The routine may be called by the names s14apf or nagf_specfun_gamma_log_real_vector.
s14apf calculates an approximate value for for an array of arguments
, 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; s14apf 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 s14apf exits with 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
1: – IntegerInput
On entry: , the number of points.
2: – Real (Kind=nag_wp) arrayInput
On entry: the argument of the function, for .
, for .
3: – Real (Kind=nag_wp) arrayOutput
On exit: , the function values.
4: – Integer arrayOutput
On exit: contains the error code for , for .
is too large and positive. The threshold value is the same as for in s14abf
, as defined in the the Users' Note for your implementation.
5: – 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, at least one value of x was invalid.
Check ivalid for more information.
On entry, .
An unexpected error has been triggered by this routine. Please
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.
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 or 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.
8Parallelism and Performance
s14apf is not threaded in any implementation.
This example reads values of x from a file, evaluates the function at each value of and prints the results.