cyclotomic (empty) → 0.1

raw patch · 4 files changed

+1074/−0 lines, 4 filesdep +arithmoidep +basedep +containerssetup-changed

Dependencies added: arithmoi, base, containers

Files

• LICENSE +674/−0
• Setup.hs +2/−0
• cyclotomic.cabal +26/−0
• src/Data/Complex/Cyclotomic.hs +372/−0

@@ -0,0 +1,674 @@+                    GNU GENERAL PUBLIC LICENSE+                       Version 3, 29 June 2007++ Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>+ Everyone is permitted to copy and distribute verbatim copies+ of this license document, but changing it is not allowed.++                            Preamble++  The GNU General Public License is a free, copyleft license for+software and other kinds of works.++  The licenses for most software and other practical works are designed+to take away your freedom to share and change the works.  By contrast,+the GNU General Public License is intended to guarantee your freedom to+share and change all versions of a program--to make sure it remains free+software for all its users.  We, the Free Software Foundation, use the+GNU General Public License for most of our software; it applies also to+any other work released this way by its authors.  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@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain

@@ -0,0 +1,26 @@+Name:                cyclotomic+Version:             0.1+Synopsis:            A subfield of the complex numbers for exact calculation+Description:         The cyclotomic numbers are a subset of the+                     complex numbers with a number of nice properties.+                     They are represented exactly, enabling exact+                     computations and equality comparisons.  They+                     contain the Gaussian rationals (complex numbers+                     of the form p + q i with p and q rational).  The+                     cyclotomic numbers contain the square roots of+                     all rational numbers.  They contain the sine and+                     cosine of all rational multiples of pi.+License:             GPL-3+License-file:        LICENSE+Author:              Scott N. Walck+Maintainer:          Scott N. Walck <walck@lvc.edu>+Category:            Math+Build-type:          Simple+Cabal-version:       >=1.6+Tested-with:         GHC == 7.4.1+Library+  Exposed-modules:     Data.Complex.Cyclotomic+  Build-depends:       base >= 4.2 && < 4.6,+                       containers >= 0.3 && < 0.5,+                       arithmoi >= 0.4 && < 0.5+  Hs-source-dirs:      src

