NAG Library Routine Document
S14ABF returns the value of the logarithm of the gamma function, , via the function name.
|REAL (KIND=nag_wp) S14ABF
S14ABF calculates an approximate value for . It is based on rational Chebyshev expansions.
a ratio of polynomials of degree
in the numerator and
in the denominator. Then:
- for ,
- for ,
- for ,
- for ,
- and for ,
For each expansion, the specific values of
are selected to be minimal such that the maximum relative error in the expansion is of the order
is the maximum number of decimal digits that can be accurately represented for the particular implementation (see X02BEF
denote machine precision
denote the largest positive model number (see X02ALF
is not defined; S14ABF returns zero and exits with
. It also exits with
, 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
term in Equation (1)
is negligible. For
there is a danger of setting overflow, and so S14ABF exits with
. 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: X – REAL (KIND=nag_wp)Input
On entry: the argument of the function.
- 2: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction 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 parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
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
(see Section 3
). On soft failure, the function value returned is the largest machine number (see X02ALF
Let and be the relative errors in the argument and result respectively, and be the absolute error in the result.
is somewhat larger than machine precision
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.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
9.1 Program Text
Program Text (s14abfe.f90)
9.2 Program Data
Program Data (s14abfe.d)
9.3 Program Results
Program Results (s14abfe.r)