{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE InstanceSigs #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE TypeFamilies #-}
module Math.Algebra.Hspray
(
HasVariables (..)
, isConstant
, isUnivariate
, isBivariate
, isTrivariate
, Powers (..)
, Spray
, QSpray
, QSpray'
, Monomial
, lone
, qlone
, unitSpray
, zeroSpray
, constantSpray
, (*^)
, (/^)
, (^+^)
, (^-^)
, (^*^)
, (^**^)
, prettySpray
, prettySpray'
, prettySpray''
, prettySprayXYZ
, prettySprayX1X2X3
, showSpray
, showSprayXYZ
, showSprayXYZ'
, showSprayX1X2X3
, showSprayX1X2X3'
, showNumSpray
, showQSpray
, showQSpray'
, prettyNumSprayX1X2X3
, prettyQSprayX1X2X3
, prettyQSprayX1X2X3'
, prettyNumSprayXYZ
, prettyQSprayXYZ
, prettyQSprayXYZ'
, prettyNumSpray
, prettyNumSpray'
, prettyQSpray
, prettyQSpray''
, prettyQSpray'
, prettyQSpray'''
, A (..)
, Rational'
, Q
, scalarQ
, Polynomial
, RatioOfPolynomials
, QPolynomial
, RatioOfQPolynomials
, (^/^)
, prettyRatioOfPolynomials
, prettyRatioOfQPolynomials
, (*.)
, constPoly
, polyFromCoeffs
, soleParameter
, constQPoly
, qpolyFromCoeffs
, qsoleParameter
, evalRatioOfPolynomials
, OneParameterSpray
, OneParameterQSpray
, prettyOneParameterSprayX1X2X3
, prettyOneParameterSprayXYZ
, prettyOneParameterSpray
, prettyOneParameterSpray'
, prettyOneParameterQSprayX1X2X3
, prettyOneParameterQSprayXYZ
, prettyOneParameterQSpray
, prettyOneParameterQSpray'
, evalOneParameterSpray
, evalOneParameterSpray'
, evalOneParameterSpray''
, RatioOfSprays (..)
, RatioOfQSprays
, (%:%)
, (%//%)
, (%/%)
, isConstantRatioOfSprays
, isPolynomialRatioOfSprays
, zeroRatioOfSprays
, zeroROS
, unitRatioOfSprays
, unitROS
, constantRatioOfSprays
, asRatioOfSprays
, evalRatioOfSprays
, substituteRatioOfSprays
, fromRatioOfPolynomials
, fromRatioOfQPolynomials
, showRatioOfSprays
, showRatioOfNumSprays
, showRatioOfQSprays
, showRatioOfSpraysXYZ
, showRatioOfSpraysXYZ'
, showRatioOfSpraysX1X2X3
, showRatioOfSpraysX1X2X3'
, prettyRatioOfQSpraysXYZ
, prettyRatioOfQSpraysX1X2X3
, prettyRatioOfQSprays
, prettyRatioOfQSprays'
, prettyRatioOfNumSpraysXYZ
, prettyRatioOfNumSpraysX1X2X3
, prettyRatioOfNumSprays
, prettyRatioOfNumSprays'
, SimpleParametricSpray
, SimpleParametricQSpray
, ParametricSpray
, ParametricQSpray
, canCoerceToSimpleParametricSpray
, asSimpleParametricSprayUnsafe
, asSimpleParametricSpray
, fromOneParameterSpray
, fromOneParameterQSpray
, parametricSprayToOneParameterSpray
, parametricQSprayToOneParameterQSpray
, gegenbauerPolynomial
, jacobiPolynomial
, numberOfParameters
, changeParameters
, substituteParameters
, evalParametricSpray
, prettyParametricQSprayABCXYZ
, prettyParametricQSpray
, prettySimpleParametricQSprayABCXYZ
, prettySimpleParametricQSpray
, getCoefficient
, getConstantTerm
, isConstantSpray
, sprayTerms
, evalSpray
, substituteSpray
, composeSpray
, evalSpraySpray
, sprayDivision
, sprayDivisionRemainder
, groebner
, reduceGroebnerBasis
, esPolynomial
, psPolynomial
, isSymmetricSpray
, resultant
, resultant'
, resultant1
, subresultants
, subresultants1
, gcdSpray
, detLaplace
, detLaplace'
, characteristicPolynomial
, (.^)
, (/>)
, fromList
, toList
, fromRationalSpray
, leadingTerm
