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@stla
stla committed May 2, 2024
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Showing 1 changed file with 45 additions and 50 deletions.
95 changes: 45 additions & 50 deletions src/Math/Algebra/Hspray.hs
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Original file line number Diff line number Diff line change
Expand Up @@ -185,6 +185,7 @@ module Math.Algebra.Hspray
, isConstantSpray
, isHomogeneousSpray
, allExponents
, allCoefficients
-- * Evaluation of a spray
, evalSpray
, substituteSpray
Expand Down Expand Up @@ -296,10 +297,10 @@ import qualified Number.Ratio as NumberRatio
-- introduce a class because it will be assigned to the ratios of sprays too.
class FunctionLike b where

-- | Number of variables
-- | Number of variables in a function-like object
numberOfVariables :: b -> Int

-- | Permutes the variables
-- | Permutes the variables of a function-like object
--
-- >>> f :: Spray Rational -> Spray Rational -> Spray Rational -> Spray Rational
-- >>> f p1 p2 p3 = p1^**^4 ^+^ (2*^p2^**^3) ^+^ (3*^p3^**^2) ^-^ (4*^unitSpray)
Expand All @@ -311,16 +312,16 @@ class FunctionLike b where
-- prop> permuteVariables [3, 1, 2] spray == f x3 x1 x2
permuteVariables ::
[Int] -- ^ permutation
-> b -- ^ the object whose variables will be permuted
-> b -- ^ the object with permuted variables
-> b -- ^ function-like object whose variables will be permuted
-> b -- ^ the function-like object with permuted variables

-- | Swaps two variables
-- | Swaps two variables of a function-like object
--
-- prop> swapVariables (1, 3) x == permuteVariables [3, 2, 1] x
swapVariables ::
(Int, Int) -- ^ the indices of the variables to be swapped (starting at 1)
-> b -- ^ the object whose variables will be swapped
-> b -- ^ the object with swapped variables
-> b -- ^ function-like object whose variables will be swapped
-> b -- ^ the function-like object with swapped variables

-- Whether a variable is involved in a function-like object
--
Expand All @@ -330,16 +331,17 @@ class FunctionLike b where
-> Int -- ^ index of the variable
-> Bool

-- | Drops the first @n@ variables; __very unsafe__, @dropVariables n x@ should
-- /not/ be used if @involvesVariable x i@ is @True@ for some @i@ in @1, ... n@
-- | Drops a given number of leading variables in a function-like object;
-- __very unsafe__, @dropVariables n x@ should /not/ be used if
-- @involvesVariable x i@ is @True@ for some @i@ in @1, ... n@
--
-- prop> dropVariables 1 (qlone 2 ^+^ qlone 3) == qlone 1 ^+^ qlone 2
dropVariables ::
Int -- ^ number of variables to drop
Int -- ^ number of leading variables to drop
-> b -- ^ a function-like object
-> b

-- | Derivative
-- | Derivative of a function-like object
--
-- >>> x = lone 1 :: Spray Int
-- >>> y = lone 2 :: Spray Int
Expand All @@ -359,43 +361,44 @@ class FunctionLike b where
type family VariablesType b

infixl 6 ^+^
-- | Addition
-- | Addition of two function-like objects
(^+^) :: (AlgAdd.C b) => b -> b -> b
(^+^) = (AlgAdd.+)

infixl 6 ^-^
-- | Substraction
-- | Substraction of two function-like objects
(^-^) :: (AlgAdd.C b) => b -> b -> b
(^-^) = (AlgAdd.-)

infixl 7 ^*^
-- | Multiplication
-- | Multiplication of two function-like objects
(^*^) :: (AlgRing.C b) => b -> b -> b
(^*^) = (AlgRing.*)

infixr 8 ^**^
-- | Power
-- | Power of a function-like object
(^**^) :: (AlgRing.C b) => b -> Int -> b
(^**^) p k = p AlgRing.^ fromIntegral k

infixr 7 *^
-- | Multiply by a scalar
-- | Multiply a function-like object by a scalar
(*^) :: BaseRing b -> b -> b

infixl 6 +>
-- | Add a constant at left
-- | Add a function-like object to a constant
--
-- prop> x +> spray == constantSpray x ^+^ spray
(+>) :: BaseRing b -> b -> b

infixr 6 <+
-- | Add a constant at right
-- | Add a constant to a function-like object
--
-- prop> object <+ x == x +> object
(<+) :: b -> BaseRing b -> b
(<+) = flip (+>)

