`AUTHOR`

- Dr. Alistair Ward
`DESCRIPTION`

- Implements the
*Brent-Salamin*(AKA*Gauss-Legendre*) algorithm; http://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_algorithm, http://mathworld.wolfram.com/Brent-SalaminFormula.html, http://www.pi314.net/eng/salamin.php. - The precision of the result approximately doubles for each iteration.

`CAVEAT`

- Assumptions on the convergence-rate result in rounding-errors, when only a small number of digits are requested.

- openR :: Algorithm squareRootAlgorithm => squareRootAlgorithm -> DecimalDigits -> Rational

# Functions

openR :: Algorithm squareRootAlgorithm => squareRootAlgorithm -> DecimalDigits -> RationalSource

- Returns
*Pi*, accurate to the specified number of decimal digits. - This algorithm is based on the
*arithmetic-geometric*mean of`1`

and`(1 / sqrt 2)`

, but there are many confusingly similar formulations. The algorithm I've used here, where`a`

is the*arithmetic mean*and`g`

is the*geometric mean*, is equivalent to other common formulations:

pi = (a[N-1] + g[N-1])^2 / (1 - sum [2^n * (a[n] - g[n])^2]) where n = [0 .. N-1] => 4*a[N]^2 / (1 - sum [2^n * (a[n]^2 - 2*a[n]*g[n] + g[n]^2)]) => 4*a[N]^2 / (1 - sum [2^n * (a[n]^2 + 2*a[n]*g[n] + g[n]^2 - 4*a[n]*g[n])]) => 4*a[N]^2 / (1 - sum [2^n * ((a[n] + g[n])^2 - 4*a[n]*g[n])]) => 4*a[N]^2 / (1 - sum [2^(n-1) * 4 * (a[n-1]^2 - g[n-1]^2)]) where n = [1 .. N] => 4*a[N]^2 / (1 - sum [2^(n+1) * (a[n-1]^2 - g[n-1]^2)])