Math.Algebra.Polynomial.Class

Indices

newtype Index

The index of a variable. This will be used as the variable type when the set of variables is a continguous set like {x_1, x_2, ... , x_N}

Constructors

Index Int

Instances

Enum Index

Defined in Math.Algebra.Polynomial.Class

Methods

succ :: Index -> Index

pred :: Index -> Index

toEnum :: Int -> Index

fromEnum :: Index -> Int

enumFrom :: Index -> [Index]

enumFromThen :: Index -> Index -> [Index]

enumFromTo :: Index -> Index -> [Index]

enumFromThenTo :: Index -> Index -> Index -> [Index]

Eq Index

Defined in Math.Algebra.Polynomial.Class

Methods

(==) :: Index -> Index -> Bool

(/=) :: Index -> Index -> Bool

Ord Index

Defined in Math.Algebra.Polynomial.Class

Methods

compare :: Index -> Index -> Ordering

(<) :: Index -> Index -> Bool

(<=) :: Index -> Index -> Bool

(>) :: Index -> Index -> Bool

(>=) :: Index -> Index -> Bool

max :: Index -> Index -> Index

min :: Index -> Index -> Index

Show Index

Defined in Math.Algebra.Polynomial.Class

Methods

showsPrec :: Int -> Index -> ShowS

show :: Index -> String

showList :: [Index] -> ShowS

Pretty Index

Defined in Math.Algebra.Polynomial.Class

Methods

pretty :: Index -> String

prettyInParens :: Index -> String

fromIndex :: Index -> Int

Rings

class (Eq c, Ord c, Num c, IsSigned c, Show c, Pretty c, Typeable c) => Ring c where

The class of coefficient rings.

Since base rings like integers or rational behave differently than say another polynomial ring as a coefficient ring, we have to be explicit about some things (mainly for pretty-printing purposes

TODO: clean this up!

Minimal complete definition

Nothing

Methods

isZeroR :: c -> Bool

signumR :: c -> Maybe Sign

absR :: c -> c

isSignedR :: Proxy c -> Bool

isAtomicR :: Proxy c -> Bool

Instances

Ring Int

Defined in Math.Algebra.Polynomial.Class

Methods

isZeroR :: Int -> Bool

signumR :: Int -> Maybe Sign

absR :: Int -> Int

isSignedR :: Proxy Int -> Bool

isAtomicR :: Proxy Int -> Bool

Ring Integer

Defined in Math.Algebra.Polynomial.Class

Methods

isZeroR :: Integer -> Bool

signumR :: Integer -> Maybe Sign

absR :: Integer -> Integer

isSignedR :: Proxy Integer -> Bool

isAtomicR :: Proxy Integer -> Bool

Ring Rational

Defined in Math.Algebra.Polynomial.Class

Methods

isZeroR :: Rational -> Bool

signumR :: Rational -> Maybe Sign

absR :: Rational -> Rational

isSignedR :: Proxy Rational -> Bool

isAtomicR :: Proxy Rational -> Bool

(Ring c, KnownSymbol v) => Ring (Poly c v)

Defined in Math.Algebra.Polynomial.Multivariate.Infinite

Methods

isZeroR :: Poly c v -> Bool

signumR :: Poly c v -> Maybe Sign

absR :: Poly c v -> Poly c v

isSignedR :: Proxy (Poly c v) -> Bool

isAtomicR :: Proxy (Poly c v) -> Bool

(Ring c, Variable v) => Ring (Poly c v)

Defined in Math.Algebra.Polynomial.Multivariate.Generic

Methods

isZeroR :: Poly c v -> Bool

signumR :: Poly c v -> Maybe Sign

absR :: Poly c v -> Poly c v

isSignedR :: Proxy (Poly c v) -> Bool

isAtomicR :: Proxy (Poly c v) -> Bool

(Ring c, KnownSymbol v) => Ring (Univariate c v)

