{- 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@] Exports functions involving integral powers. -} module Factory.Math.Power( -- * Functions square, squaresFrom, cube, cubeRoot, raiseModulo ) where -- | Mainly for convenience. square :: Num n => n -> n square x = x ^ (2 :: Int) -- CAVEAT: this could be eta-reduced, but it won't then inline when called with a single argument. {-# INLINE square #-} -- | Just for convenience. cube :: Num n => n -> n cube = (^ (3 :: Int)) {- | * Iteratively generate sequential /squares/, from the specified initial value, based on the fact that @(x + 1)^2 = x^2 + 2 * x + 1@. * The initial value doesn't need to be either positive or integral. -} squaresFrom :: (Enum n, Num n) => n -- ^ Lower bound. -> [(n, n)] -- ^ @ [(n, n^2)] @. squaresFrom from = iterate (\(x, y) -> (succ x, succ $ y + 2 * x)) (from, square from) -- | Just for convenience. cubeRoot :: Double -> Double cubeRoot = (** recip 3) {- | * Raise an arbitrary number to the specified positive integral power, using /modular/ arithmetic. * Implements exponentiation as a sequence of either /squares/ or multiplications by the base; <http://en.wikipedia.org/wiki/Exponentiation_by_squaring>. * <http://en.wikipedia.org/wiki/Modular_exponentiation>. -} raiseModulo :: (Integral i, Integral power, Show power) => i -- ^ Base. -> power -> i -- ^ Modulus. -> i -- ^ Result. raiseModulo _ _ 0 = error "Factory.Math.Power.raiseModulo:\tzero modulus." raiseModulo _ _ 1 = 0 raiseModulo _ 0 modulus = 1 `mod` modulus raiseModulo base power modulus | base < 0 = (`mod` modulus) . (if even power then id else negate) $ raiseModulo (negate base) power modulus -- Recurse. | power < 0 = error $ "Factory.Math.Power.raiseModulo:\tnegative power; " ++ show power | first `elem` [0, 1] = first | otherwise = slave power where first = base `mod` modulus slave 1 = first slave e = (`mod` modulus) . (if r == 0 {-even-} then id else (* base)) . square $ slave q {-recurse-} where (q, r) = e `quotRem` 2