s14abf calculates an approximate value for . It is based on rational Chebyshev expansions.
Denote by a ratio of polynomials of degree in the numerator and in the denominator. Then:
and for ,
For each expansion, the specific values of and are selected to be minimal such that the maximum relative error in the expansion is of the order , where is the maximum number of decimal digits that can be accurately represented for the particular implementation (see x02bef).
Let denote machine precision and let denote the largest positive model number (see x02alf). For the value is not defined; s14abf returns zero and exits with . It also exits with when , and in this case the value is returned. For in the interval , the function to machine accuracy.
Now denote by the largest allowable argument for on the machine. For the term in Equation (1) is negligible. For there is a danger of setting overflow, and so s14abf exits with and returns . The value of is given in the Users' Note for your implementation.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Cody W J and Hillstrom K E (1967) Chebyshev approximations for the natural logarithm of the gamma function Math.Comp.21 198–203
1: – Real (Kind=nag_wp)Input
On entry: the argument of the function.
2: – IntegerInput/Output
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).
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:
On entry, . If the function is undefined; on soft failure, the function value returned is zero. If and soft failure is selected, the function value returned is the largest machine number (see x02alf).
On entry, (see Section 3). On soft failure, the function value returned is the largest machine number (see x02alf).
An unexpected error has been triggered by this routine. Please
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
Let and be the relative errors in the argument and result respectively, and be the absolute error in the result.
If is somewhat larger than machine precision, then
where is the digamma function . Figure 1 and Figure 2 show the behaviour of these error amplification factors.
These show that relative error can be controlled, since except near relative error is attenuated by the function or at least is not greatly amplified.
For large , and for small , .
The function has zeros at and and hence relative accuracy is not maintainable near those points. However absolute accuracy can still be provided near those zeros as is shown above.
If however, is of the order of machine precision, then rounding errors in the routine's internal arithmetic may result in errors which are slightly larger than those predicted by the equalities. It should be noted that even in areas where strong attenuation of errors is predicted the relative precision is bounded by the effective machine precision.
Parallelism and Performance
s14abf is not threaded in any implementation.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.