Defined in Math.Algebra.Polynomial.Univariate

Methods

isZeroR :: Univariate c v -> Bool

signumR :: Univariate c v -> Maybe Sign

absR :: Univariate c v -> Univariate c v

isSignedR :: Proxy (Univariate c v) -> Bool

isAtomicR :: Proxy (Univariate c v) -> Bool

(Ring c, KnownSymbol v, KnownNat n) => Ring (Poly c v n)

Defined in Math.Algebra.Polynomial.Multivariate.Indexed

Methods

isZeroR :: Poly c v n -> Bool

signumR :: Poly c v n -> Maybe Sign

absR :: Poly c v n -> Poly c v n

isSignedR :: Proxy (Poly c v n) -> Bool

isAtomicR :: Proxy (Poly c v n) -> Bool

(Ring c, KnownSymbol v, KnownNat n) => Ring (Poly c v n)

Defined in Math.Algebra.Polynomial.Multivariate.Compact

Methods

isZeroR :: Poly c v n -> Bool

signumR :: Poly c v n -> Maybe Sign

absR :: Poly c v n -> Poly c v n

isSignedR :: Proxy (Poly c v n) -> Bool

isAtomicR :: Proxy (Poly c v n) -> Bool

(Ring c, KnownSymbol v, KnownNat n) => Ring (ExtAlg c v n)

Defined in Math.Algebra.Polynomial.Exterior.Indexed

Methods

isZeroR :: ExtAlg c v n -> Bool

signumR :: ExtAlg c v n -> Maybe Sign

absR :: ExtAlg c v n -> ExtAlg c v n

isSignedR :: Proxy (ExtAlg c v n) -> Bool

isAtomicR :: Proxy (ExtAlg c v n) -> Bool

type Field c = (Ring c, Fractional c)

The class of coefficient fields (this is just a constraint synonym for now)

type Variable v = (Ord v, Show v, Pretty v, Typeable v)

The class of types whose inhabitants can serve as variables (this is just a constraint synonym for now)

Monomials

class Pretty m => Monomial m where

The class of (multivariate) monomials

The Maybe-s are there to allow truncated and exterior polynomial rings

Associated Types

type VarM m :: *

the type of variables

Methods

normalizeM

Arguments

:: m

-> m

enforce the invariant

isNormalM

Arguments

:: m

-> Bool

check the invariant construction and deconstruction

fromListM

Arguments

:: [(VarM m, Int)]

-> m

construction from (variable,exponent) pairs

toListM

Arguments

:: m

-> [(VarM m, Int)]

extracting (variable,exponent) pairs simple monomials

emptyM

Arguments

:: m

the empty monomial (corresponding to the polynomial 1)

isEmptyM

Arguments

:: m

-> Bool

checks whether it is empty

variableM

Arguments

:: VarM m

-> m

a single variable

singletonM

Arguments

:: VarM m

-> Int

-> m

a single variable raised to a power algebra

mulM

Arguments

:: m

-> m

-> m

multiplication of monomials

productM

Arguments

:: [m]

-> m

product of several monomials

powM

Arguments

:: m

-> Int

-> m

raising to a power

divM

Arguments

:: m

-> m

-> Maybe m

division of monomials calculus

diffM

Arguments

:: Num c

=> VarM m

-> Int

-> m

-> Maybe (m, c)

differentiation degrees

maxDegM

Arguments

:: m

-> Int

maximum degree

totalDegM

Arguments

:: m

-> Int

total degree substitution and evaluation

evalM

Arguments

:: Num c

=> (VarM m -> c)

-> m

-> c

ring substitution (evaluation)

varSubsM

Arguments

:: (VarM m -> VarM m)

-> m

-> m

simple variable substitution

termSubsM

Arguments

:: Num c

=> (VarM m -> Maybe c)

-> (m, c)

-> (m, c)

term substitution

Instances

KnownSymbol var => Monomial (U var)

Defined in Math.Algebra.Polynomial.Monomial.Univariate

Associated Types

type VarM (U var)

