{- 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@] * Provides implementations of the class 'Math.Factorial.Algorithmic'. * Provides additional functions related to /factorials/, but which depends on a specific implementation, and which therefore can't be accessed throught the class-interface. * <http://en.wikipedia.org/wiki/Factorial>. * <http://mathworld.wolfram.com/Factorial.html>. * <http://www.luschny.de/math/factorial/FastFactorialFunctions.htm>. -} module Factory.Math.Implementations.Factorial( -- * Types -- ** Data-types Algorithm(..), -- * Functions primeFactors, -- primeMultiplicity, risingFactorial, fallingFactorial, -- ** Operators (!/!) ) where import qualified Data.Numbers.Primes import qualified Factory.Data.Interval as Data.Interval import qualified Factory.Data.PrimeFactors as Data.PrimeFactors import qualified Factory.Math.Factorial as Math.Factorial import qualified ToolShed.Defaultable infixl 7 !/! --Same as (/). -- | The algorithms by which /factorial/ has been implemented. data Algorithm = Bisection -- ^ The integers from which the /factorial/ is composed, are multiplied using @Data.Interval.product'@. | PrimeFactorisation -- ^ The /prime factors/ of the /factorial/ are extracted, then raised to the appropriate power, before multiplication. deriving (Eq, Read, Show) instance ToolShed.Defaultable.Defaultable Algorithm where defaultValue = Bisection instance Math.Factorial.Algorithmic Algorithm where factorial algorithm n | n < 2 = 1 | otherwise = case algorithm of Bisection -> risingFactorial 2 $ pred n PrimeFactorisation -> Data.PrimeFactors.product' (recip 5) {-empirical-} 10 {-empirical-} $ primeFactors n {- | * Returns the /prime factors/, of the /factorial/ of the specifed integer. * Precisely all the primes less than or equal to the specified integer /n/, are included in /n!/; only the multiplicity of each of these known prime components need be determined. * <http://en.wikipedia.org/wiki/Factorial#Number_theory>. * CAVEAT: currently a hotspot. -} primeFactors :: Integral base => base -- ^ The number, whose /factorial/ is to be factorised. -> Data.PrimeFactors.Factors base base -- ^ The /base/ and /exponent/ of each /prime factor/ in the /factorial/, ordered by increasing /base/ (and decreasing /exponent/). primeFactors n = takeWhile ((> 0) . snd) $ map (\prime -> (prime, primeMultiplicity prime n)) Data.Numbers.Primes.primes {- | * The number of times a specific /prime/, can be factored from the /factorial/ of the specified integer. * General purpose /prime-factorisation/ has /exponential time-complexity/, so use /Legendre's Theorem/, which relates only to the /prime factors/ of /factorials/. * <http://www.proofwiki.org/wiki/Multiplicity_of_Prime_Factor_in_Factorial>. -} primeMultiplicity :: Integral i => i -- ^ A prime number. -> i -- ^ The integer, the factorial of which the prime is a factor. -> i -- ^ The number of times the prime occurs in the factorial. primeMultiplicity prime = sum . takeWhile (> 0) . tail . iterate (`div` prime) -- | Returns the /rising factorial/; <http://mathworld.wolfram.com/RisingFactorial.html> risingFactorial :: (Integral i, Show i) => i -- ^ The lower bound of the integer-range, whose product is returned. -> i -- ^ The number of integers in the range above. -> i -- ^ The result. risingFactorial _ 0 = 1 risingFactorial 0 _ = 0 risingFactorial x n = Data.Interval.product' (recip 2) 64 $ Data.Interval.normalise (x, pred $ x + n) -- | Returns the /falling factorial/; <http://mathworld.wolfram.com/FallingFactorial.html> fallingFactorial :: (Integral i, Show i) => i -- ^ The upper bound of the integer-range, whose product is returned. -> i -- ^ The number of integers in the range beneath. -> i -- ^ The result. fallingFactorial _ 0 = 1 fallingFactorial 0 _ = 0 fallingFactorial x n = Data.Interval.product' (recip 2) 64 $ Data.Interval.normalise (x, succ $ x - n) {- | * Returns the ratio of two factorials. * It is more efficient than evaluating both factorials, and then dividing. * For more complex combinations of factorials, such as in the /Binomial coefficient/, extract the /prime factors/ using 'primeFactors' then manipulate them using the module "Data.PrimeFactors", and evaluate it using by /Data.PrimeFactors.product'/. -} (!/!) :: (Integral i, Fractional f, Show i) => i -- ^ The /numerator/. -> i -- ^ The /denominator/. -> f -- ^ The resulting fraction. numerator !/! denominator | numerator <= 1 = recip . fromIntegral $ Math.Factorial.factorial (ToolShed.Defaultable.defaultValue :: Algorithm) denominator | denominator <= 1 = fromIntegral $ Math.Factorial.factorial (ToolShed.Defaultable.defaultValue :: Algorithm) numerator | numerator == denominator = 1 | numerator < denominator = recip $ denominator !/! numerator --Recurse. | otherwise = fromIntegral $ Data.Interval.product' (recip 2) 64 (succ denominator, numerator)