-- | Evaluation (replacing the variables with some values)
-- | Evaluation (replacing the variables with some values) of a
-- function-like object
--
-- >>> x = lone 1 :: Spray Int
-- >>> y = lone 2 :: Spray Int
Expand All @@ -413,7 +416,8 @@ class FunctionLike b where
evaluateAt :: [BaseRing b] -> b -> BaseRing b
evaluateAt = flip evaluate

-- | Partial evaluation
-- | Partial evaluation of a function-like object (replace some variables
-- with some values)
--
-- >>> x1 = lone 1 :: Spray Int
-- >>> x2 = lone 2 :: Spray Int
Expand All @@ -427,7 +431,7 @@ class FunctionLike b where
-> b -- ^ function-like object to be partially evaluated
-> b

-- | Polynomial change of variables
-- | Polynomial change of variables of a function-like object
--
-- >>> x = lone 1 :: Spray Int
-- >>> y = lone 2 :: Spray Int
Expand Down Expand Up @@ -609,12 +613,6 @@ instance (Eq a, AlgField.C a) => FunctionLike (RatioOfPolynomials a) where
changeVariables r ps = changeVariables (NumberRatio.numerator r) ps %
changeVariables (NumberRatio.denominator r) ps

{- -- | Division of univariate polynomials; this is an application of `:%`
-- followed by a simplification of the obtained fraction of the two polynomials
(^/^) :: (Eq a, AlgField.C a)
=> Polynomial a -> Polynomial a -> RatioOfPolynomials a
(^/^) pol1 pol2 = simplifyRatioOfPolynomials $ pol1 :% pol2
-}
instance (Eq a, AlgField.C a) => AlgZT.C (A a) where
isZero :: A a -> Bool
isZero (A r) = r == AlgAdd.zero
Expand Down Expand Up @@ -1253,16 +1251,7 @@ instance (AlgRing.C a, Eq a) => AlgRing.C (Spray a) where
(*) :: Spray a -> Spray a -> Spray a
p * q = multSprays p q
one :: Spray a
one = lone 0

{- instance (AlgRing.C a, Eq a) => Num (Spray a) where
p + q = addSprays p q
negate = negateSpray
p * q = multSprays p q
fromInteger n = fromInteger n .^ AlgRing.one
abs _ = error "Prelude.Num.abs: inappropriate abstraction"
signum _ = error "Prelude.Num.signum: inappropriate abstraction"
-}
one = HM.singleton (Powers S.empty 0) AlgRing.one

infixr 7 /^
-- | Divides a spray by a scalar; you can equivalently use `(/>)` if the type
Expand Down Expand Up @@ -1377,16 +1366,20 @@ constantSpray c = if c == AlgAdd.zero
-- >>> 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
-- >>> getCoefficient [3, 2] p -- coefficient of x^3.y^2
-- 4
-- >>> getCoefficient [3, 2, 0] p -- same as getCoefficient [3, 2] p
-- 4
-- >>> getCoefficient [0, 4] p
-- >>> getCoefficient [0, 4] p -- coefficient of y^4
-- 0
getCoefficient :: AlgAdd.C a => [Int] -> Spray a -> a
getCoefficient ::
AlgAdd.C a
=> [Int] -- ^ sequence of exponents; trailing zeros are dropped
-> Spray a -- ^ spray
-> a -- ^ coefficient of the monomial given by the sequence of exponents
getCoefficient expnts = getCoefficient' powers
where
powers = makePowers (S.fromList expnts)
-- expnts' = S.dropWhileR (== 0) (S.fromList expnts)
-- powers = Powers expnts' (S.length expnts')

