{-
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@] Exports functions involving integral powers.
-}
module Factory.Math.Power(
-- * Functions
square,
squaresFrom,
maybeSquareNumber,
cube,
cubeRoot,
raiseModulo,
-- ** Predicates
isPerfectPower
) where
import qualified Data.Set
-- | Mainly for convenience.
{-# INLINE square #-}
square :: Num n => n -> n
square = (^ (2 :: Int))
-- | 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 :: Num n => n -> [(n, n)]
squaresFrom from = iterate (\(x, y) -> (x + 1, y + 2 * x + 1)) (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;
.
* .
-}
raiseModulo :: (Integral i, Integral 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 odd power then negate else id) $ 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
{- |
* Returns @(Just . sqrt)@ if the specified integer is a /square number/ (AKA /perfect square/).
* .
* .
* @(square . sqrt)@ is expensive, so the modulus of the operand is tested first, in an attempt to prove it isn't a /perfect square/.
The set of tests, and the valid moduli within each test, are ordered to maximize the rate of failure-detection.
-}
maybeSquareNumber :: Integral i => i -> Maybe i
maybeSquareNumber i
-- | i < 0 = Nothing --This function is performance-sensitive, but this test is neither strictly nor frequently required.
| all (\(modulus, valid) -> mod i modulus `elem` valid) [
-- --Distribution of moduli amongst perfect squares Cumulative failure-detection.
(16, [0,1,4,9]), --All moduli are equally likely. 75%
(9, [0,1,4,7]), --Zero occurs 33%, the others only 22%. 88%
(17, [1,2,4,8,9,13,15,16,0]), --Zero only occurs 5.8%, the others 11.8%. 94%
-- These additional tests, aren't always cost-effective.
(13, [1,3,4,9,10,12,0]), --Zero only occurs 7.7%, the others 15.4%. 97%
(7, [1,2,4,0]), --Zero only occurs 14.3%, the others 28.6%. 98%
(5, [1,4,0]) --Zero only occurs 20%, the others 40%. 99%
-- ] && fromIntegral iSqrt == sqrt' = Just iSqrt --CAVEAT: erroneously True for 187598574531033120, whereas 187598574531033121 is square.
] && square iSqrt == i = Just iSqrt
| otherwise = Nothing
where
sqrt' :: Double
sqrt' = sqrt $ fromIntegral i
iSqrt = round sqrt'
{- |
* An integer @(> 1)@ which can be expressed as an integral power @(> 1)@ of a smaller /natural/ number.
* CAVEAT: /zero/ and /one/ are normally excluded from this set.
* .
* .
-}
isPerfectPower :: Integral i => i -> Bool
isPerfectPower i
| i < square 2 = False
| otherwise = i `Data.Set.member` foldr (
\n set -> if n `Data.Set.member` set
then set
-- else Data.Set.union set . Data.Set.fromList . takeWhile (<= i) . iterate (* n) $ square n --TODO: test relative speed.
else foldr Data.Set.insert set . takeWhile (<= i) . iterate (* n) $ square n
) Data.Set.empty [2 .. round $ sqrt (fromIntegral i :: Double)]