s14bac) is not threaded in any implementation.
The time taken for a call of
nag_incomplete_gamma (s14bac) depends on the precision requested through
tol, and also varies slightly with the input arguments
and
.
10
Example
This example reads values of the argument and from a file, evaluates the function and prints the results.
10.1
Program Text
Program Text (s14bace.c)
10.2
Program Data
Program Data (s14bace.d)
10.3
Program Results
Program Results (s14bace.r)