{-# OPTIONS -fno-implicit-prelude -fglasgow-exts #-} {- | Polynomials and rational functions in a single indeterminate. Polynomials are represented by a list of coefficients. All non-zero coefficients are listed, but there may be extra '0's at the end. Usage: Say you have the ring of 'Integer' numbers and you want to add a transcendental element @x@, that is an element, which does not allow for simplifications. More precisely, for all positive integer exponents @n@ the power @x^n@ cannot be rewritten as a sum of powers with smaller exponents. The element @x@ must be represented by the polynomial @[0,1]@. In principle, you can have more than one transcendental element by using polynomials whose coefficients are polynomials as well. However, most algorithms on multi-variate polynomials prefer a different (sparse) representation, where the ordering of elements is not so fixed. If you want division, you need "Number.Ratio"s of polynomials with coefficients from a "Algebra.Field". You can also compute with an algebraic element, that is an element which satisfies an algebraic equation like @x^3-x-1==0@. Actually, powers of @x@ with exponents above @3@ can be simplified, since it holds @x^3==x+1@. You can perform these computations with "Number.ResidueClass" of polynomials, where the divisor is the polynomial equation that determines @x@. If the polynomial is irreducible (in our case @x^3-x-1@ cannot be written as a non-trivial product) then the residue classes also allow unrestricted division (except by zero, of course). That is, using residue classes of polynomials you can work with roots of polynomial equations without representing them by radicals (powers with fractional exponents). It is well-known, that roots of polynomials of degree above 4 may not be representable by radicals. -} module MathObj.Polynomial(T(..), fromCoeffs, showsExpressionPrec, const, eval, compose, equal, add, sub, negate, shift, unShift, mul, scale, divMod, tensorProduct, tensorProduct', mulShear, mulShearTranspose, horner, horner', progression, differentiate, integrate, integrateInt, fromRoots, alternate) where import qualified Algebra.Differential as Differential import qualified Algebra.VectorSpace as VectorSpace import qualified Algebra.Module as Module import qualified Algebra.Vector as Vector import qualified Algebra.Field as Field import qualified Algebra.PrincipalIdealDomain as PID import qualified Algebra.Units as Units import qualified Algebra.IntegralDomain as Integral import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import qualified Algebra.ZeroTestable as ZeroTestable import qualified Algebra.Indexable as Indexable import Algebra.Module((*>)) import Algebra.ZeroTestable(isZero) import Control.Monad (liftM) import qualified Data.List as List import NumericPrelude.List (zipWithOverlap, dropWhileRev, shear, shearTranspose, outerProduct) import Test.QuickCheck (Arbitrary(arbitrary,coarbitrary)) import qualified Prelude as P98 import qualified PreludeBase as P import qualified NumericPrelude as NP import PreludeBase hiding (const) import NumericPrelude hiding (divMod, negate, stdUnit) newtype T a = Cons {coeffs :: [a]} fromCoeffs :: [a] -> T a fromCoeffs = lift0 lift0 :: [a] -> T a lift0 = Cons lift1 :: ([a] -> [a]) -> (T a -> T a) lift1 f (Cons x0) = Cons (f x0) lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a) lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1) {- Functor instance is e.g. useful for showing polynomials in residue rings. @fmap (ResidueClass.concrete 7) (polynomial [1,4,4::ResidueClass.T Integer] * polynomial [1,5,6])@ -} instance Functor T where fmap f (Cons xs) = Cons (map f xs) plusPrec, appPrec :: Int plusPrec = 6 appPrec = 10 instance (Show a) => Show (T a) where showsPrec p (Cons xs) = showParen (p >= appPrec) (showString "Polynomial.fromCoeffs " . shows xs) showsExpressionPrec :: (Show a, ZeroTestable.C a, Additive.C a) => Int -> String -> T a -> String -> String showsExpressionPrec p var poly = if isZero poly then showString "0" else let terms = filter (not . isZero . fst) (zip (coeffs poly) monomials) monomials = id : showString "*" . showString var : map (\k -> showString "*" . showString var . showString "^" . shows k) [(2::Int)..] showsTerm x showsMon = showsPrec (plusPrec+1) x . showsMon in showParen (p > plusPrec) (foldl (.) id $ List.intersperse (showString " + ") $ map (uncurry showsTerm) terms) {- | Horner's scheme for evaluating an polynomial -} horner' :: Ring.C a => a -> [a] -> a horner' x = foldr (\c val -> c+x*val) zero {- ***** Module types are more general, but most times this flexibility is not needed and let type inference fail. -} horner :: Module.C a b => a -> [b] -> b horner x = foldr (\c val -> c+x*>val) zero eval :: Module.C a b => T b -> a -> b eval (Cons y) x = horner x y {- | 'compose' is the functional composition of polynomials. It fulfills @ eval x . eval y == eval (compose x y) @ -} -- compose :: Module.C a b => T b -> T a -> T a -- compose (Cons x) y = horner y (map const x) compose :: (Ring.C a) => T a -> T a -> T a compose (Cons x) y = horner' y (map const x) {- | It's also helpful to put a polynomial in canonical form. 