, isPolynomialOf
, bombieriSpray
, collinearSprays
) where
import qualified Algebra.Additive as AlgAdd
import qualified Algebra.Differential as AlgDiff
import qualified Algebra.Field as AlgField
import qualified Algebra.Module as AlgMod
import qualified Algebra.RightModule as AlgRightMod
import qualified Algebra.Ring as AlgRing
import qualified Algebra.ZeroTestable as AlgZT
import qualified Data.Foldable as DF
import Data.Function ( on )
import Data.HashMap.Strict ( HashMap )
import qualified Data.HashMap.Strict as HM
import Data.Hashable ( Hashable(hashWithSalt) )
import qualified Data.IntMap.Strict as IM
import Data.List ( sortBy
, maximumBy
, (\\)
, findIndices
, elemIndices
, nub
, foldl1'
, uncons
)
import Data.Matrix ( Matrix
, fromLists
, minorMatrix
, nrows
, ncols
, submatrix
)
import qualified Data.Matrix as DM
import Data.Maybe ( isJust
, isNothing
, fromJust
, fromMaybe
)
import Data.Ord ( comparing )
import qualified Data.Ratio as DR
import qualified GHC.Real as DR
import qualified Data.Sequence as S
import Data.Sequence ( (><)
, Seq
, dropWhileR
, (|>)
, index
, adjust
, fromFunction
)
import Data.Text ( Text
, append
, cons
, intercalate
, pack
, snoc
, unpack
)
import Data.Tuple.Extra ( both )
import qualified MathObj.Matrix as MathMatrix
import qualified MathObj.Polynomial as MathPol
import Number.Ratio ( T ( (:%) ), (%) )
import qualified Number.Ratio as NumberRatio
class HasVariables b where
numberOfVariables :: b -> Int
permuteVariables ::
[Int]
-> b
-> b
swapVariables ::
(Int, Int)
-> b
-> b
derivative ::
Int
-> b
-> b
type family BaseRing b
type family VariablesType b
evaluate :: b -> [BaseRing b] -> BaseRing b
evaluateAt :: [BaseRing b] -> b -> BaseRing b
evaluateAt = (b -> [BaseRing b] -> BaseRing b)
-> [BaseRing b] -> b -> BaseRing b
forall a b c. (a -> b -> c) -> b -> a -> c
flip b -> [BaseRing b] -> BaseRing b
forall b. HasVariables b => b -> [BaseRing b] -> BaseRing b
evaluate
substitute :: [Maybe (BaseRing b)] -> b -> b
changeVariables ::
b
-> [VariablesType b]
-> b
isConstant :: HasVariables b => b -> Bool
isConstant :: forall b. HasVariables b => b -> Bool
isConstant b
f = b -> Int
forall b. HasVariables b => b -> Int
numberOfVariables b
f Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0
isUnivariate :: HasVariables b => b -> Bool
isUnivariate :: forall b. HasVariables b => b -> Bool
isUnivariate b
f = b -> Int
forall b. HasVariables b => b -> Int
numberOfVariables b
f Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
1
isBivariate :: HasVariables b => b -> Bool
isBivariate :: forall b. HasVariables b => b -> Bool
isBivariate b
f = b -> Int
forall b. HasVariables b => b -> Int
numberOfVariables b
f Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
2
isTrivariate :: HasVariables b => b -> Bool
isTrivariate :: forall b. HasVariables b => b -> Bool
isTrivariate b
f = b -> Int
forall b. HasVariables b => b -> Int
numberOfVariables b
f Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
3
infixr 7 />
(/>) :: (AlgField.C k, AlgMod.C k a) => a -> k -> a
a
x /> :: forall k a. (C k, C k a) => a -> k -> a
/> k
lambda = k -> k
forall a. C a => a -> a
AlgField.recip k
lambda k -> a -> a
