-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | Multivariate polynomials.
--
-- Manipulation of multivariate polynomials on a ring and Gröbner bases.
@package hspray
@version 0.2.2.0
-- | Deals with multivariate polynomials on a commutative ring. See README
-- for examples.
module Math.Algebra.Hspray
data Powers
Powers :: Seq Int -> Int -> Powers
[exponents] :: Powers -> Seq Int
[nvariables] :: Powers -> Int
type Spray a = HashMap Powers a
type Monomial a = (Powers, a)
-- | Spray corresponding to the basic monomial x_n
--
--
-- >>> x :: lone 1 :: Spray Int
--
-- >>> y :: lone 2 :: Spray Int
--
-- >>> p = 2*^x^**^2 ^-^ 3*^y
--
-- >>> putStrLn $ prettySpray' p
-- (2) x1^2 + (-3) x2
--
--
--
-- lone 0 == unitSpray
--
lone :: C a => Int -> Spray a
-- | The unit spray
--
--
-- p ^*^ unitSpray == p
--
unitSpray :: C a => Spray a
-- | The null spray
--
--
-- p ^+^ zeroSpray == p
--
zeroSpray :: (Eq a, C a) => Spray a
-- | Constant spray
--
--
-- constantSpray 3 == 3 *^ unitSpray
--
constantSpray :: (C a, Eq a) => a -> Spray a
-- | Scale spray by a scalar
(*^) :: (C a, Eq a) => a -> Spray a -> Spray a
infixr 7 *^
-- | Scale spray by an integer
--
--
-- 3 .^ p == p ^+^ p ^+^ p
--
(.^) :: (C a, Eq a) => Int -> Spray a -> Spray a
infixr 7 .^
-- | 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 6 ^-^
-- | Multiply two sprays
(^*^) :: (C a, Eq a) => Spray a -> Spray a -> Spray a
infixl 7 ^*^
-- | Power of a spray
(^**^) :: (C a, Eq a) => Spray a -> Int -> Spray a
infixr 8 ^**^
-- | Pretty form of a spray
--
--
-- >>> x :: lone 1 :: Spray Int
--
-- >>> y :: lone 2 :: Spray Int
--
-- >>> z :: lone 3 :: Spray Int
--
-- >>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
--
-- >>> putStrLn $ prettySpray show "x" p
-- (2) * x^(1) + (3) * x^(0, 2) + (-4) * x^(0, 0, 3)
--
prettySpray :: (a -> String) -> String -> Spray a -> String
-- | Pretty form of a spray, with monomials showed as "x1x3^2"
--
--
-- >>> x :: lone 1 :: Spray Int
--
-- >>> y :: lone 2 :: Spray Int
--
-- >>> z :: lone 3 :: Spray Int
--
-- >>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
--
-- >>> putStrLn $ prettySpray' p
-- (2) x1 + (3) x2^2 + (-4) x3^3
--
prettySpray' :: Show a => Spray a -> String
-- | Pretty form of a spray having at more three variables
--
--
-- >>> x :: lone 1 :: Spray Int
--
-- >>> y :: lone 2 :: Spray Int
--
-- >>> z :: lone 3 :: Spray Int
--
-- >>> p = 2*^x ^+^ 3*^y^**^2 ^-^ 4*^z^**^3
--
-- >>> putStrLn $ prettySprayXYZ p
-- (2) X + (3) Y^2 + (-4) Z^3
--
prettySprayXYZ :: Show a => Spray a -> String
-- | Get coefficient of a term of a spray
--
--
-- >>> x = lone 1 :: Spray Int
--
-- >>> y = lone 2 :: Spray Int
--
-- >>> z = lone 3 :: Spray Int
--
-- >>> p = 2 *^ (2 *^ (x^**^3 ^*^ y^**^2)) ^+^ 4*^z ^+^ 5*^unitSpray
--
-- >>> getCoefficient [3, 2, 0] p
-- 4
--
-- >>> getCoefficient [0, 4] p
-- 0
--
getCoefficient :: C a => [Int] -> Spray a -> a
-- | Terms of a spray
sprayTerms :: Spray a -> HashMap (Seq Int) a
-- | Evaluates a spray
--
--
-- >>> x :: lone 1 :: Spray Int
--
-- >>> y :: lone 2 :: Spray Int
--
-- >>> p = 2*^x^**^2 ^-^ 3*^y
--
-- >>> evalSpray p [2, 1]
-- 5
--
evalSpray :: C a => Spray a -> [a] -> a
-- | Substitutes some variables in a spray
--
--
-- >>> x1 :: lone 1 :: Spray Int
--
-- >>> x2 :: lone 2 :: Spray Int
--
-- >>> x3 :: lone 3 :: Spray Int
--
-- >>> p = x1^**^2 ^-^ x2 ^+^ x3 ^-^ unitSpray
--
-- >>> p' = substituteSpray [Just 2, Nothing, Just 3] p
--
-- >>> putStrLn $ prettySpray' p'
-- (-1) x2 + (6)
--
substituteSpray :: (Eq a, C a) => [Maybe a] -> Spray a -> Spray a
