Math.Algebra.Hspray

Deals with multivariate polynomials on a ring. See README for examples.

Documentation

data Powers

Constructors

Powers

Fields

• exponents :: Seq Int
• nvariables :: Int

Instances

Show Powers

Defined in Math.Algebra.Hspray

Methods

showsPrec :: Int -> Powers -> ShowS

show :: Powers -> String

showList :: [Powers] -> ShowS

Eq Powers

Defined in Math.Algebra.Hspray

Methods

(==) :: Powers -> Powers -> Bool

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

Hashable Powers

Defined in Math.Algebra.Hspray

Methods

hashWithSalt :: Int -> Powers -> Int

hash :: Powers -> Int

(C a, Eq a) => C a (Spray a)

Defined in Math.Algebra.Hspray

Methods

(*>) :: a -> Spray a -> Spray a

(C a, Eq a) => C (Spray a)

Defined in Math.Algebra.Hspray

Methods

zero :: Spray a

(+) :: Spray a -> Spray a -> Spray a

(-) :: Spray a -> Spray a -> Spray a

negate :: Spray a -> Spray a

(C a, Eq a) => C (Spray a)

Defined in Math.Algebra.Hspray

Methods

(*) :: Spray a -> Spray a -> Spray a

one :: Spray a

fromInteger :: Integer -> Spray a

(^) :: Spray a -> Integer -> Spray a

type Spray a = HashMap Powers a

type Monomial a = (Powers, a)

lone :: C a => Int -> Spray a

Spray corresponding to the basic monomial x_n

unitSpray :: C a => Spray a

Unit spray

constantSpray :: (C a, Eq a) => a -> Spray a

Constant spray

fromList :: (C a, Eq a) => [([Int], a)] -> Spray a

Create a spray from list of terms

toList :: Spray a -> [([Int], a)]

Spray as list

sprayTerms :: Spray a -> HashMap (Seq Int) a

Terms of a spray

(*^) :: (C a, Eq a) => a -> Spray a -> Spray a infixr 7

Scale spray by a scalar

(.^) :: (C a, Eq a) => Int -> Spray a -> Spray a infixr 7

Scale spray by an integer

(^+^) :: (C a, Eq a) => Spray a -> Spray a -> Spray a infixl 6

Addition of two sprays

(^-^) :: (C a, Eq a) => Spray a -> Spray a -> Spray a infixl 6

Substraction of two sprays

(^*^) :: (C a, Eq a) => Spray a -> Spray a -> Spray a infixl 7

Multiply two sprays

(^**^) :: (C a, Eq a) => Spray a -> Int -> Spray a infixr 8

Power of a spray

evalSpray :: C a => Spray a -> [a] -> a

Evaluate a spray

substituteSpray :: (Eq a, C a) => [Maybe a] -> Spray a -> Spray a

Substitute some variables in a spray

fromRationalSpray :: Spray Rational -> Spray Double

Convert a spray with rational coefficients to a spray with double coefficients

prettySpray

Arguments

:: (a -> String)

function mapping a coefficient to a string, typically show

-> String

a string denoting the variable, e.g. "x"

-> Spray a

the spray

-> String

Pretty form of a spray

prettySpray' :: Show a => Spray a -> String

Pretty form of a spray, with monomials showed as "x1x3^2"

prettySprayXYZ :: Show a => Spray a -> String

Pretty form of a spray having at more three variables

composeSpray :: (C a, Eq a) => Spray a -> [Spray a] -> Spray a

Compose a spray with a change of variables

bombieriSpray :: C a => Spray a -> Spray a

Bombieri spray (for internal usage in the scubature library)

derivSpray

Arguments

:: (C a, Eq a)

=> Int

index of the variable of differentiation

-> Spray a

the spray

-> Spray a

Derivative of a spray

leadingTerm :: Spray a -> Monomial a

Leading term of a spray

sprayDivision :: forall a. (Eq a, C a) => Spray a -> [Spray a] -> Spray a

Remainder of the division of a spray by a list of divisors, using the lexicographic ordering of the monomials

groebner

Arguments

:: forall a. (Eq a, C a)

=> [Spray a]

list of sprays

-> Bool

whether to return the reduced basis

-> [Spray a]

Groebner basis (always minimal and possibly reduced)

esPolynomial

Arguments

:: (C a, Eq a)

=> Int

number of variables

-> Int

index

-> Spray a

Elementary symmetric polynomial

isSymmetricSpray :: forall a. (C a, Eq a) => Spray a -> Bool

Whether a spray is a symmetric polynomial

isPolynomialOf :: forall a. (C a, Eq a) => Spray a -> [Spray a] -> (Bool, Maybe (Spray a))

Whether a spray can be written as a polynomial of a given list of sprays; the sprays in the list must belong to the same polynomial ring as the spray