forall a v. C a v => a -> v -> v
AlgMod.*> a
x
infixr 7 .^
(.^) :: (AlgAdd.C a, Eq a) => Int -> a -> a
Int
k .^ :: forall a. (C a, Eq a) => Int -> a -> a
.^ a
x = if Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
0
then (a -> a -> a) -> a -> a -> Int -> a
forall {t} {t}. Integral t => (t -> t -> t) -> t -> t -> t -> t
powerOperation a -> a -> a
forall a. C a => a -> a -> a
(AlgAdd.+) a
forall a. C a => a
AlgAdd.zero a
x Int
k
else Int -> a -> a
forall a. (C a, Eq a) => Int -> a -> a
(.^) (-Int
k) (a -> a
forall a. C a => a -> a
AlgAdd.negate a
x)
where
powerOperation :: (t -> t -> t) -> t -> t -> t -> t
powerOperation t -> t -> t
op =
let go :: t -> t -> t -> t
go t
acc t
_ t
0 = t
acc
go t
acc t
a t
n = t -> t -> t -> t
go (if t -> Bool
forall a. Integral a => a -> Bool
even t
n then t
acc else t -> t -> t
op t
acc t
a) (t -> t -> t
op t
a t
a) (t -> t -> t
forall a. Integral a => a -> a -> a
div t
n t
2)
in t -> t -> t -> t
go
newtype A a = A a
deriving
(A a -> A a -> Bool
(A a -> A a -> Bool) -> (A a -> A a -> Bool) -> Eq (A a)
forall a. Eq a => A a -> A a -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
$c== :: forall a. Eq a => A a -> A a -> Bool
== :: A a -> A a -> Bool
$c/= :: forall a. Eq a => A a -> A a -> Bool
/= :: A a -> A a -> Bool
Eq, Int -> A a -> ShowS
[A a] -> ShowS
A a -> String
(Int -> A a -> ShowS)
-> (A a -> String) -> ([A a] -> ShowS) -> Show (A a)
forall a. Show a => Int -> A a -> ShowS
forall a. Show a => [A a] -> ShowS
forall a. Show a => A a -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
$cshowsPrec :: forall a. Show a => Int -> A a -> ShowS
showsPrec :: Int -> A a -> ShowS
$cshow :: forall a. Show a => A a -> String
show :: A a -> String
$cshowList :: forall a. Show a => [A a] -> ShowS
showList :: [A a] -> ShowS
Show, A a
A a -> A a
A a -> A a -> A a
A a
-> (A a -> A a -> A a)
-> (A a -> A a -> A a)
-> (A a -> A a)
-> C (A a)
forall a. a -> (a -> a -> a) -> (a -> a -> a) -> (a -> a) -> C a
forall a. C a => A a
forall a. C a => A a -> A a
forall a. C a => A a -> A a -> A a
$czero :: forall a. C a => A a
zero :: A a
$c+ :: forall a. C a => A a -> A a -> A a
+ :: A a -> A a -> A a
$c- :: forall a. C a => A a -> A a -> A a
- :: A a -> A a -> A a
$cnegate :: forall a. C a => A a -> A a
negate :: A a -> A a
AlgAdd.C, C (A a)
A a
Integer -> A a
C (A a) =>
(A a -> A a -> A a)
-> A a -> (Integer -> A a) -> (A a -> Integer -> A a) -> C (A a)
A a -> Integer -> A a
A a -> A a -> A a
forall a.
C a =>
(a -> a -> a) -> a -> (Integer -> a) -> (a -> Integer -> a) -> C a
forall a. C a => C (A a)
forall a. C a => A a
forall a. C a => Integer -> A a
forall a. C a => A a -> Integer -> A a
forall a. C a => A a -> A a -> A a
$c* :: forall a. C a => A a -> A a -> A a
* :: A a -> A a -> A a
$cone :: forall a. C a => A a
one :: A a
$cfromInteger :: forall a. C a => Integer -> A a
fromInteger :: Integer -> A a
$c^ :: forall a. C a => A a -> Integer -> A a
^ :: A a -> Integer -> A a
AlgRing.C, C (A a)
C (A a) =>
(A a -> A a -> A a)
-> (A a -> A a)
-> (Rational' -> A a)
-> (A a -> Integer -> A a)
-> C (A a)
Rational' -> A a
A a -> A a
A a -> Integer -> A a
A a -> A a -> A a
forall a.
