Alternating Series Acceleration

Computes the value of an (infinite) alternating sum applying an acceleration method found by Cohen et al.

sumalt(f_alt, n)

Arguments

f_alt

a funktion of k=0..Inf that returns element a_k of the infinite alternating series.

n

number of elements of the series used for calculating.

Details

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!

Value

Returns an approximation of the series value.

References

Henri Cohen, F. Rodriguez Villegas, and Don Zagier. Convergence Acceleration of Alternating Series. Experimental Mathematics, Vol. 9 (2000).

Author

Implemented by Hans W Borchers.

See also

aitken

Examples

# 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()
#> [1] 0.7853982

# 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
#> [1] 0

# 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()
#> [1] 0.822467

if (FALSE) { # \dontrun{
# 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
} # }