| Safe Haskell | None |
|---|---|
| Language | Haskell98 |
Factory.Math.Implementations.Pi.AGM.BrentSalamin
Contents
Description
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 :: Algorithmic squareRootAlgorithm => squareRootAlgorithm -> DecimalDigits -> Rational
Functions
openR :: Algorithmic squareRootAlgorithm => squareRootAlgorithm -> DecimalDigits -> Rational Source
- Returns Pi, accurate to the specified number of decimal digits.
- This algorithm is based on the arithmetic-geometric mean of
1and(1 / sqrt 2), but there are many confusingly similar formulations. The algorithm I've used here, whereais the arithmetic mean andgis 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)])