C a =>
(a -> a -> a)
-> (a -> a) -> (Rational' -> a) -> (a -> Integer -> a) -> C a
forall a. C a => C (A a)
forall a. C a => Rational' -> A a
forall a. C a => A a -> A a
forall a. C a => A a -> Integer -> A a
forall a. C a => A a -> A a -> A a
$c/ :: forall a. C a => A a -> A a -> A a
/ :: A a -> A a -> A a
$crecip :: forall a. C a => A a -> A a
recip :: A a -> A a
$cfromRational' :: forall a. C a => Rational' -> A a
fromRational' :: Rational' -> A a
$c^- :: forall a. C a => A a -> Integer -> A a
^- :: A a -> Integer -> A a
AlgField.C)
type Rational' = NumberRatio.Rational
type Q = A Rational'
scalarQ :: Rational' -> Q
scalarQ :: Rational' -> A Rational'
scalarQ = Rational' -> A Rational'
forall a. a -> A a
A
type Polynomial a = MathPol.T (A a)
type RatioOfPolynomials a = NumberRatio.T (Polynomial a)
type QPolynomial = Polynomial Rational'
type RatioOfQPolynomials = RatioOfPolynomials Rational'
instance (Eq a, AlgField.C a) => HasVariables (Polynomial a) where
numberOfVariables :: Polynomial a -> Int
numberOfVariables :: Polynomial a -> Int
numberOfVariables Polynomial a
p = case Polynomial a -> Maybe Int
forall a. C a => T a -> Maybe Int
MathPol.degree Polynomial a
p of
Maybe Int
Nothing -> Int
0
Just Int
d -> Int -> Int -> Int
forall a. Ord a => a -> a -> a
min Int
1 Int
d
type BaseRing (Polynomial a) = a
type VariablesType (Polynomial a) = Polynomial a
evaluate :: Polynomial a -> [a] -> a
evaluate :: Polynomial a -> [a] -> a
evaluate Polynomial a
p [a]
xs = A a -> a
forall {a}. A a -> a
get (Polynomial a -> A a -> A a
forall a. C a => T a -> a -> a
MathPol.evaluate Polynomial a
p (a -> A a
forall a. a -> A a
A ([a]
xs [a] -> Int -> a
forall a. HasCallStack => [a] -> Int -> a
!! Int
0)))
where
get :: A a -> a
get (A a
x) = a
x
substitute :: [Maybe a] -> Polynomial a -> Polynomial a
substitute :: [Maybe a] -> Polynomial a -> Polynomial a
substitute [Maybe a]
x Polynomial a
p =
if Maybe a -> Bool
forall a. Maybe a -> Bool
isNothing ([Maybe a]
x [Maybe a] -> Int -> Maybe a
forall a. HasCallStack => [a] -> Int -> a
!! Int
0)
then Polynomial a
p
else a -> Polynomial a
forall a. a -> Polynomial a
constPoly (Polynomial a
-> [BaseRing (Polynomial a)] -> BaseRing (Polynomial a)
forall b. HasVariables b => b -> [BaseRing b] -> BaseRing b
evaluate Polynomial a
p [Maybe (BaseRing (Polynomial a)) -> BaseRing (Polynomial a)
forall a. HasCallStack => Maybe a -> a
fromJust (Maybe (BaseRing (Polynomial a)) -> BaseRing (Polynomial a))
-> Maybe (BaseRing (Polynomial a)) -> BaseRing (Polynomial a)
forall a b. (a -> b) -> a -> b
$ [Maybe a]
x [Maybe a] -> Int -> Maybe a
forall a. HasCallStack => [a] -> Int -> a
!! Int
0])
permuteVariables :: [Int] -> Polynomial a -> Polynomial a
permuteVariables :: [Int] -> Polynomial a -> Polynomial a
permuteVariables = String -> [Int] -> Polynomial a -> Polynomial a
forall a. HasCallStack => String -> a
error String
"permuteVariables: there is only one variable."
