nag_log_gamma (s14abc) 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 nag_decimal_digits (X02BEC)).
Let denote machine precision and let denote the largest positive model number (see nag_real_largest_number (X02ALC)). For the value is not defined; nag_log_gamma (s14abc) returns zero and exits with NE_REAL_ARG_LE. It also exits with NE_REAL_ARG_LE 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 nag_log_gamma (s14abc) exits with NE_REAL_ARG_GT and returns . The value of is given in the Users' Note for your implementation.
Cody W J and Hillstrom K E (1967) Chebyshev approximations for the natural logarithm of the gamma function Math.Comp.21 198–203
On entry: the argument of the function.
– NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 188.8.131.52 in How to Use the NAG Library and its Documentation for further information.
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.
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.
On entry, .
On entry, .
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 function'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
nag_log_gamma (s14abc) 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.