R: Alternating Series Acceleration

sumalt {pracma} | R Documentation

## Alternating Series Acceleration

### Description

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

### Usage

    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.

### Author(s)

Implemented by Hans W Borchers.

### References

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

### 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()

    # 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

    ## End(Not run)

[Package pracma version 2.0.4 Index]