swapVariables :: (Int, Int) -> Polynomial a -> Polynomial a
swapVariables :: (Int, Int) -> Polynomial a -> Polynomial a
swapVariables = String -> (Int, Int) -> Polynomial a -> Polynomial a
forall a. HasCallStack => String -> a
error String
"swapVariables: there is only one variable."
derivative :: Int -> Polynomial a -> Polynomial a
derivative :: Int -> Polynomial a -> Polynomial a
derivative Int
i Polynomial a
p =
if Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1
then Polynomial a -> Polynomial a
forall a. C a => a -> a
AlgDiff.differentiate Polynomial a
p
else a -> Polynomial a
forall a. a -> Polynomial a
constPoly a
forall a. C a => a
AlgAdd.zero
changeVariables :: Polynomial a -> [Polynomial a] -> Polynomial a
changeVariables :: Polynomial a -> [Polynomial a] -> Polynomial a
changeVariables Polynomial a
p [Polynomial a]
ps = Polynomial a -> Polynomial a -> Polynomial a
forall a. C a => T a -> T a -> T a
MathPol.compose Polynomial a
p ([Polynomial a]
ps [Polynomial a] -> Int -> Polynomial a
forall a. HasCallStack => [a] -> Int -> a
!! Int
0)
instance (Eq a, AlgField.C a) => HasVariables (RatioOfPolynomials a) where
numberOfVariables :: RatioOfPolynomials a -> Int
numberOfVariables :: RatioOfPolynomials a -> Int
numberOfVariables (Polynomial a
p :% Polynomial a
q) =
Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Polynomial a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Polynomial a
p) (Polynomial a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Polynomial a
q)
type BaseRing (RatioOfPolynomials a) = a
type VariablesType (RatioOfPolynomials a) = Polynomial a
evaluate :: RatioOfPolynomials a -> [a] -> a
evaluate :: RatioOfPolynomials a -> [a] -> a
evaluate RatioOfPolynomials a
r [a]
xs = Polynomial a
-> [BaseRing (Polynomial a)] -> BaseRing (Polynomial a)
forall b. HasVariables b => b -> [BaseRing b] -> BaseRing b
evaluate (RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.numerator RatioOfPolynomials a
r) [a]
[BaseRing (Polynomial a)]
xs a -> a -> a
forall a. C a => a -> a -> a
AlgField./
Polynomial a
-> [BaseRing (Polynomial a)] -> BaseRing (Polynomial a)
forall b. HasVariables b => b -> [BaseRing b] -> BaseRing b
evaluate (RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.denominator RatioOfPolynomials a
r) [a]
[BaseRing (Polynomial a)]
xs
substitute :: [Maybe a] -> RatioOfPolynomials a -> RatioOfPolynomials a
substitute :: [Maybe a] -> RatioOfPolynomials a -> RatioOfPolynomials a
substitute [Maybe a]
x RatioOfPolynomials a
r =
if Maybe a -> Bool
forall a. Maybe a -> Bool
isNothing ([Maybe a]
x [Maybe a] -> Int -> Maybe a
forall a. HasCallStack => [a] -> Int -> a
!! Int
0)
then RatioOfPolynomials a
r
else [Maybe (BaseRing (Polynomial a))] -> Polynomial a -> Polynomial a
forall b. HasVariables b => [Maybe (BaseRing b)] -> b -> b
substitute [Maybe a]
[Maybe (BaseRing (Polynomial a))]
x (RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.numerator RatioOfPolynomials a
r) Polynomial a -> Polynomial a -> RatioOfPolynomials a
forall a. C a => a -> a -> T a
%
[Maybe (BaseRing (Polynomial a))] -> Polynomial a -> Polynomial a
forall b. HasVariables b => [Maybe (BaseRing b)] -> b -> b
substitute [Maybe a]
[Maybe (BaseRing (Polynomial a))]
x (RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.denominator RatioOfPolynomials a
r)
permuteVariables :: [Int] -> RatioOfPolynomials a -> RatioOfPolynomials a
permuteVariables :: [Int] -> RatioOfPolynomials a -> RatioOfPolynomials a
permuteVariables = String -> [Int] -> RatioOfPolynomials a -> RatioOfPolynomials a
forall a. HasCallStack => String -> a
error String
"permuteVariables: there is only one variable."
