Safe Haskell	None
Language	Haskell2010

Algebra.Ring.Fraction.Decomp

Description

Partial fraction decomposition for unary polynomials

Synopsis

data PartialFractionDecomp k = PartialFraction {
- residualPoly :: Unipol k
- partialFracs :: NonEmpty (Unipol k, IntMap (Unipol k))
}
partialFractionDecomposition :: (Field k, CoeffRing k, Functor m) => (Unipol k -> m (k, NonEmpty (Unipol k, Natural))) -> Fraction (Unipol k) -> m (PartialFractionDecomp k)
scalePF :: CoeffRing k => k -> PartialFractionDecomp k -> PartialFractionDecomp k
partialFractionDecompositionWith :: (CoeffRing k, Field k) => Unipol k -> NonEmpty (Unipol k, Natural) -> PartialFractionDecomp k
pAdicExpansion :: (Field k, CoeffRing k) => Unipol k -> Unipol k -> IntMap (Unipol k)
poweredPartialFraction :: forall k. (Field k, CoeffRing k) => Unipol k -> NonEmpty (Unipol k, Natural) -> NonEmpty ((Unipol k, Natural), Unipol k)
scanZipper :: (NonEmpty a -> [a] -> b) -> NonEmpty a -> NonEmpty b

Documentation

data PartialFractionDecomp k Source #

Constructors

PartialFraction
Fields residualPoly :: Unipol k partialFracs :: NonEmpty (Unipol k, IntMap (Unipol k))

Instances

Instances details

(Eq k, DecidableZero k) => Eq (PartialFractionDecomp k) Source #
Instance details Defined in Algebra.Ring.Fraction.Decomp Methods (==) :: PartialFractionDecomp k -> PartialFractionDecomp k -> Bool # (/=) :: PartialFractionDecomp k -> PartialFractionDecomp k -> Bool #
(Ord k, DecidableZero k) => Ord (PartialFractionDecomp k) Source #
Instance details Defined in Algebra.Ring.Fraction.Decomp Methods compare :: PartialFractionDecomp k -> PartialFractionDecomp k -> Ordering # (<) :: PartialFractionDecomp k -> PartialFractionDecomp k -> Bool # (<=) :: PartialFractionDecomp k -> PartialFractionDecomp k -> Bool # (>) :: PartialFractionDecomp k -> PartialFractionDecomp k -> Bool # (>=) :: PartialFractionDecomp k -> PartialFractionDecomp k -> Bool # max :: PartialFractionDecomp k -> PartialFractionDecomp k -> PartialFractionDecomp k # min :: PartialFractionDecomp k -> PartialFractionDecomp k -> PartialFractionDecomp k #
(DecidableZero k, Ring k, Commutative k, Eq k, PrettyCoeff k) => Show (PartialFractionDecomp k) Source #
Instance details Defined in Algebra.Ring.Fraction.Decomp Methods showsPrec :: Int -> PartialFractionDecomp k -> ShowS # show :: PartialFractionDecomp k -> String # showList :: [PartialFractionDecomp k] -> ShowS #

partialFractionDecomposition :: (Field k, CoeffRing k, Functor m) => (Unipol k -> m (k, NonEmpty (Unipol k, Natural))) -> Fraction (Unipol k) -> m (PartialFractionDecomp k) Source #

scalePF :: CoeffRing k => k -> PartialFractionDecomp k -> PartialFractionDecomp k Source #

partialFractionDecompositionWith Source #

Arguments

:: (CoeffRing k, Field k)
=> Unipol k	Numerator
-> NonEmpty (Unipol k, Natural)	Pairwise coprime (partial) factorisation of denominator \(f = \sum_i f_i^{e_i}\) with each \(f_i\) monic and non-constant.
-> PartialFractionDecomp k

Calculates the partial fraction decomposition with respect to the given pairwise coprime monic factorisation of the denominator.

>>> partialFractionDecompositionWith (#x ^ 3 + 4 * #x^2 - #x - 2 :: Unipol Rational) ((#x, 2) :| [(#x - 1, 1), (#x + 1, 1)])
(x,fromList [(1,1),(2,2)]) :| [(x - 1,fromList [(1,1)]),(x + 1,fromList [(1,-1)])]

pAdicExpansion :: (Field k, CoeffRing k) => Unipol k -> Unipol k -> IntMap (Unipol k) Source #

pAdicExpansion f p gives a p-adic expansion of f by a monic nonconstant polynomial p; i.e. \(a_i \in k[X]\) such that \(\deg a_i < \deg p\)_{i < k}, \(\deg f < km\), and:

\[ f = a_{k - 1} p^{k-1} + \cdots + a_1 p + a_0. \]

>>> pAdicExpansion (#x + 2 :: Unipol Rational) #x
fromList [(0,2),(1,1)]

poweredPartialFraction Source #

Arguments

:: forall k. (Field k, CoeffRing k)
=> Unipol k	Numerator `g` with (deg g < deg f).
-> NonEmpty (Unipol k, Natural)	Pairwise coprime (partial) factorisation of denominator \(f = \sum_i f_i^{e_i}\) with each \(f_i\) monic and non-constant.
-> NonEmpty ((Unipol k, Natural), Unipol k)

Pseudo-partial fraction decomposition

scanZipper :: (NonEmpty a -> [a] -> b) -> NonEmpty a -> NonEmpty b Source #