{-
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 .
-}
{- |
[@AUTHOR@] Dr. Alistair Ward
[@DESCRIPTION@]
* Implements the /Brent-Salamin/ (AKA /Gauss-Legendre/) algorithm;
,
,
.
* 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))