```{-
Copyright (C) 2011 Dr. Alistair Ward

This program is free software: you can redistribute it and/or modify
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	Data.Ratio
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 -> Data.Ratio.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 (