swapVariables :: (Int, Int) -> RatioOfPolynomials a -> RatioOfPolynomials a
swapVariables :: (Int, Int) -> RatioOfPolynomials a -> RatioOfPolynomials a
swapVariables = String
-> (Int, Int) -> RatioOfPolynomials a -> RatioOfPolynomials a
forall a. HasCallStack => String -> a
error String
"swapVariables: there is only one variable."
derivative :: Int -> RatioOfPolynomials a -> RatioOfPolynomials a
derivative :: Int -> RatioOfPolynomials a -> RatioOfPolynomials a
derivative Int
i RatioOfPolynomials a
r =
if Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1
then
(Polynomial a
p' Polynomial a -> Polynomial a -> Polynomial a
forall a. C a => a -> a -> a
AlgRing.* Polynomial a
q Polynomial a -> Polynomial a -> Polynomial a
forall a. C a => a -> a -> a
AlgAdd.- Polynomial a
p Polynomial a -> Polynomial a -> Polynomial a
forall a. C a => a -> a -> a
AlgRing.* Polynomial a
q') Polynomial a -> Polynomial a -> RatioOfPolynomials a
forall a. C a => a -> a -> T a
% Polynomial a
q Polynomial a -> Integer -> Polynomial a
forall a. C a => a -> Integer -> a
AlgRing.^ Integer
2
else a -> Polynomial a
forall a. a -> Polynomial a
constPoly a
forall a. C a => a
AlgAdd.zero Polynomial a -> Polynomial a -> RatioOfPolynomials a
forall a. a -> a -> T a
:% a -> Polynomial a
forall a. a -> Polynomial a
constPoly a
forall a. C a => a
AlgRing.one
where
p :: Polynomial a
p = RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.numerator RatioOfPolynomials a
r
q :: Polynomial a
q = RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.denominator RatioOfPolynomials a
r
p' :: Polynomial a
p' = Polynomial a -> Polynomial a
forall a. C a => a -> a
AlgDiff.differentiate Polynomial a
p
q' :: Polynomial a
q' = Polynomial a -> Polynomial a
forall a. C a => a -> a
AlgDiff.differentiate Polynomial a
q
changeVariables :: RatioOfPolynomials a -> [Polynomial a] -> RatioOfPolynomials a
changeVariables :: RatioOfPolynomials a -> [Polynomial a] -> RatioOfPolynomials a
changeVariables RatioOfPolynomials a
r [Polynomial a]
ps = Polynomial a -> [VariablesType (Polynomial a)] -> Polynomial a
forall b. HasVariables b => b -> [VariablesType b] -> b
changeVariables (RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.numerator RatioOfPolynomials a
r) [Polynomial a]
[VariablesType (Polynomial a)]
ps Polynomial a -> Polynomial a -> RatioOfPolynomials a
forall a. C a => a -> a -> T a
%
Polynomial a -> [VariablesType (Polynomial a)] -> Polynomial a
forall b. HasVariables b => b -> [VariablesType b] -> b
changeVariables (RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.denominator RatioOfPolynomials a
r) [Polynomial a]
[VariablesType (Polynomial a)]
ps
instance (Eq a, AlgField.C a) => AlgZT.C (A a) where
isZero :: A a -> Bool
isZero :: A a -> Bool
isZero (A a
r) = a
r a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
forall a. C a => a
AlgAdd.zero
instance (Eq a, AlgField.C a) => AlgMod.C (A a) (RatioOfPolynomials a) where
(*>) :: A a -> RatioOfPolynomials a -> RatioOfPolynomials a
A a
r *> :: A a -> RatioOfPolynomials a -> RatioOfPolynomials a
*> RatioOfPolynomials a
rop = T (A a) -> RatioOfPolynomials a -> RatioOfPolynomials a
forall a. C a => a -> T a -> T a
NumberRatio.scale (A a -> T (A a)
forall a. a -> T a
MathPol.const A a
r) RatioOfPolynomials a
rop
instance (Eq a, AlgField.C a) => AlgMod.C (Polynomial a) (RatioOfPolynomials a) where
(*>) :: Polynomial a -> RatioOfPolynomials a -> RatioOfPolynomials a
Polynomial a
p *> :: Polynomial a -> RatioOfPolynomials a -> RatioOfPolynomials a
*> RatioOfPolynomials a
r = Polynomial a -> RatioOfPolynomials a -> RatioOfPolynomials a
forall a. C a => a -> T a -> T a
NumberRatio.scale Polynomial a
p RatioOfPolynomials a
r
instance (Eq a, AlgField.C a) => AlgMod.C (Polynomial a) (OneParameterSpray a) where
(*>) :: Polynomial a -> OneParameterSpray a -> OneParameterSpray a
Polynomial a
p *> :: Polynomial a -> OneParameterSpray a -> OneParameterSpray a
*> OneParameterSpray a
r = T (Polynomial a) -> OneParameterSpray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray (Polynomial a
p Polynomial a -> Polynomial a -> T (Polynomial a)
forall a. a -> a -> T a
NumberRatio.:% Polynomial a
forall a. C a => a
AlgRing.one) OneParameterSpray a -> OneParameterSpray a -> OneParameterSpray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ OneParameterSpray a
r
infixr 7 *.