Methods

normalizeM :: U var -> U var

isNormalM :: U var -> Bool

fromListM :: [(VarM (U var), Int)] -> U var

toListM :: U var -> [(VarM (U var), Int)]

emptyM :: U var

isEmptyM :: U var -> Bool

variableM :: VarM (U var) -> U var

singletonM :: VarM (U var) -> Int -> U var

mulM :: U var -> U var -> U var

productM :: [U var] -> U var

powM :: U var -> Int -> U var

divM :: U var -> U var -> Maybe (U var)

diffM :: Num c => VarM (U var) -> Int -> U var -> Maybe (U var, c)

maxDegM :: U var -> Int

totalDegM :: U var -> Int

evalM :: Num c => (VarM (U var) -> c) -> U var -> c

varSubsM :: (VarM (U var) -> VarM (U var)) -> U var -> U var

termSubsM :: Num c => (VarM (U var) -> Maybe c) -> (U var, c) -> (U var, c)

KnownSymbol v => Monomial (XInf v)

Defined in Math.Algebra.Polynomial.Monomial.Infinite

Associated Types

type VarM (XInf v)

Methods

normalizeM :: XInf v -> XInf v

isNormalM :: XInf v -> Bool

fromListM :: [(VarM (XInf v), Int)] -> XInf v

toListM :: XInf v -> [(VarM (XInf v), Int)]

emptyM :: XInf v

isEmptyM :: XInf v -> Bool

variableM :: VarM (XInf v) -> XInf v

singletonM :: VarM (XInf v) -> Int -> XInf v

mulM :: XInf v -> XInf v -> XInf v

productM :: [XInf v] -> XInf v

powM :: XInf v -> Int -> XInf v

divM :: XInf v -> XInf v -> Maybe (XInf v)

diffM :: Num c => VarM (XInf v) -> Int -> XInf v -> Maybe (XInf v, c)

maxDegM :: XInf v -> Int

totalDegM :: XInf v -> Int

evalM :: Num c => (VarM (XInf v) -> c) -> XInf v -> c

varSubsM :: (VarM (XInf v) -> VarM (XInf v)) -> XInf v -> XInf v

termSubsM :: Num c => (VarM (XInf v) -> Maybe c) -> (XInf v, c) -> (XInf v, c)

(Ord v, Pretty v) => Monomial (Monom v)

Defined in Math.Algebra.Polynomial.Monomial.Generic

Associated Types

type VarM (Monom v)

Methods

normalizeM :: Monom v -> Monom v

isNormalM :: Monom v -> Bool

fromListM :: [(VarM (Monom v), Int)] -> Monom v

toListM :: Monom v -> [(VarM (Monom v), Int)]

emptyM :: Monom v

isEmptyM :: Monom v -> Bool

variableM :: VarM (Monom v) -> Monom v

singletonM :: VarM (Monom v) -> Int -> Monom v

mulM :: Monom v -> Monom v -> Monom v

productM :: [Monom v] -> Monom v

powM :: Monom v -> Int -> Monom v

divM :: Monom v -> Monom v -> Maybe (Monom v)

diffM :: Num c => VarM (Monom v) -> Int -> Monom v -> Maybe (Monom v, c)

maxDegM :: Monom v -> Int

totalDegM :: Monom v -> Int

evalM :: Num c => (VarM (Monom v) -> c) -> Monom v -> c

varSubsM :: (VarM (Monom v) -> VarM (Monom v)) -> Monom v -> Monom v

termSubsM :: Num c => (VarM (Monom v) -> Maybe c) -> (Monom v, c) -> (Monom v, c)

(KnownNat n, KnownSymbol v) => Monomial (XS v n)

Defined in Math.Algebra.Polynomial.Monomial.Indexed

Associated Types

type VarM (XS v n)