@@ -0,0 +1,372 @@+{-# OPTIONS_GHC -Wall #-}++-- | The cyclotomic numbers are a subset of the complex numbers with+--   the following properties:+--+--   1.  The cyclotomic numbers are represented exactly, enabling exact+--   computations and equality comparisons.+--+--   2.  The cyclotomic numbers contain the Gaussian rationals+--   (complex numbers of the form 'p' + 'q' 'i' with 'p' and 'q' rational).+--   As a consequence, the cyclotomic numbers are a dense subset of the+--   complex numbers.+--+--   3.  The cyclotomic numbers contain the square roots of all rational numbers.+--+--   4.  The cyclotomic numbers form a field:  they are closed under addition, subtraction,+--   multiplication, and division.+--+--   5.  The cyclotomic numbers contain the sine and cosine of all rational+--   multiples of pi.+--+--   6.  The cyclotomic numbers can be thought of as the rational field extended+--   with 'n'th roots of unity for arbitrarily large integers 'n'.+--+--   This algorithm for cyclotomic numbers is adapted from code by+--   Martin Schoenert and Thomas Breuer in the GAP project <http://www.gap-system.org/> .+--   See in particular source files gap4r4\/src\/cyclotom.c and+--   gap4r4\/lib\/cyclotom.gi .++module Data.Complex.Cyclotomic+    (Cyclotomic+    ,i+    ,e+    ,sqrtInteger+    ,sqrtRat+    ,sinDeg+    ,cosDeg+    ,conj+    ,real+    ,imag+    ,modSq+    ,toComplex+    ,isReal+    ,isRational+    ,isGaussianRational+    ,toRat+    )+    where++import Data.List (nub)+import Data.Ratio+import Data.Complex+import qualified Data.Map as M+import Math.NumberTheory.Primes.Factorisation (factorise)++-- | A cyclotomic number.+data Cyclotomic = Cyclotomic { order  :: Integer+                             , coeffs :: M.Map Integer Rational+                             } deriving (Eq)++instance Num Cyclotomic where+    (+) = sumCyc+    (*) = prodCyc+    (-) c1 c2 = sumCyc c1 (aInvCyc c2)+    negate = aInvCyc+    abs = sqrtRat . modSq+    signum = error "signum not defined for cyclotomic numbers"+    fromInteger 0 = zeroCyc+    fromInteger n = Cyclotomic 1 (M.singleton 0 (fromIntegral n))++instance Fractional Cyclotomic where+    recip = invCyc+    fromRational r = Cyclotomic 1 (M.singleton 0 r)++-- | The primitive 'n'th root of unity.+--   For example, 'e'(4) = 'i' is the primitive 4th root of unity,+--   and 'e'(5) = exp(2*pi*i/5) is the primitive 5th root of unity.+--   In general, 'e' 'n' = exp(2*pi*i/'n').+e :: Integer -> Cyclotomic+e n+    | n < 1      = error "e requires a positive integer"+    | n == 1     = Cyclotomic 1 (M.singleton 0 1)+    | otherwise  = cyclotomic n $ convertToBase n (M.singleton 1 1)++instance Show Cyclotomic where+    show (Cyclotomic n mp)+        | mp == M.empty  = "0"+        | otherwise      = leadingTerm rat n ex ++ followingTerms n xs+        where ((ex,rat):xs) = M.toList mp++showBaseExp :: Integer -> Integer -> String+showBaseExp n 1  = "e(" ++ show n ++ ")"+showBaseExp n ex = "e(" ++ show n ++ ")^" ++ show ex++leadingTerm :: Rational -> Integer -> Integer -> String+leadingTerm r _ 0 = showRat r+leadingTerm r n ex+    | r == 1     = t+    | r == (-1)  = "-" ++ t+    | r > 0      = showRat r ++ "*" ++ t+    | r < 0      = "-" ++ showRat (abs r) ++ "*" ++ t+    | otherwise  = ""+    where t = showBaseExp n ex++followingTerms :: Integer -> [(Integer,Rational)] -> String+followingTerms _ [] = ""+followingTerms n ((ex,rat):xs) = followingTerm rat n ex ++ followingTerms n xs++followingTerm :: Rational -> Integer -> Integer -> String+followingTerm r n ex+    | r == 1     = " + " ++ t+    | r == (-1)  = " - " ++ t+    | r > 0      = " + " ++ showRat r ++ "*" ++ t+    | r < 0      = " - " ++ showRat (abs r) ++ "*" ++ t+    | otherwise  = ""+    where t = showBaseExp n ex++showRat :: Rational -> String+showRat r+    | d == 1     = show n+    | otherwise  = show n ++ "/" ++ show d+    where+      n = numerator r+      d = denominator r++-- GAP function EB from gap4r4/lib/cyclotom.gi+eb :: Integer -> Cyclotomic+eb n+    | n < 1           = error "eb needs a positive integer"+    | n `mod` 2 /= 1  = error "eb needs an odd integer"+    | n == 1          = zeroCyc+    | otherwise       = let en = e n+                        in sum [en^(k*k `mod` n) | k <- [1..