{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances #-}
{- |
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, coeffs,
showsExpressionPrec, const,
evaluate, evaluateCoeffVector, evaluateArgVector,
compose, equal, add, sub, negate,
horner, hornerCoeffVector, hornerArgVector,
shift, unShift,
mul, scale, divMod,
tensorProduct, tensorProductAlt,
mulShear, mulShearTranspose,
progression, differentiate, integrate, integrateInt,
fromRoots, alternate, reverse, )
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, )
import Data.List.HT
(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, reverse, )
import NumericPrelude hiding (divMod, negate, stdUnit, )
newtype T a = Cons {coeffs :: [a]}
{-# INLINE fromCoeffs #-}
fromCoeffs :: [a] -> T a
fromCoeffs = lift0
{-# INLINE lift0 #-}
lift0 :: [a] -> T a
lift0 = Cons
{-# INLINE lift1 #-}
lift1 :: ([a] -> [a]) -> (T a -> T a)
lift1 f (Cons x0) = Cons (f x0)
{-# INLINE lift2 #-}
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)
{-# INLINE plusPrec #-}
{-# INLINE appPrec #-}
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)
{-# INLINE showsExpressionPrec #-}
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 a polynomial in a ring.
-}
{-# INLINE horner #-}
horner :: Ring.C a => a -> [a] -> a
horner x = foldr (\c val -> c+x*val) zero
{- |
Horner's scheme for evaluating a polynomial in a module.
-}
{-# INLINE hornerCoeffVector #-}
hornerCoeffVector :: Module.C a v => a -> [v] -> v
hornerCoeffVector x = foldr (\c val -> c+x*>val) zero
{-# INLINE hornerArgVector #-}
hornerArgVector :: (Module.C a v, Ring.C v) => v -> [a] -> v
hornerArgVector x = foldr (\c val -> c*>one+val*x) zero
{-# INLINE evaluate #-}
evaluate :: Ring.C a => T a -> a -> a
evaluate (Cons y) x = horner x y
{- |
Here the coefficients are vectors,
for example the coefficients are real and the coefficents are real vectors.
-}
{-# INLINE evaluateCoeffVector #-}
evaluateCoeffVector :: Module.C a v => T v -> a -> v
evaluateCoeffVector (Cons y) x = hornerCoeffVector x y
{- |
Here the argument is a vector,
for example the coefficients are complex numbers or square matrices
and the coefficents are reals.
-}
{-# INLINE evaluateArgVector #-}
evaluateArgVector :: (Module.C a v, Ring.C v) => T a -> v -> v
evaluateArgVector (Cons y) x = hornerArgVector 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)
{-# INLINE compose #-}
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.
-}
{-# INLINE normalize #-}
normalize :: (ZeroTestable.C a) => [a] -> [a]
normalize = dropWhileRev isZero
{- |
Multiply by the variable, used internally.
-}
{-# INLINE shift #-}
shift :: (Additive.C a) => [a] -> [a]
shift [] = []
shift l = zero : l
{-# INLINE unShift #-}
unShift :: [a] -> [a]
unShift [] = []
unShift (_:xs) = xs
{-# INLINE const #-}
const :: a -> T a
const x = lift0 [x]
{-# INLINE equal #-}
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 = (-)
{-# INLINE negate #-}
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
{-# INLINE scale #-}
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)
{-# INLINE tensorProduct #-}
tensorProduct :: Ring.C a => [a] -> [a] -> [[a]]
tensorProduct = outerProduct (*)
tensorProductAlt :: Ring.C a => [a] -> [a] -> [[a]]
tensorProductAlt xs ys = map (flip scale ys) xs
{- |
'mul' is fast if the second argument is a short polynomial,
'MathObj.PowerSeries.**' relies on that fact.
-}
{-# INLINE mul #-}
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)) []
{-# INLINE mulShear #-}
mulShear :: Ring.C a => [a] -> [a] -> [a]
mulShear xs ys = map sum (shear (tensorProduct xs ys))
{-# INLINE mulShearTranspose #-}
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 (List.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) (List.reverse x)
in if isZero y
then error "MathObj.Polynomial: division by zero"
else (List.reverse d, List.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)
{-# INLINE stdUnit #-}
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)
{-# INLINE progression #-}
progression :: Ring.C a => [a]
progression = iterate (one+) one
{-# INLINE differentiate #-}
differentiate :: (Ring.C a) => [a] -> [a]
differentiate = zipWith (*) progression . tail
{-# INLINE integrate #-}
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.
-}
{-# INLINE integrateInt #-}
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
{-# INLINE fromRoots #-}
fromRoots :: (Ring.C a) => [a] -> T a
fromRoots = Cons . foldl (flip mulLinearFactor) [1]
{-# INLINE mulLinearFactor #-}
mulLinearFactor :: Ring.C a => a -> [a] -> [a]
mulLinearFactor x yt@(y:ys) = Additive.negate (x*y) : yt - scale x ys
mulLinearFactor _ [] = []
{-# INLINE alternate #-}
alternate :: Additive.C a => [a] -> [a]
alternate = zipWith ($) (cycle [id, Additive.negate])
{-# INLINE reverse #-}
reverse :: Additive.C a => T a -> T a
reverse = lift1 alternate
{-
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.
-}
{-# INLINE legacyInstance #-}
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