Methods

normalizeM :: XS v n -> XS v n

isNormalM :: XS v n -> Bool

fromListM :: [(VarM (XS v n), Int)] -> XS v n

toListM :: XS v n -> [(VarM (XS v n), Int)]

emptyM :: XS v n

isEmptyM :: XS v n -> Bool

variableM :: VarM (XS v n) -> XS v n

singletonM :: VarM (XS v n) -> Int -> XS v n

mulM :: XS v n -> XS v n -> XS v n

productM :: [XS v n] -> XS v n

powM :: XS v n -> Int -> XS v n

divM :: XS v n -> XS v n -> Maybe (XS v n)

diffM :: Num c => VarM (XS v n) -> Int -> XS v n -> Maybe (XS v n, c)

maxDegM :: XS v n -> Int

totalDegM :: XS v n -> Int

evalM :: Num c => (VarM (XS v n) -> c) -> XS v n -> c

varSubsM :: (VarM (XS v n) -> VarM (XS v n)) -> XS v n -> XS v n

termSubsM :: Num c => (VarM (XS v n) -> Maybe c) -> (XS v n, c) -> (XS v n, c)

(KnownNat n, KnownSymbol v) => Monomial (Compact v n)

Defined in Math.Algebra.Polynomial.Monomial.Compact

Associated Types

type VarM (Compact v n)

Methods

normalizeM :: Compact v n -> Compact v n

isNormalM :: Compact v n -> Bool

fromListM :: [(VarM (Compact v n), Int)] -> Compact v n

toListM :: Compact v n -> [(VarM (Compact v n), Int)]

emptyM :: Compact v n

isEmptyM :: Compact v n -> Bool

variableM :: VarM (Compact v n) -> Compact v n

singletonM :: VarM (Compact v n) -> Int -> Compact v n

mulM :: Compact v n -> Compact v n -> Compact v n

productM :: [Compact v n] -> Compact v n

powM :: Compact v n -> Int -> Compact v n

divM :: Compact v n -> Compact v n -> Maybe (Compact v n)

diffM :: Num c => VarM (Compact v n) -> Int -> Compact v n -> Maybe (Compact v n, c)

maxDegM :: Compact v n -> Int

totalDegM :: Compact v n -> Int

evalM :: Num c => (VarM (Compact v n) -> c) -> Compact v n -> c

varSubsM :: (VarM (Compact v n) -> VarM (Compact v n)) -> Compact v n -> Compact v n

termSubsM :: Num c => (VarM (Compact v n) -> Maybe c) -> (Compact v n, c) -> (Compact v n, c)

(KnownSymbol sym, Monomial a, Monomial b) => Monomial (Tensor sym a b)

Defined in Math.Algebra.Polynomial.Monomial.Tensor

Associated Types

type VarM (Tensor sym a b)

Methods

normalizeM :: Tensor sym a b -> Tensor sym a b

isNormalM :: Tensor sym a b -> Bool

fromListM :: [(VarM (Tensor sym a b), Int)] -> Tensor sym a b

toListM :: Tensor sym a b -> [(VarM (Tensor sym a b), Int)]

emptyM :: Tensor sym a b

isEmptyM :: Tensor sym a b -> Bool

variableM :: VarM (Tensor sym a b) -> Tensor sym a b

singletonM :: VarM (Tensor sym a b) -> Int -> Tensor sym a b

mulM :: Tensor sym a b -> Tensor sym a b -> Tensor sym a b

productM :: [Tensor sym a b] -> Tensor sym a b

powM :: Tensor sym a b -> Int -> Tensor sym a b

divM :: Tensor sym a b -> Tensor sym a b -> Maybe (Tensor sym a b)

diffM :: Num c => VarM (Tensor sym a b) -> Int -> Tensor sym a b -> Maybe (Tensor sym a b, c)

maxDegM :: Tensor sym a b -> Int

totalDegM :: Tensor sym a b -> Int

evalM :: Num c => (VarM (Tensor sym a b) -> c) -> Tensor sym a b -> c

varSubsM :: (VarM (Tensor sym a b) -> VarM (Tensor sym a b)) -> Tensor sym a b -> Tensor sym a b

termSubsM :: Num c => (VarM (Tensor sym a b) -> Maybe c) -> (Tensor sym a b, c) -> (Tensor sym a b, c)

proxyVarM :: Monomial m => m -> Proxy (VarM m)