(n-1) `div` 2]]++sqrt2 :: Cyclotomic+sqrt2 = e 8 - e 8 ^ (3 :: Int)++-- | The square root of an 'Integer'.+sqrtInteger :: Integer -> Cyclotomic+sqrtInteger n+    | n == 0     = zeroCyc+    | n < 0      = i * sqrtPositiveInteger (-n)+    | otherwise  = sqrtPositiveInteger n++sqrtPositiveInteger :: Integer -> Cyclotomic+sqrtPositiveInteger n+    | n < 1      = error "sqrtPositiveInteger needs a positive integer"+    | otherwise  = let factors = factorise n+                       factor = product [p^(m `div` 2) | (p,m) <- factors]+                       nn     = product [p^(m `mod` 2) | (p,m) <- factors]+                   in case nn `mod` 4 of+                        1 -> fromInteger factor * (2 * eb nn + 1)+                        2 -> fromInteger factor * sqrt2 * sqrtPositiveInteger (nn `div` 2)+                        3 -> fromInteger factor * (-i) * (2 * eb nn + 1)+                        _ -> fromInteger factor * 2 * sqrtPositiveInteger (nn `div` 4)++-- | The square root of a 'Rational' number.+sqrtRat :: Rational -> Cyclotomic+sqrtRat r = prodRatCyc (1 % fromInteger den) (sqrtInteger (numerator r * den))+    where+      den = denominator r++-- | The square root of -1.+i :: Cyclotomic+i = e 4++-- | Complex conjugate.+conj :: Cyclotomic -> Cyclotomic+conj (Cyclotomic n mp)+    = mkCyclotomic n (M.mapKeys (\k -> (n-k) `mod` n) mp)++-- | Real part of the cyclotomic number.+real :: Cyclotomic -> Cyclotomic+real z = (z + conj z) / 2++-- | Imaginary part of the cyclotomic number.+imag :: Cyclotomic -> Cyclotomic+imag z = (z - conj z) / (2*i)++-- | Modulus squared.+modSq :: Cyclotomic -> Rational+modSq z = case toRat (z * conj z) of+            Just msq -> msq+            Nothing  -> error $ "modSq:  tried z = " ++ show z++-- | Export as an inexact complex number.+toComplex :: Cyclotomic -> Complex Double+toComplex c = sum [fromRational r * en^p | (p,r) <- M.toList (coeffs c)]+    where en = exp (0 :+ 2*pi/n)+          n = fromIntegral (order c)++convertToBase :: Integer -> M.Map Integer Rational -> M.Map Integer Rational+convertToBase n mp = foldr (\(p,r) m -> replace n p r m) mp (extraneousPowers n)++removeZeros :: M.Map Integer Rational -> M.Map Integer Rational+removeZeros = M.filter (/= 0)++-- Corresponds to GAP implementation.+-- Expects that convertToBase has already been done.+cyclotomic :: Integer -> M.Map Integer Rational -> Cyclotomic+cyclotomic ord = tryReduce . tryRational . gcdReduce . Cyclotomic ord++mkCyclotomic :: Integer -> M.Map Integer Rational -> Cyclotomic+mkCyclotomic ord = cyclotomic ord . removeZeros . convertToBase ord++-- | Step 1 of cyclotomic is gcd reduction.+gcdReduce :: Cyclotomic -> Cyclotomic+gcdReduce cyc@(Cyclotomic n mp) = case gcdCyc cyc of+                                    1 -> cyc+                                    d -> Cyclotomic (n `div` d) (M.mapKeys (\k -> k `div` d) mp)++gcdCyc :: Cyclotomic -> Integer+gcdCyc (Cyclotomic n mp) = gcdList (n:M.keys mp)++-- | Step 2 of cyclotomic is reduction to a rational if possible.+tryRational :: Cyclotomic -> Cyclotomic+tryRational c+    | lenCyc c == fromIntegral phi && sqfree+        = case equalCoefficients c of+            Nothing -> c+            Just r  -> fromRational $ (-1)^(nrp `mod` 2)*r+    | otherwise+        = c+    where+      (phi,nrp,sqfree) = phiNrpSqfree (order c)++-- | Compute phi(n), the number of prime factors, and test if n is square-free.+--   We do these all together for efficiency, so we only call factorise once.+phiNrpSqfree :: Integer -> (Integer,Int,Bool)+phiNrpSqfree n = (phi,nrp,sqfree)+    where+      factors = factorise n+      phi = foldr (\p n' -> n' `div` p * (p-1)) n [p | (p,_) <- factors]+      nrp = length (factors)+      sqfree = all (<=1) [m | (_,m) <- factors]++equalCoefficients :: Cyclotomic -> Maybe Rational+equalCoefficients (Cyclotomic _ mp)+    = case ts of+        []    -> Nothing+        (x:_) -> case equal ts of+                   True  -> Just x+                   False -> Nothing+      where+        ts = M.elems mp++lenCyc :: Cyclotomic -> Int+lenCyc (Cyclotomic _ mp) = M.size $ removeZeros mp++-- | Step 3 of cyclotomic is base reduction+tryReduce :: Cyclotomic -> Cyclotomic+tryReduce c+    = foldr reduceByPrime c squareFreeOddFactors+      where+        squareFreeOddFactors = [p | (p,m) <- factorise (order c), p > 2, m <= 1]++reduceByPrime :: Integer -> Cyclotomic -> Cyclotomic+reduceByPrime p c@(Cyclotomic n _)+    = case sequence $ map (\r -> equalReplacements p r c) [0,p..n-p] of+        Just cfs -> Cyclotomic (n `div` p) $ removeZeros $ M.fromList $ zip [0..