getCoefficient' :: AlgAdd.C a => Powers -> Spray a -> a
getCoefficient' powers spray = fromMaybe AlgAdd.zero (HM.lookup powers spray)
Expand Down Expand Up @@ -1649,7 +1642,7 @@ prettySpray'' var = showSpray show ("(", ")") (showMonomialsOld var)
showNumSpray ::
(Num a, Ord a)
=> ([Exponents] -> [String]) -- ^ function mapping a list of monomial exponents to a list of strings representing the monomials
-> (a -> String) -- ^ function mapping a positive coefficient to a string
-> (a -> String) -- ^ function mapping a positive coefficient to a string
-> Spray a
-> String
showNumSpray showMonomials showCoeff spray =
Expand Down Expand Up @@ -1884,13 +1877,13 @@ orderedTerms spray =
toList :: Spray a -> [([Int], a)]
toList p = HM.toList $ HM.mapKeys (DF.toList . exponents) p

-- | Bombieri spray (for internal usage in the \'scubature\' library)
-- | Bombieri spray (for internal usage in the \'__scubature__\' package)
bombieriSpray :: (Eq a, AlgAdd.C a) => Spray a -> Spray a
bombieriSpray = HM.mapWithKey f
where
f pows = times (pfactorial $ exponents pows)
pfactorial pows = product $ DF.toList $ factorial <$> S.filter (/= 0) pows
factorial n = product [1 .. n]
factorial n = product [2 .. n]
times k x = k .^ x

-- | Whether two sprays are equal up to a scalar factor
Expand All @@ -1916,6 +1909,9 @@ isHomogeneousSpray spray
allExponents :: Spray a -> [Exponents]
allExponents spray = map exponents (HM.keys spray)

-- | Get all the coefficients of a spray
allCoefficients :: Spray a -> [a]
allCoefficients = HM.elems

-- division stuff -------------------------------------------------------------

Expand Down Expand Up @@ -3286,7 +3282,7 @@ numberOfParameters pspray =
if isZeroSpray pspray
then 0
else
maximum (map numberOfVariables (HM.elems pspray))
maximum (map numberOfVariables (allCoefficients pspray))

-- | Apply polynomial transformations to the parameters of a parametric spray;
-- e.g. you have a two-parameters polynomial \(P_{a, b}(X, Y, Z)\) and you want
Expand Down Expand Up @@ -3363,7 +3359,7 @@ evalParametricSpray' spray as xs =
canCoerceToSimpleParametricSpray ::
(Eq a, AlgRing.C a) => ParametricSpray a -> Bool
canCoerceToSimpleParametricSpray spray =
all isPolynomialRatioOfSprays (HM.elems spray)
all isPolynomialRatioOfSprays (allCoefficients spray)

-- | Coerces a parametric spray to a simple parametric spray, without
-- checking this makes sense with `canCoerceToSimpleParametricSpray`
Expand Down Expand Up @@ -3482,13 +3478,12 @@ gegenbauerPolynomial n
| n == 0 = unitSpray
| n == 1 = (2.^a) *^ x
| otherwise =
(2.^(n'' ^+^ a) /^ n') *^ (x ^*^ gegenbauerPolynomial (n - 1)) ^-^
((n'' ^+^ 2.^a ^-^ unitSpray) /^ n') *^ gegenbauerPolynomial (n - 2)
(2.^((n'-1) +> a) /^ n') *^ (x ^*^ gegenbauerPolynomial (n - 1)) ^-^
(((n'-1) +> 2.^a ^-^ unitSpray) /^ n') *^ gegenbauerPolynomial (n - 2)
where
x = lone 1 :: SimpleParametricQSpray
a = lone 1 :: QSpray
n' = toRational n
n'' = constantSpray (n' - 1)

-- | [Jacobi polynomial](https://en.wikipedia.org/wiki/Jacobi_polynomials);
-- the @n@-th Jacobi polynomial is a univariate polynomial of degree @n@ with
Expand Down

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