Polynomial rings

class (Pretty p, Ring (CoeffP p), FreeModule p, CoeffP p ~ CoeffF p, MonomP p ~ BaseF p) => AlmostPolynomial p where

The class of almost polynomial rings

Minimal complete definition

fromListP, toListP, zeroP, isZeroP, oneP, variableP, singletonP, monomP, monomP', scalarP, addP, subP, negP, mulP, coeffOfP, mulByMonomP, scaleP

Associated Types

type CoeffP p :: *

Type of coefficients

type MonomP p :: *

Type of monomials

type VarP p :: *

Type of variables

Methods

fromListP

Arguments

:: [(MonomP p, CoeffP p)]

-> p

construction from (variable,exponent) pairs

toListP

Arguments

:: p

-> [(MonomP p, CoeffP p)]

extracting (variable,exponent) pairs

zeroP :: p

isZeroP :: p -> Bool

oneP :: p

variableP

Arguments

:: VarP p

-> p

a single variable

singletonP

Arguments

:: VarP p

-> Int

-> p

a single variable raised to a power

monomP :: MonomP p -> p

monomP' :: MonomP p -> CoeffP p -> p

scalarP :: CoeffP p -> p

addP :: p -> p -> p

subP :: p -> p -> p

negP :: p -> p

sumP :: [p] -> p

mulP :: p -> p -> p

productP :: [p] -> p

coeffOfP :: MonomP p -> p -> CoeffP p

mulByMonomP :: MonomP p -> p -> p

scaleP :: CoeffP p -> p -> p

Instances

(Ring c, KnownSymbol v) => AlmostPolynomial (Poly c v)

Defined in Math.Algebra.Polynomial.Multivariate.Infinite

Associated Types

type CoeffP (Poly c v)

type MonomP (Poly c v)

type VarP (Poly c v)

Methods

fromListP :: [(MonomP (Poly c v), CoeffP (Poly c v))] -> Poly c v

toListP :: Poly c v -> [(MonomP (Poly c v), CoeffP (Poly c v))]

zeroP :: Poly c v

isZeroP :: Poly c v -> Bool

oneP :: Poly c v

variableP :: VarP (Poly c v) -> Poly c v

singletonP :: VarP (Poly c v) -> Int -> Poly c v

monomP :: MonomP (Poly c v) -> Poly c v

monomP' :: MonomP (Poly c v) -> CoeffP (Poly c v) -> Poly c v

scalarP :: CoeffP (Poly c v) -> Poly c v

addP :: Poly c v -> Poly c v -> Poly c v

subP :: Poly c v -> Poly c v -> Poly c v

negP :: Poly c v -> Poly c v

sumP :: [Poly c v] -> Poly c v

mulP :: Poly c v -> Poly c v -> Poly c v

productP :: [Poly c v] -> Poly c v

coeffOfP :: MonomP (Poly c v) -> Poly c v -> CoeffP (Poly c v)

mulByMonomP :: MonomP (Poly c v) -> Poly c v -> Poly c v

scaleP :: CoeffP (Poly c v) -> Poly c v -> Poly c v

(Ring c, Ord v, Pretty v) => AlmostPolynomial (Poly c v)

Defined in Math.Algebra.Polynomial.Multivariate.Generic

Associated Types

type CoeffP (Poly c v)

type MonomP (Poly c v)

type VarP (Poly c v)

Methods

fromListP :: [(MonomP (Poly c v), CoeffP (Poly c v))] -> Poly c v

toListP :: Poly c v -> [(MonomP (Poly c v), CoeffP (Poly c v))]

zeroP :: Poly c v

isZeroP :: Poly c v -> Bool

oneP :: Poly c v

variableP :: VarP (Poly c v) -> Poly c v

singletonP :: VarP (Poly c v) -> Int -> Poly c v

monomP :: MonomP (Poly c v) -> Poly c v

monomP' :: MonomP (Poly c v) -> CoeffP (Poly c v) -> Poly c v

scalarP :: CoeffP (Poly c v) -> Poly c v

addP :: Poly c v -> Poly c v -> Poly c v

subP :: Poly c v -> Poly c v -> Poly c v

negP :: Poly c v -> Poly c v

sumP :: [Poly c v] -> Poly c v

mulP :: Poly c v -> Poly c v -> Poly c v

productP :: [Poly c v] -> Poly c v

coeffOfP :: MonomP (Poly c v) -> Poly c v -> CoeffP (Poly c v)

