Enhanced Computation of Polylogarithm aka de Jonquieres function

Fast Computation of polylogarithmic function using closed form approximations

Uses closed form approximations to compute the polylogarithm Li_n(z) of a complex array z base n.

Description: % polylog - Computes the n-based polylogarithm of z: Li_n(z)

Approximate closed form expressions for the Polylogarithm aka de Jonquière's function are used. Computes reasonably faster than direct calculation given by SUM_{k=1 to Inf}[z^k / k^n] = z + z^2/2^n + ...

Usage: [y errors] = PolyLog(n,z)

Input: z < 1 : real/complex number or array or array

n > -4 : base of polylogarithm

Output: y ... value of polylogarithm

errors ... number of errors

Approximation should be correct up to at least 5 digits for |z| > 0.55
and on the order of 10 digits for |z| <= 0.55!

Please Note: z array input is possible but not recommended as precision might drop for big ranged z inputs (unresolved Matlab issue unknown to the author).

following V. Bhagat, et al., On the evaluation of generalized Bose–Einstein and Fermi–Dirac integrals, Computer Physics Communications, Vol. 155, p.7, 2003

v3 20120616