NAG CL Interface
s14adc (polygamma_deriv)
1
Purpose
s14adc returns a sequence of values of scaled derivatives of the psi function (also known as the digamma function).
2
Specification
| void |
s14adc (double x,
Integer n,
Integer m,
double ans[],
NagError *fail) |
|
The function may be called by the names: s14adc, nag_specfun_polygamma_deriv or nag_polygamma_deriv.
3
Description
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:
– double
Input
-
On entry: the argument of the function.
Constraint:
.
-
2:
– Integer
Input
-
On entry: the index of the first member of the sequence of functions.
Constraint:
.
-
3:
– Integer
Input
-
On entry: the number of members required in the sequence
, for .
Constraint:
.
-
4:
– double
Output
-
On exit: the first
elements of
ans contain the required values
, for
.
-
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: .
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_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 8 in the Introduction to the NAG Library CL Interface 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 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 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
s14adc is not threaded in any implementation.
The time taken for a call of s14adc is approximately proportional to , plus a constant. In general, it is much cheaper to call s14adc with greater than to evaluate the function , for , rather than to make separate calls of 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
10.2
Program Data
10.3
Program Results