{-# OPTIONS -fno-implicit-prelude -fglasgow-exts #-} {- | Power series, either finite or unbounded. (zipWith does exactly the right thing to make it work almost transparently.) -} module MathObj.PowerSeries where import qualified MathObj.Polynomial as Poly import qualified Algebra.Differential as Differential import qualified Algebra.IntegralDomain as Integral import qualified Algebra.VectorSpace as VectorSpace import qualified Algebra.Module as Module import qualified Algebra.Vector as Vector import qualified Algebra.Transcendental as Transcendental import qualified Algebra.Algebraic as Algebraic import qualified Algebra.Field as Field import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import qualified Algebra.ZeroTestable as ZeroTestable import Algebra.Module((*>)) import Algebra.ZeroTestable(isZero) import NumericPrelude.List(splitAtMatch) import qualified NumericPrelude as NP import qualified PreludeBase as P import PreludeBase hiding (const) import NumericPrelude hiding (negate, stdUnit, divMod, sqrt, exp, log, sin, cos, tan, asin, acos, atan) newtype T a = Cons {coeffs :: [a]} deriving (Ord) {-# 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) {-# INLINE const #-} const :: a -> T a const x = lift0 [x] {- Functor instance is e.g. useful for showing power series in residue rings. @fmap (ResidueClass.concrete 7) (powerSeries [1,4,4::ResidueClass.T Integer] * powerSeries [1,5,6])@ -} instance Functor T where fmap f (Cons xs) = Cons (map f xs) {-# INLINE appPrec #-} appPrec :: Int appPrec = 10 instance (Show a) => Show (T a) where showsPrec p (Cons xs) = showParen (p >= appPrec) (showString "PowerSeries.fromCoeffs " . shows xs) {-# INLINE truncate #-} truncate :: Int -> T a -> T a truncate n = lift1 (take n) {- | Evaluate (truncated) power series. -} {-# INLINE eval #-} eval :: Ring.C a => [a] -> a -> a eval = flip Poly.horner {-# INLINE evaluate #-} evaluate :: Ring.C a => T a -> a -> a evaluate (Cons y) = eval y {- | Evaluate (truncated) power series. -} {-# INLINE evalCoeffVector #-} evalCoeffVector :: Module.C a v => [v] -> a -> v evalCoeffVector = flip Poly.hornerCoeffVector {-# INLINE evaluateCoeffVector #-} evaluateCoeffVector :: Module.C a v => T v -> a -> v evaluateCoeffVector (Cons y) = evalCoeffVector y {-# INLINE evalArgVector #-} evalArgVector :: (Module.C a v, Ring.C v) => [a] -> v -> v evalArgVector = flip Poly.hornerArgVector {-# INLINE evaluateArgVector #-} evaluateArgVector :: (Module.C a v, Ring.C v) => T a -> v -> v evaluateArgVector (Cons y) = evalArgVector y {- | Evaluate approximations that is evaluate all truncations of the series. -} {-# INLINE approx #-} approx :: Ring.C a => [a] -> a -> [a] approx y x = scanl (+) zero (zipWith (*) (iterate (x*) 1) y) {-# INLINE approximate #-} approximate :: Ring.C a => T a -> a -> [a] approximate (Cons y) = approx y {- | Evaluate approximations that is evaluate all truncations of the series. -} {-# INLINE approxCoeffVector #-} approxCoeffVector :: Module.C a v => [v] -> a -> [v] approxCoeffVector y x = scanl (+) zero (zipWith (*>) (iterate (x*) 1) y) {-# INLINE approximateCoeffVector #-} approximateCoeffVector :: Module.C a v => T v -> a -> [v] approximateCoeffVector (Cons y) = approxCoeffVector y {- | Evaluate approximations that is evaluate all truncations of the series. -} {-# INLINE approxArgVector #-} approxArgVector :: (Module.C a v, Ring.C v) => [a] -> v -> [v] approxArgVector y x = scanl (+) zero (zipWith (*>) y (iterate (x*) 1)) {-# INLINE approximateArgVector #-} approximateArgVector :: (Module.C a v, Ring.C v) => T a -> v -> [v] approximateArgVector (Cons y) = approxArgVector y {- * Simple series manipulation -} {- | For the series of a real function @f@ compute the series for @\x -> f (-x)@ -} alternate :: Additive.C a => [a] -> [a] alternate = zipWith id (cycle [id, NP.negate]) {- | For the series of a real function @f@ compute the series for @\x -> (f x + f (-x)) \/ 2@ -} holes2 :: Additive.C a => [a] -> [a] holes2 = zipWith id (cycle [id, P.const zero]) {- | For the series of a real function @f@ compute the real series for @\x -> (f (i*x) + f (-i*x)) \/ 2@ -} holes2alternate :: Additive.C a => [a] -> [a] holes2alternate = zipWith id (cycle [id, P.const zero, NP.negate, P.const zero]) {- * Series arithmetic -} add, sub :: (Additive.C a) => [a] -> [a] -> [a] add = Poly.add sub = Poly.sub negate :: (Additive.C a) => [a] -> [a] negate = Poly.negate scale :: Ring.C a => a -> [a] -> [a] scale = Poly.scale mul :: Ring.C a => [a] -> [a] -> [a] mul = Poly.mul {- Note that the derived instances only make sense for finite series. -} instance (Eq a, ZeroTestable.C a) => Eq (T a) where (Cons x) == (Cons y) = Poly.equal x y instance (Additive.C a) => Additive.C (T a) where negate = lift1 Poly.negate (+) = lift2 Poly.add (-) = lift2 Poly.sub zero = lift0 [] instance (Ring.C a) => Ring.C (T a) where one = const one fromInteger n = const (fromInteger n) (*) = lift2 mul 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) stripLeadZero :: (ZeroTestable.C a) => [a] -> [a] -> ([a],[a]) stripLeadZero (x:xs) (y:ys) = if isZero x && isZero y then stripLeadZero xs ys else (x:xs,y:ys) stripLeadZero xs ys = (xs,ys) {- | Divide two series where the absolute term of the divisor is non-zero. That is, power series with leading non-zero terms are the units in the ring of power series. Knuth: Seminumerical algorithms -} divide :: (Field.C a) => [a] -> [a] -> [a] divide (x:xs) (y:ys) = let zs = map (/y) (x : sub xs (mul zs ys)) in zs divide [] _ = [] divide _ [] = error "PowerSeries.divide: division by empty series" {- | Divide two series also if the divisor has leading zeros. -} divideStripZero :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> [a] divideStripZero x' y' = let (x0,y0) = stripLeadZero x' y' in if null y0 || isZero (head y0) then error "PowerSeries.divideStripZero: Division by zero." else divide x0 y0 instance (Field.C a) => Field.C (T a) where (/) = lift2 divide divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a],[a]) divMod xs ys = let (yZero,yRem) = span isZero ys (xMod, xRem) = splitAtMatch yZero xs in (divide xRem yRem, xMod) 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) progression :: Ring.C a => [a] progression = Poly.progression recipProgression :: (Field.C a) => [a] recipProgression = map recip progression differentiate :: (Ring.C a) => [a] -> [a] differentiate = Poly.differentiate integrate :: (Field.C a) => a -> [a] -> [a] integrate = Poly.integrate instance (Ring.C a) => Differential.C (T a) where differentiate = lift1 differentiate {- | We need to compute the square root only of the first term. That is, if the first term is rational, then all terms of the series are rational. -} sqrt :: Field.C a => (a -> a) -> [a] -> [a] sqrt _ [] = [] sqrt f0 (x:xs) = let y = f0 x ys = map (/(y+y)) (xs - (0 : mul ys ys)) in y:ys {- pow alpha t = t^alpha (pow alpha . x)' = alpha * (pow (alpha-1) . x) * x' alpha * (pow alpha . x) = x * x' * (pow alpha . x)' y = pow alpha . x alpha * y = x * x' * y' -} {- | Input series must start with non-zero term. -} pow :: (Field.C a) => (a -> a) -> a -> [a] -> [a] pow f0 expon x = let y = integrate (f0 (head x)) y' y' = scale expon (divide y (mul x (differentiate x))) in y instance (Algebraic.C a) => Algebraic.C (T a) where sqrt = lift1 (sqrt Algebraic.sqrt) x ^/ y = lift1 (pow (Algebraic.