{- Copyright (C) 2011 Dr. Alistair Ward This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see <http://www.gnu.org/licenses/>. -} {- | [@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. -} module Factory.Math.Implementations.Pi.AGM.BrentSalamin( -- * Functions openR ) where import Control.Arrow((&&&)) import qualified Factory.Math.ArithmeticGeometricMean as Math.ArithmeticGeometricMean import qualified Factory.Math.Power as Math.Power import qualified Factory.Math.Precision as Math.Precision import qualified Factory.Math.SquareRoot as Math.SquareRoot {- | * 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)]) -} openR :: Math.SquareRoot.Algorithmic squareRootAlgorithm => squareRootAlgorithm -> Math.Precision.DecimalDigits -> Rational openR squareRootAlgorithm decimalDigits = uncurry (/) . ( Math.Power.square . uncurry (+) . last &&& negate . pred . sum . zipWith (*) (iterate (* 2) 1) . map (Math.Power.square . Math.ArithmeticGeometricMean.spread) ) . take ( Math.Precision.getIterationsRequired Math.Precision.quadraticConvergence 1 decimalDigits ) $ Math.ArithmeticGeometricMean.convergeToAGM squareRootAlgorithm decimalDigits (1, Math.SquareRoot.squareRoot squareRootAlgorithm decimalDigits (recip 2 :: Rational))