NAG Library Function Document

nag_polygamma_fun (s14acc)

is large, sums or differences of psi functions may be computed without unnecessary loss of precision, by analytically combining the logarithmic terms. For example, the difference

d = ψ (x + \frac{1}{2}) - ψ (x)

has an asymptotic behaviour for large

x

given by

d \sim \ln (x + \frac{1}{2}) - \ln x + O (\frac{1}{x^{2}}) \sim \ln (1 + \frac{1}{2 x}) \sim \frac{1}{2 x}

Computing

d

directly would amount to subtracting two large numbers which are close to

\ln (x + \frac{1}{2})

and

\ln x

to produce a small number close to

\frac{1}{2 x}

, resulting in a loss of significant digits. However, using this function to compute

f (x) = ψ (x) - \ln x

, we can compute

d = f (x + \frac{1}{2}) - f (x) + \ln (1 + \frac{1}{2 x})

, and the dominant logarithmic term may be computed accurately from its power series when

x

is large. Thus we avoid the unnecessary loss of precision.

The function is derived from the function PSIFN in Amos (1983).

4 References

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

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.
Constraint: $x > 0.0$ .
2: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
NE_OVERFLOW_LIKELY: Computation halted due to likelihood of overflow. x may be too small. $x = ⟨value⟩$ .
NE_REAL: On entry, $x = ⟨value⟩$ .
Constraint: $x > 0.0$ .
NE_UNDERFLOW_LIKELY: Computation halted due to likelihood of underflow. x may be too large. $x = ⟨value⟩$ .

7 Accuracy

All constants in nag_polygamma_fun (s14acc) are given to approximately

18

digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used

t

, then clearly the maximum number of correct digits in the results obtained is limited by

p = \min (t, 18)

With the above proviso, results returned by this function should be accurate almost to full precision, except at points close to the zero of

ψ (x)

x ≃ 1.461632

, where only absolute rather than relative accuracy can be obtained.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

The example program reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Function Documentnag_polygamma_fun (s14acc)

+− Contents

1 Purpose

2 Specification

3 Description