'normalize' strips leading coefficients that are zero. -} normalize :: (ZeroTestable.C a) => [a] -> [a] normalize = dropWhileRev isZero {- | Multiply by the variable, used internally. -} shift :: (Additive.C a) => [a] -> [a] shift [] = [] shift l = zero : l unShift :: [a] -> [a] unShift [] = [] unShift (_:xs) = xs const :: a -> T a const x = lift0 [x] equal :: (Eq a, ZeroTestable.C a) => [a] -> [a] -> Bool equal x y = and (zipWithOverlap isZero isZero (==) x y) instance (Eq a, ZeroTestable.C a) => Eq (T a) where (Cons x) == (Cons y) = equal x y instance (Indexable.C a, ZeroTestable.C a) => Indexable.C (T a) where compare = Indexable.liftCompare coeffs instance (ZeroTestable.C a) => ZeroTestable.C (T a) where isZero (Cons x) = isZero x add, sub :: (Additive.C a) => [a] -> [a] -> [a] add = (+) sub = (-) negate :: (Additive.C a) => [a] -> [a] negate = map NP.negate instance (Additive.C a) => Additive.C (T a) where (+) = lift2 add (-) = lift2 sub zero = lift0 [] negate = lift1 negate scale :: Ring.C a => a -> [a] -> [a] scale s = map (s*) instance Vector.C T where zero = zero (<+>) = (+) (*>) = Vector.functorScale instance (Module.C a b) => Module.C a (T b) where (*>) x = lift1 (x *>) instance (Field.C a, Module.C a b) => VectorSpace.C a (T b) tensorProduct :: Ring.C a => [a] -> [a] -> [[a]] tensorProduct = outerProduct (*) tensorProduct' :: Ring.C a => [a] -> [a] -> [[a]] tensorProduct' xs ys = map (flip scale ys) xs {- | 'mul' is fast if the second argument is a short polynomial, 'MathObj.PowerSeries.**' relies on that fact. -} mul :: Ring.C a => [a] -> [a] -> [a] {- prevent from generation of many zeros if the first operand is the empty list -} mul [] = P.const [] mul xs = foldr (\y zs -> let (v:vs) = scale y xs in v : add vs zs) [] -- this one fails on infinite lists -- mul xs = foldr (\y zs -> add (scale y xs) (shift zs)) [] mulShear :: Ring.C a => [a] -> [a] -> [a] mulShear xs ys = map sum (shear (tensorProduct xs ys)) mulShearTranspose :: Ring.C a => [a] -> [a] -> [a] mulShearTranspose xs ys = map sum (shearTranspose (tensorProduct xs ys)) instance (Ring.C a) => Ring.C (T a) where one = const one fromInteger = const . fromInteger (*) = lift2 mul divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a]) divMod x y = let (y0:ys) = dropWhile isZero (reverse y) aux l xs' = if l < 0 then ([], xs') else let (x0:xs) = xs' q0 = x0/y0 (d',m') = aux (l-1) (sub xs (scale q0 ys)) in (q0:d',m') (d, m) = aux (length x - length y) (reverse x) in if isZero y then error "MathObj.Polynomial: division by zero" else (reverse d, reverse m) instance (ZeroTestable.C a, Field.C a) => Integral.C (T a) where divMod (Cons x) (Cons y) = let (d,m) = divMod x y in (Cons d, Cons m) stdUnit :: (ZeroTestable.C a, Ring.C a) => [a] -> a stdUnit x = case normalize x of [] -> one l -> last l instance (ZeroTestable.C a, Field.C a) => Units.C (T a) where isUnit (Cons []) = False isUnit (Cons (x0:xs)) = not (isZero x0) && all isZero xs stdUnit (Cons x) = const (stdUnit x) stdUnitInv (Cons x) = const (recip (stdUnit x)) {- Polynomials are a Euclidean domain, so no instance is necessary (although it might be faster). -} instance (ZeroTestable.C a, Field.C a) => PID.C (T a) progression :: Ring.C a => [a] progression = iterate (one+) one differentiate :: (Ring.C a) => [a] -> [a] differentiate = zipWith (*) progression . tail integrate :: (Field.C a) => a -> [a] -> [a] integrate c x = c : zipWith (/) x progression {- | Integrates if it is possible to represent the integrated polynomial in the given ring. Otherwise undefined coefficients occur. -} integrateInt :: (ZeroTestable.C a, Integral.C a) => a -> [a] -> [a] integrateInt c x = c : zipWith Integral.safeDiv x progression instance (Ring.C a) => Differential.C (T a) where differentiate = lift1 differentiate fromRoots :: (Ring.C a) => [a] -> T a fromRoots = Cons . foldl (flip mulLinearFactor) [1] mulLinearFactor :: Ring.C a => a -> [a] -> [a] mulLinearFactor x yt@(y:ys) = Additive.negate (x*y) : yt - scale x ys mulLinearFactor _ [] = [] alternate :: Additive.C a => [a] -> [a] alternate = zipWith ($) (cycle [id, Additive.negate]) {- see htam: Wavelet/DyadicResultant resultant :: Ring.C a => [a] -> [a] -> [a] resultant xs ys = discriminant :: Ring.C a => [a] -> [a] discriminant xs = let degree = genericLength xs in parityFlip (safeDiv (degree*(degree-1)) 2) (resultant xs (differentiate xs)) `safeDiv` last xs -} instance (Arbitrary a, ZeroTestable.C a) => Arbitrary (T a) where arbitrary = liftM (fromCoeffs . normalize) arbitrary coarbitrary = undefined {- * legacy instances -} {- | It is disputable whether polynomials shall be represented by number literals or not. An advantage is, that one can write let x = polynomial [0,1] in (x^2+x+1)*(x-1) However the output looks much different. -} legacyInstance :: a legacyInstance = error "legacy Ring.C instance for simple input of numeric literals" instance (Ring.C a, Eq a, Show a, ZeroTestable.C a) => P98.Num (T a) where fromInteger = const . fromInteger negate = Additive.negate -- for unary minus (+) = legacyInstance (*) = legacyInstance abs = legacyInstance signum = legacyInstance instance (Field.C a, Eq a, Show a, ZeroTestable.C a) => P98.Fractional (T a) where fromRational = const . fromRational (/) = legacyInstance