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