NAG C Library Function Document
nag_polygamma_deriv (s14adc)
1
Purpose
nag_polygamma_deriv (s14adc) returns a sequence of values of scaled derivatives of the psi function (also known as the digamma function).
2
Specification
| #include <nag.h> |
| #include <nags.h> |
| void |
nag_polygamma_deriv (double x,
Integer n,
Integer m,
double ans[],
NagError *fail) |
|
3
Description
nag_polygamma_deriv (s14adc) computes
values of the function
for
,
,
, where
is the psi function
and
denotes the
th derivative of
.
The function is derived from the function PSIFN in
Amos (1983). The basic method of evaluation of
is the asymptotic series
for large
greater than a machine-dependent value
, followed by backward recurrence using
for smaller values of
, where
when
,
when
, and
,
, are the Bernoulli numbers.
When
is large, the above procedure may be inefficient, and the expansion
which converges rapidly for large
, is used instead.
4
References
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software 9 494–502
5
Arguments
- 1:
– doubleInput
-
On entry: the argument of the function.
Constraint:
.
- 2:
– IntegerInput
-
On entry: the index of the first member of the sequence of functions.
Constraint:
.
- 3:
– IntegerInput
-
On entry: the number of members required in the sequence
, for .
Constraint:
.
- 4:
– doubleOutput
-
On exit: the first
elements of
ans contain the required values
, for
.
- 5:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_INTERNAL_WORKSPACE
-
There is not enough internal workspace to continue computation.
m is probably too large.
- NE_NO_LICENCE
-
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.
- NE_OVERFLOW_LIKELY
-
Computation abandoned due to the likelihood of overflow.
- NE_REAL
-
On entry, .
Constraint: .
- NE_UNDERFLOW_LIKELY
-
Computation abandoned due to the likelihood of underflow.
7
Accuracy
All constants in nag_polygamma_deriv (s14adc) are given to approximately digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used , then clearly the maximum number of correct digits in the results obtained is limited by . Empirical tests of nag_polygamma_deriv (s14adc), taking values of in the range , and in the range , have shown that the maximum relative error is a loss of approximately two decimal places of precision. Tests with , i.e., testing the function , have shown somewhat better accuracy, except at points close to the zero of , , where only absolute accuracy can be obtained.
8
Parallelism and Performance
nag_polygamma_deriv (s14adc) is not threaded in any implementation.
The time taken for a call of nag_polygamma_deriv (s14adc) is approximately proportional to , plus a constant. In general, it is much cheaper to call nag_polygamma_deriv (s14adc) with greater than to evaluate the function , for , rather than to make separate calls of nag_polygamma_deriv (s14adc).
10
Example
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
10.1
Program Text
Program Text (s14adce.c)
10.2
Program Data
Program Data (s14adce.d)
10.3
Program Results
Program Results (s14adce.r)