^/ y) (fromRational' y)) x {- | The first term needs a transcendent computation but the others do not. That's why we accept a function which computes the first term. > (exp . x)' = (exp . x) * x' > (sin . x)' = (cos . x) * x' > (cos . x)' = - (sin . x) * x' -} exp :: Field.C a => (a -> a) -> [a] -> [a] exp f0 x = let x' = differentiate x y = integrate (f0 (head x)) (mul y x') in y sinCos :: Field.C a => (a -> (a,a)) -> [a] -> ([a],[a]) sinCos f0 x = let (y0Sin, y0Cos) = f0 (head x) x' = differentiate x ySin = integrate y0Sin (mul yCos x') yCos = integrate y0Cos (negate (mul ySin x')) in (ySin, yCos) sinCosScalar :: Transcendental.C a => a -> (a,a) sinCosScalar x = (Transcendental.sin x, Transcendental.cos x) sin, cos :: Field.C a => (a -> (a,a)) -> [a] -> [a] sin f0 = fst . sinCos f0 cos f0 = snd . sinCos f0 tan :: (Field.C a) => (a -> (a,a)) -> [a] -> [a] tan f0 = uncurry divide . sinCos f0 {- (log x)' == x'/x (asin x)' == (acos x) == x'/sqrt(1-x^2) (atan x)' == x'/(1+x^2) -} {- | Input series must start with non-zero term. -} log :: (Field.C a) => (a -> a) -> [a] -> [a] log f0 x = integrate (f0 (head x)) (derivedLog x) {- | Computes @(log x)'@, that is @x'\/x@ -} derivedLog :: (Field.C a) => [a] -> [a] derivedLog x = divide (differentiate x) x atan :: (Field.C a) => (a -> a) -> [a] -> [a] atan f0 x = let x' = differentiate x in integrate (f0 (head x)) (divide x' ([1] + mul x x)) asin, acos :: (Field.C a) => (a -> a) -> (a -> a) -> [a] -> [a] asin sqrt0 f0 x = let x' = differentiate x in integrate (f0 (head x)) (divide x' (sqrt sqrt0 ([1] - mul x x))) acos = asin instance (Transcendental.C a) => Transcendental.C (T a) where pi = const NP.pi exp = lift1 (exp Transcendental.exp) sin = lift1 (sin sinCosScalar) cos = lift1 (cos sinCosScalar) tan = lift1 (tan sinCosScalar) x ** y = Transcendental.exp (Transcendental.log x * y) {- This order of multiplication is especially fast when y is a singleton. -} log = lift1 (log Transcendental.log) asin = lift1 (asin Algebraic.sqrt Transcendental.asin) acos = lift1 (acos Algebraic.sqrt Transcendental.acos) atan = lift1 (atan Transcendental.atan) {- | It fulfills @ evaluate x . evaluate y == evaluate (compose x y) @ -} compose :: (Ring.C a, ZeroTestable.C a) => T a -> T a -> T a compose (Cons []) (Cons []) = Cons [] compose (Cons (x:_)) (Cons []) = Cons [x] compose (Cons x) (Cons (y:ys)) = if isZero y then Cons (comp x ys) else error "PowerSeries.compose: inner series must not have an absolute term." {- | Since the inner series must start with a zero, the first term is omitted in y. -} comp :: (Ring.C a) => [a] -> [a] -> [a] comp xs y = foldr (\x acc -> x : mul y acc) [] xs {- | Compose two power series where the outer series can be developed for any expansion point. To be more precise: The outer series must be expanded with respect to the leading term of the inner series. -} composeTaylor :: Ring.C a => (a -> [a]) -> [a] -> [a] composeTaylor x (y:ys) = comp (x y) ys composeTaylor x [] = x 0 {- (x . y) = id (x' . y) * y' = 1 y' = 1 / (x' . y) -} {- | This function returns the series of the function in the form: (point of the expansion, power series) This is exceptionally slow and needs cubic run-time. -} inv :: (Field.C a) => [a] -> (a, [a]) inv x = let y' = divide [1] (comp (differentiate x) (tail y)) y = integrate 0 y' -- the first term is zero, which is required for composition in (head x, y)