mulByMonomP :: MonomP (Poly c v) -> Poly c v -> Poly c v

scaleP :: CoeffP (Poly c v) -> Poly c v -> Poly c v

(Ring coeff, KnownSymbol var) => AlmostPolynomial (Univariate coeff var)

Defined in Math.Algebra.Polynomial.Univariate

Associated Types

type CoeffP (Univariate coeff var)

type MonomP (Univariate coeff var)

type VarP (Univariate coeff var)

Methods

fromListP :: [(MonomP (Univariate coeff var), CoeffP (Univariate coeff var))] -> Univariate coeff var

toListP :: Univariate coeff var -> [(MonomP (Univariate coeff var), CoeffP (Univariate coeff var))]

zeroP :: Univariate coeff var

isZeroP :: Univariate coeff var -> Bool

oneP :: Univariate coeff var

variableP :: VarP (Univariate coeff var) -> Univariate coeff var

singletonP :: VarP (Univariate coeff var) -> Int -> Univariate coeff var

monomP :: MonomP (Univariate coeff var) -> Univariate coeff var

monomP' :: MonomP (Univariate coeff var) -> CoeffP (Univariate coeff var) -> Univariate coeff var

scalarP :: CoeffP (Univariate coeff var) -> Univariate coeff var

addP :: Univariate coeff var -> Univariate coeff var -> Univariate coeff var

subP :: Univariate coeff var -> Univariate coeff var -> Univariate coeff var

negP :: Univariate coeff var -> Univariate coeff var

sumP :: [Univariate coeff var] -> Univariate coeff var

mulP :: Univariate coeff var -> Univariate coeff var -> Univariate coeff var

productP :: [Univariate coeff var] -> Univariate coeff var

coeffOfP :: MonomP (Univariate coeff var) -> Univariate coeff var -> CoeffP (Univariate coeff var)

mulByMonomP :: MonomP (Univariate coeff var) -> Univariate coeff var -> Univariate coeff var

scaleP :: CoeffP (Univariate coeff var) -> Univariate coeff var -> Univariate coeff var

(Ring c, KnownSymbol v, KnownNat n) => AlmostPolynomial (Poly c v n)

Defined in Math.Algebra.Polynomial.Multivariate.Indexed

Associated Types

type CoeffP (Poly c v n)

type MonomP (Poly c v n)

type VarP (Poly c v n)

Methods

fromListP :: [(MonomP (Poly c v n), CoeffP (Poly c v n))] -> Poly c v n

toListP :: Poly c v n -> [(MonomP (Poly c v n), CoeffP (Poly c v n))]

zeroP :: Poly c v n

isZeroP :: Poly c v n -> Bool

oneP :: Poly c v n

variableP :: VarP (Poly c v n) -> Poly c v n

singletonP :: VarP (Poly c v n) -> Int -> Poly c v n

monomP :: MonomP (Poly c v n) -> Poly c v n

monomP' :: MonomP (Poly c v n) -> CoeffP (Poly c v n) -> Poly c v n

scalarP :: CoeffP (Poly c v n) -> Poly c v n

addP :: Poly c v n -> Poly c v n -> Poly c v n

subP :: Poly c v n -> Poly c v n -> Poly c v n

negP :: Poly c v n -> Poly c v n

sumP :: [Poly c v n] -> Poly c v n

mulP :: Poly c v n -> Poly c v n -> Poly c v n

productP :: [Poly c v n] -> Poly c v n

coeffOfP :: MonomP (Poly c v n) -> Poly c v n -> CoeffP (Poly c v n)

mulByMonomP :: MonomP (Poly c v n) -> Poly c v n -> Poly c v n

scaleP :: CoeffP (Poly c v n) -> Poly c v n -> Poly c v n

(Ring c, KnownSymbol v, KnownNat n) => AlmostPolynomial (Poly c v n)

