NAG CL Interface
s14apc (gamma_log_real_vector)
1
Purpose
s14apc returns an array of values of the logarithm of the gamma function, .
2
Specification
void |
s14apc (Integer n,
const double x[],
double f[],
Integer ivalid[],
NagError *fail) |
|
The function may be called by the names: s14apc, nag_specfun_gamma_log_real_vector or nag_log_gamma_vector.
3
Description
s14apc 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:
- for ,
- for ,
- for ,
- for ,
- 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
X02BEC).
Let
denote
machine precision and let
denote the largest positive model number (see
X02ALC). For
the value
is not defined;
s14apc returns zero and exits with
NW_IVALID. It also exits with
NW_IVALID 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
s14apc exits with
NE_INT and returns
. The value of
is given in the
Users' Note for your implementation.
4
References
Cody W J and Hillstrom K E (1967) Chebyshev approximations for the natural logarithm of the gamma function Math.Comp. 21 198–203
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of points.
Constraint:
.
-
2:
– const double
Input
-
On entry:
the argument of the function, for .
Constraint:
, for .
-
3:
– double
Output
-
On exit: , the function values.
-
4:
– Integer
Output
-
On exit:
contains the error code for
, for
.
-
No error.
-
.
-
is too large and positive. The threshold value is the same as for NE_REAL_ARG_GT in s14abc, as defined in the Users' Note for your implementation.
-
5:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NW_IVALID
-
On entry, at least one value of
x was invalid.
Check
ivalid for more information.
7
Accuracy
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.
8
Parallelism and Performance
s14apc is not threaded in any implementation.
None.
10
Example
This example reads values of
x from a file, evaluates the function at each value of
and prints the results.
10.1
Program Text
10.2
Program Data
10.3
Program Results