NAG Library Function Document
nag_polygamma_deriv (s14adc) returns a sequence of values of scaled derivatives of the psi function (also known as the digamma function).
||nag_polygamma_deriv (double x,
nag_polygamma_deriv (s14adc) computes
values of the function
is the psi function
th derivative of
The function is derived from the function PSIFN in Amos (1983)
. The basic method of evaluation of
is the asymptotic series
greater than a machine-dependent value
, followed by backward recurrence using
for smaller values of
, are the Bernoulli numbers.
is large, the above procedure may be inefficient, and the expansion
which converges rapidly for large
, is used instead.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software 9 494–502
x – doubleInput
On entry: the argument of the function.
n – IntegerInput
On entry: the index of the first member of the sequence of functions.
m – IntegerInput
the number of members required in the sequence , for .
ans[m] – doubleOutput
: the first
elements of ans
contain the required values
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, .
On entry, .
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
There is not enough internal workspace to continue computation. m
is probably too large.
Computation abandoned due to the likelihood of overflow.
On entry, .
Computation abandoned due to the likelihood of underflow.
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.
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).
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 (s14adce.c)
9.2 Program Data
Program Data (s14adce.d)
9.3 Program Results
Program Results (s14adce.r)