Defined in Math.Algebra.Polynomial.Multivariate.Compact

Associated Types

type CoeffP (Poly c v n)

type MonomP (Poly c v n)

type VarP (Poly c v n)

Methods

fromListP :: [(MonomP (Poly c v n), CoeffP (Poly c v n))] -> Poly c v n

toListP :: Poly c v n -> [(MonomP (Poly c v n), CoeffP (Poly c v n))]

zeroP :: Poly c v n

isZeroP :: Poly c v n -> Bool

oneP :: Poly c v n

variableP :: VarP (Poly c v n) -> Poly c v n

singletonP :: VarP (Poly c v n) -> Int -> Poly c v n

monomP :: MonomP (Poly c v n) -> Poly c v n

monomP' :: MonomP (Poly c v n) -> CoeffP (Poly c v n) -> Poly c v n

scalarP :: CoeffP (Poly c v n) -> Poly c v n

addP :: Poly c v n -> Poly c v n -> Poly c v n

subP :: Poly c v n -> Poly c v n -> Poly c v n

negP :: Poly c v n -> Poly c v n

sumP :: [Poly c v n] -> Poly c v n

mulP :: Poly c v n -> Poly c v n -> Poly c v n

productP :: [Poly c v n] -> Poly c v n

coeffOfP :: MonomP (Poly c v n) -> Poly c v n -> CoeffP (Poly c v n)

mulByMonomP :: MonomP (Poly c v n) -> Poly c v n -> Poly c v n

scaleP :: CoeffP (Poly c v n) -> Poly c v n -> Poly c v n

(Ring c, KnownSymbol v, KnownNat n) => AlmostPolynomial (ExtAlg c v n)

Defined in Math.Algebra.Polynomial.Exterior.Indexed

Associated Types

type CoeffP (ExtAlg c v n)

type MonomP (ExtAlg c v n)

type VarP (ExtAlg c v n)

Methods

fromListP :: [(MonomP (ExtAlg c v n), CoeffP (ExtAlg c v n))] -> ExtAlg c v n

toListP :: ExtAlg c v n -> [(MonomP (ExtAlg c v n), CoeffP (ExtAlg c v n))]

zeroP :: ExtAlg c v n

isZeroP :: ExtAlg c v n -> Bool

oneP :: ExtAlg c v n

variableP :: VarP (ExtAlg c v n) -> ExtAlg c v n

singletonP :: VarP (ExtAlg c v n) -> Int -> ExtAlg c v n

monomP :: MonomP (ExtAlg c v n) -> ExtAlg c v n

monomP' :: MonomP (ExtAlg c v n) -> CoeffP (ExtAlg c v n) -> ExtAlg c v n

scalarP :: CoeffP (ExtAlg c v n) -> ExtAlg c v n

addP :: ExtAlg c v n -> ExtAlg c v n -> ExtAlg c v n

subP :: ExtAlg c v n -> ExtAlg c v n -> ExtAlg c v n

negP :: ExtAlg c v n -> ExtAlg c v n

sumP :: [ExtAlg c v n] -> ExtAlg c v n

mulP :: ExtAlg c v n -> ExtAlg c v n -> ExtAlg c v n

productP :: [ExtAlg c v n] -> ExtAlg c v n

coeffOfP :: MonomP (ExtAlg c v n) -> ExtAlg c v n -> CoeffP (ExtAlg c v n)

mulByMonomP :: MonomP (ExtAlg c v n) -> ExtAlg c v n -> ExtAlg c v n

scaleP :: CoeffP (ExtAlg c v n) -> ExtAlg c v n -> ExtAlg c v n

class (AlmostPolynomial p, Num p, Monomial (MonomP p), VarM (MonomP p) ~ VarP p) => Polynomial p where