-- | Composes a spray with a change of variables
--
--
-- >>> x :: lone 1 :: Spray Int
--
-- >>> y :: lone 2 :: Spray Int
--
-- >>> z :: lone 3 :: Spray Int
--
-- >>> p = x ^+^ y
--
-- >>> q = composeSpray p [z, x ^+^ y ^+^ z]
--
-- >>> putStrLn $ prettySprayXYZ q
-- (1) X + (1) Y + (2) Z
--
composeSpray :: (C a, Eq a) => Spray a -> [Spray a] -> Spray a
-- | Derivative of a spray
derivSpray :: (C a, Eq a) => Int -> Spray a -> Spray a
-- | Permutes the variables of a spray
--
--
-- >>> f :: Spray Rational -> Spray Rational -> Spray Rational -> Spray Rational
--
-- >>> f p1 p2 p3 = p1^**^4 ^+^ (2*^p2^**^3) ^+^ (3*^p3^**^2) ^-^ (4*^unitSpray)
--
-- >>> x1 = lone 1 :: Spray Rational
--
-- >>> x2 = lone 2 :: Spray Rational
--
-- >>> x3 = lone 3 :: Spray Rational
--
-- >>> p = f x1 x2 x3
--
--
--
-- permuteVariables p [3, 1, 2] == f x3 x1 x2
--
permuteVariables :: Spray a -> [Int] -> Spray a
-- | Swaps two variables of a spray
--
--
-- swapVariables (1, 3) p == permuteVariables p [3, 2, 1]
--
swapVariables :: Spray a -> (Int, Int) -> Spray a
-- | Remainder of the division of a spray by a list of divisors, using the
-- lexicographic ordering of the monomials
sprayDivision :: forall a. (Eq a, C a) => Spray a -> [Spray a] -> Spray a
-- | Groebner basis (always minimal and possibly reduced)
--
--
-- groebner ps True = reduceGroebnerBasis (groebner ps False)
--
groebner :: forall a. (Eq a, C a) => [Spray a] -> Bool -> [Spray a]
-- | Reduces a Groebner basis
reduceGroebnerBasis :: forall a. (Eq a, C a) => [Spray a] -> [Spray a]
-- | Elementary symmetric polynomial
--
--
-- >>> putStrLn $ prettySpray' (esPolynomial 3 2)
-- (1) x1x2 + (1) x1x3 + (1) x2x3
--
esPolynomial :: (C a, Eq a) => Int -> Int -> Spray a
-- | Whether a spray is a symmetric polynomial
isSymmetricSpray :: forall a. (C a, Eq a) => Spray a -> Bool
-- | Resultant of two sprays
resultant :: (Eq a, C a) => Int -> Spray a -> Spray a -> Spray a
-- | Resultant of two univariate sprays
resultant1 :: (Eq a, C a) => Spray a -> Spray a -> a
-- | Subresultants of two sprays
subresultants :: (Eq a, C a) => Int -> Spray a -> Spray a -> [Spray a]
-- | Subresultants of two univariate sprays
subresultants1 :: (Eq a, C a) => Spray a -> Spray a -> [a]
-- | Creates a spray from list of terms
fromList :: (C a, Eq a) => [([Int], a)] -> Spray a
-- | Spray as a list
toList :: Spray a -> [([Int], a)]
-- | Converts a spray with rational coefficients to a spray with double
-- coefficients (useful for evaluation)
fromRationalSpray :: Spray Rational -> Spray Double
-- | Leading term of a spray
leadingTerm :: Spray a -> Monomial 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); this polynomial is returned if this is true
--
--
-- >>> x = lone 1 :: Spray Rational
--
-- >>> y = lone 2 :: Spray Rational
--
-- >>> p1 = x ^+^ y
--
-- >>> p2 = x ^-^ y
--
-- >>> p = p1 ^*^ p2
--
--
--
-- isPolynomialOf p [p1, p2] == (True, Just $ x ^*^ y)
--
isPolynomialOf :: forall a. (C a, Eq a) => Spray a -> [Spray a] -> (Bool, Maybe (Spray a))
-- | Bombieri spray (for internal usage in the 'scubature' library)
bombieriSpray :: C a => Spray a -> Spray a
instance GHC.Show.Show Math.Algebra.Hspray.Powers
instance (Algebra.Additive.C a, GHC.Classes.Eq a) => Algebra.Additive.C (Math.Algebra.Hspray.Spray a)
instance (Algebra.Ring.C a, GHC.Classes.Eq a) => Algebra.Module.C a (Math.Algebra.Hspray.Spray a)
instance (Algebra.Ring.C a, GHC.Classes.Eq a) => Algebra.Ring.C (Math.Algebra.Hspray.Spray a)
instance GHC.Classes.Eq Math.Algebra.Hspray.Powers
instance Data.Hashable.Class.Hashable Math.Algebra.Hspray.Powers