Alternating Series Acceleration
Computes the value of an (infinite) alternating sum applying an acceleration method found by Cohen et al.
sumalt(f_alt, n)
f_alt |
a funktion of |
n |
number of elements of the series used for calculating. |
Computes the sum of an alternating series (whose entries are strictly decreasing), applying the acceleration method developped by H. Cohen, F. Rodriguez Villegas, and Don Zagier.
For example, to compute the Leibniz series (see below) to 15 digits
exactly, 10^15 summands of the series will be needed. This
accelleration approach here will only need about 20 of them!
Returns an approximation of the series value.
Implemented by Hans W Borchers.
Henri Cohen, F. Rodriguez Villegas, and Don Zagier. Convergence Acceleration of Alternating Series. Experimental Mathematics, Vol. 9 (2000).
# Beispiel: Leibniz-Reihe 1 - 1/3 + 1/5 - 1/7 +- ...
a_pi4 <- function(k) (-1)^k / (2*k + 1)
sumalt(a_pi4, 20) # 0.7853981633974484 = pi/4 + eps()
# Beispiel: Van Wijngaarden transform needs 60 terms
n <- 60; N <- 0:n
a <- cumsum((-1)^N / (2*N+1))
for (i in 1:n) {
a <- (a[1:(n-i+1)] + a[2:(n-i+2)]) / 2
}
a - pi/4 # 0.7853981633974483
# Beispiel: 1 - 1/2^2 + 1/3^2 - 1/4^2 +- ...
b_alt <- function(k) (-1)^k / (k+1)^2
sumalt(b_alt, 20) # 0.8224670334241133 = pi^2/12 + eps()
## Not run:
# Dirichlet eta() function: eta(s) = 1/1^s - 1/2^s + 1/3^s -+ ...
eta_ <- function(s) {
eta_alt <- function(k) (-1)^k / (k+1)^s
sumalt(eta_alt, 30)
}
eta_(1) # 0.6931471805599453 = log(2)
abs(eta_(1+1i) - eta(1+1i)) # 1.24e-16
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