nag_polygamma_deriv (s14adc) returns a sequence of values of scaled derivatives of the psi function .

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.

"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 consult NAG for assistance.

"NE_INTERNAL_WORKSPACE"

 There is not enough internal workspace to continue computation. m is probably too large.

"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.

All constants in nag_polygamma_deriv (s14adc) are given to approximately 18 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 1 to evaluate the function , for , rather than to make  separate calls of nag_polygamma_deriv (s14adc).