pracma (version 0.7.5)

agm: Arithmetic-geometric Mean

Description

The arithmetic-geometric mean of positive numbers.

Usage

agm(a, b, maxiter = 25, tol = .Machine$double.eps^(1/2))

Arguments

a, b

Positive numbers.

maxiter

Maximum number of iterations.

tol

tolerance; stops when abs(a-b) < tol.

Value

• Returnes one value, the mean of the last two values a, b.

Details

The arithmetic-geometric mean is defined as the common limit of the two sequences $a_{n+1} = (a_n + b_n)/2$ and $b_{n+1} = \sqrt(a_n b_n)$.

References

Borwein, J. M., and P. B. Borwein (1998). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Second, reprinted Edition, A Wiley-interscience publication.

See Also

Arithmetic, geometric, and harmonic mean.

Examples

##  Gauss constant
1 / agm(1, sqrt(2), tol = 1e-15)$agm  # 0.834626841674073

##  Calculate the (elliptic) integral 2/pi \int_0^1 dt / sqrt(1 - t^4)
f <- function(t) 1 / sqrt(1-t^4)
2 / pi * integrate(f, 0, 1)$value
1 / agm(1, sqrt(2))$agm

##  Calculate Pi with quadratic convergence (modified AGM)
#   See algorithm 2.1 in Borwein and Borwein
y <- sqrt(sqrt(2))
x <- (y+1/y)/2
p <- 2+sqrt(2)
for (i in 1:6){
  cat(format(p, digits=16), "")
  p <- p * (1+x) / (1+y)
  s <- sqrt(x)
  y <- (y*s + 1/s) / (1+y)
  x <- (s+1/s)/2
  }