(n `div` p)-1] (map negate cfs)+        Nothing  -> c++equalReplacements :: Integer -> Integer -> Cyclotomic -> Maybe Rational+equalReplacements p r (Cyclotomic n mp)+    =  case [M.findWithDefault 0 k mp | k <- replacements n p r] of+         [] -> error "equalReplacements generated empty list"+         (x:xs) | equal (x:xs) -> Just x+         _ -> Nothing++replacements :: Integer -> Integer -> Integer -> [Integer]+replacements n p r = takeWhile (>= 0) [r-s,r-2*s..] ++ takeWhile (< n) [r+s,r+2*s..]+    where s = n `div` p++replace :: Integer -> Integer -> Integer -> M.Map Integer Rational -> M.Map Integer Rational+replace n p r mp = case M.lookup r mp of+                     Nothing  -> mp+                     Just rat -> foldr (\k m -> M.insertWith (+) k (-rat) m) (M.delete r mp) (replacements n p r)++includeMods :: Integer -> Integer -> Integer -> [Integer]+includeMods n q start = [start] ++ takeWhile (>= 0) [start-q,start-2*q..] ++ takeWhile (< n) [start+q,start+2*q..]++removeExps :: Integer -> Integer -> Integer -> [Integer]+removeExps n 2 q = concat $ map (includeMods n q) $ map ((n `div` q) *) [q `div` 2..q-1]+removeExps n p q = concat $ map (includeMods n q) $ map ((n `div` q) *) [-m..m]+    where m = (q `div` p - 1) `div` 2++pqPairs :: Integer -> [(Integer,Integer)]+pqPairs n = map (\(p,k) -> (p,p^k)) (factorise n)++extraneousPowers :: Integer -> [(Integer,Integer)]+extraneousPowers n+    | n < 1      = error "extraneousPowers needs a postive integer"+    | otherwise  = nub $ concat $ [[(p,r) | r <- removeExps n p q] | (p,q) <- pqPairs n]++-- | Sum of two cyclotomic numbers.+sumCyc :: Cyclotomic -> Cyclotomic -> Cyclotomic+sumCyc (Cyclotomic o1 map1) (Cyclotomic o2 map2)+    = let ord = lcm o1 o2+          m1 = ord `div` o1+          m2 = ord `div` o2+          map1' = M.mapKeys (m1*) map1+          map2' = M.mapKeys (m2*) map2+      in mkCyclotomic ord $ M.unionWith (+) map1' map2'++-- | Product of two cyclotomic numbers.+prodCyc :: Cyclotomic -> Cyclotomic -> Cyclotomic+prodCyc (Cyclotomic o1 m1) (Cyclotomic o2 m2)+    = mkCyclotomic ord $ M.fromListWith (+)+      [((o2*e1+o1*e2) `mod` ord,c1*c2) | (e1,c1) <- M.toList m1, (e2,c2) <- M.toList m2]+      where ord = o1 * o2++-- | Product of a rational number and a cyclotomic number.+prodRatCyc :: Rational -> Cyclotomic -> Cyclotomic+prodRatCyc 0 _                   = zeroCyc+prodRatCyc r (Cyclotomic ord mp) = Cyclotomic ord $ M.map (r*) mp++-- | Additive identity.+zeroCyc :: Cyclotomic+zeroCyc = Cyclotomic 1 (M.empty)++-- | Additive inverse.+aInvCyc :: Cyclotomic -> Cyclotomic+aInvCyc = prodRatCyc (-1)++-- | Multiplicative inverse.+invCyc :: Cyclotomic -> Cyclotomic+invCyc z = prodRatCyc (1 / modSq z) (conj z)++-- | Is the cyclotomic a real number?+isReal :: Cyclotomic -> Bool+isReal c = c == conj c++-- | Is the cyclotomic a rational?+isRational :: Cyclotomic -> Bool+isRational (Cyclotomic 1 _) = True+isRational _                = False++-- | Is the cyclotomic a Gaussian rational?+isGaussianRational :: Cyclotomic -> Bool+isGaussianRational c = isRational (real c) && isRational (imag c)++-- | Return Just rational if the cyclotomic is rational, Nothing otherwise.+toRat :: Cyclotomic -> Maybe Rational+toRat (Cyclotomic 1 mp)+    | mp == M.empty  = Just 0+    | otherwise      = M.lookup 0 mp+toRat _ = Nothing++-- | Sine function with argument in degrees.+sinDeg :: Rational -> Cyclotomic+sinDeg d = let n = d / 360+               nm = abs (numerator n)+               dn = denominator n+               a = e dn^nm+           in fromRational(signum d) * (a - conj a) / (2*i)++-- | Cosine function with argument in degrees.+cosDeg :: Rational -> Cyclotomic+cosDeg d = let n = d / 360+               nm = abs (numerator n)+               dn = denominator n+               a = e dn^nm+           in (a + conj a) / 2++gcdList :: [Integer] -> Integer+gcdList [] = error "gcdList called on empty list"+gcdList (n:ns) = foldr gcd n ns++equal :: Eq a => [a] -> Bool+equal [] = True+equal [_] = True+equal (x:y:ys) = x == y && equal (y:ys)