The routine may be called by the names s14baf or nagf_specfun_gamma_incomplete.
3Description
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).
4References
Gautschi W (1979a) A computational procedure for incomplete gamma functions ACM Trans. Math. Software5 466–481
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
5Arguments
1: – Real (Kind=nag_wp)Input
On entry: the argument of the functions.
Constraint:
.
2: – Real (Kind=nag_wp)Input
On entry: the argument of the functions.
Constraint:
.
3: – 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: – Real (Kind=nag_wp)Output
5: – Real (Kind=nag_wp)Output
On exit: the values of the functions and respectively.
6: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or 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).
6Error 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:
On entry, .
Constraint: .
On entry, .
Constraint: .
Algorithm fails to terminate in iterations.
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
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 routine are given to this precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s14baf is not threaded in any implementation.
9Further Comments
The time taken for a call of s14baf depends on the precision requested through tol, and also varies slightly with the input arguments and .
10Example
This example reads values of the argument and from a file, evaluates the function and prints the results.