1Function: sumnumapinit 2Section: sums 3C-Name: sumnumapinit 4Prototype: DGp 5Help: sumnumapinit({asymp}): initialize tables for Abel-Plana 6 summation of a series. 7Doc: initialize tables for Abel--Plana summation of a series $\sum f(n)$, 8 where $f$ is holomorphic in a right half-plane. 9 If given, \kbd{asymp} is of the form $[\kbd{+oo}, \alpha]$, 10 as in \tet{intnum} and indicates the decrease rate at infinity of functions 11 to be summed. A positive 12 $\alpha > 0$ encodes an exponential decrease of type $\exp(-\alpha n)$ and 13 a negative $-2 < \alpha < -1$ encodes a slow polynomial decrease of type 14 $n^{\alpha}$. 15 \bprog 16 ? \p200 17 ? sumnumap(n=1, n^-2); 18 time = 163 ms. 19 ? tab = sumnumapinit(); 20 time = 160 ms. 21 ? sumnumap(n=1, n^-2, tab); \\ faster 22 time = 7 ms. 23 24 ? tab = sumnumapinit([+oo, log(2)]); \\ decrease like 2^-n 25 time = 164 ms. 26 ? sumnumap(n=1, 2^-n, tab) - 1 27 time = 36 ms. 28 %5 = 3.0127431466707723218 E-282 29 30 ? tab = sumnumapinit([+oo, -4/3]); \\ decrease like n^(-4/3) 31 time = 166 ms. 32 ? sumnumap(n=1, n^(-4/3), tab); 33 time = 181 ms. 34 @eprog 35