The class of polynomial rings

Methods

evalP :: Num d => (CoeffP p -> d) -> (VarP p -> d) -> p -> d

varSubsP :: (VarP p -> VarP p) -> p -> p

coeffSubsP :: (VarP p -> Maybe (CoeffP p)) -> p -> p

subsP :: (VarP p -> p) -> p -> p

Instances

(Ring c, KnownSymbol v) => Polynomial (Poly c v)

Defined in Math.Algebra.Polynomial.Multivariate.Infinite

Methods

evalP :: Num d => (CoeffP (Poly c v) -> d) -> (VarP (Poly c v) -> d) -> Poly c v -> d

varSubsP :: (VarP (Poly c v) -> VarP (Poly c v)) -> Poly c v -> Poly c v

coeffSubsP :: (VarP (Poly c v) -> Maybe (CoeffP (Poly c v))) -> Poly c v -> Poly c v

subsP :: (VarP (Poly c v) -> Poly c v) -> Poly c v -> Poly c v

(Ring c, Ord v, Pretty v) => Polynomial (Poly c v)

Defined in Math.Algebra.Polynomial.Multivariate.Generic

Methods

evalP :: Num d => (CoeffP (Poly c v) -> d) -> (VarP (Poly c v) -> d) -> Poly c v -> d

varSubsP :: (VarP (Poly c v) -> VarP (Poly c v)) -> Poly c v -> Poly c v

coeffSubsP :: (VarP (Poly c v) -> Maybe (CoeffP (Poly c v))) -> Poly c v -> Poly c v

subsP :: (VarP (Poly c v) -> Poly c v) -> Poly c v -> Poly c v

(Ring coeff, KnownSymbol var) => Polynomial (Univariate coeff var)

Defined in Math.Algebra.Polynomial.Univariate

Methods

evalP :: Num d => (CoeffP (Univariate coeff var) -> d) -> (VarP (Univariate coeff var) -> d) -> Univariate coeff var -> d

varSubsP :: (VarP (Univariate coeff var) -> VarP (Univariate coeff var)) -> Univariate coeff var -> Univariate coeff var

coeffSubsP :: (VarP (Univariate coeff var) -> Maybe (CoeffP (Univariate coeff var))) -> Univariate coeff var -> Univariate coeff var

subsP :: (VarP (Univariate coeff var) -> Univariate coeff var) -> Univariate coeff var -> Univariate coeff var

(Ring c, KnownSymbol v, KnownNat n) => Polynomial (Poly c v n)

Defined in Math.Algebra.Polynomial.Multivariate.Indexed

Methods

evalP :: Num d => (CoeffP (Poly c v n) -> d) -> (VarP (Poly c v n) -> d) -> Poly c v n -> d

varSubsP :: (VarP (Poly c v n) -> VarP (Poly c v n)) -> Poly c v n -> Poly c v n

coeffSubsP :: (VarP (Poly c v n) -> Maybe (CoeffP (Poly c v n))) -> Poly c v n -> Poly c v n

subsP :: (VarP (Poly c v n) -> Poly c v n) -> Poly c v n -> Poly c v n

(Ring c, KnownSymbol v, KnownNat n) => Polynomial (Poly c v n)

Defined in Math.Algebra.Polynomial.Multivariate.Compact

Methods

evalP :: Num d => (CoeffP (Poly c v n) -> d) -> (VarP (Poly c v n) -> d) -> Poly c v n -> d

varSubsP :: (VarP (Poly c v n) -> VarP (Poly c v n)) -> Poly c v n -> Poly c v n

coeffSubsP :: (VarP (Poly c v n) -> Maybe (CoeffP (Poly c v n))) -> Poly c v n -> Poly c v n

subsP :: (VarP (Poly c v n) -> Poly c v n) -> Poly c v n -> Poly c v n

proxyCoeffP :: AlmostPolynomial p => p -> Proxy (CoeffP p)

proxyMonomP :: AlmostPolynomial p => p -> Proxy (MonomP p)

proxyVarP :: AlmostPolynomial p => p -> Proxy (VarP p)