(*.) :: (Eq a, AlgField.C a) => a -> RatioOfPolynomials a -> RatioOfPolynomials a
*. :: forall a.
(Eq a, C a) =>
a -> RatioOfPolynomials a -> RatioOfPolynomials a
(*.) a
scalar RatioOfPolynomials a
rop = a -> A a
forall a. a -> A a
A a
scalar A a -> RatioOfPolynomials a -> RatioOfPolynomials a
forall a v. C a v => a -> v -> v
AlgMod.*> RatioOfPolynomials a
rop
constPoly :: a -> Polynomial a
constPoly :: forall a. a -> Polynomial a
constPoly a
x = A a -> T (A a)
forall a. a -> T a
MathPol.const (a -> A a
forall a. a -> A a
A a
x)
polyFromCoeffs :: [a] -> Polynomial a
polyFromCoeffs :: forall a. [a] -> Polynomial a
polyFromCoeffs [a]
as = [A a] -> T (A a)
forall a. [a] -> T a
MathPol.fromCoeffs ((a -> A a) -> [a] -> [A a]
forall a b. (a -> b) -> [a] -> [b]
map a -> A a
forall a. a -> A a
A [a]
as)
soleParameter :: AlgRing.C a => Polynomial a
soleParameter :: forall a. C a => Polynomial a
soleParameter = [a] -> Polynomial a
forall a. [a] -> Polynomial a
polyFromCoeffs [a
forall a. C a => a
AlgAdd.zero, a
forall a. C a => a
AlgRing.one]
constQPoly :: Rational' -> QPolynomial
constQPoly :: Rational' -> QPolynomial
constQPoly = Rational' -> QPolynomial
forall a. a -> Polynomial a
constPoly
qpolyFromCoeffs :: [Rational'] -> QPolynomial
qpolyFromCoeffs :: [Rational'] -> QPolynomial
qpolyFromCoeffs = [Rational'] -> QPolynomial
forall a. [a] -> Polynomial a
polyFromCoeffs
qsoleParameter :: QPolynomial
qsoleParameter :: QPolynomial
qsoleParameter = [Rational'] -> QPolynomial
qpolyFromCoeffs [Rational'
0, Rational'
1]
polynomialToSpray :: forall a. (Eq a, AlgRing.C a) => Polynomial a -> Spray a
polynomialToSpray :: forall a. (Eq a, C a) => Polynomial a -> Spray a
polynomialToSpray Polynomial a
pol = [Spray a] -> Spray a
forall a. C a => [a] -> a
AlgAdd.sum [Spray a]
terms
where
coeffs :: [A a]
coeffs = Polynomial a -> [A a]
forall a. T a -> [a]
MathPol.coeffs Polynomial a
pol
indices :: [Int]
indices = (A a -> Bool) -> [A a] -> [Int]
forall a. (a -> Bool) -> [a] -> [Int]
findIndices (A a -> A a -> Bool
forall a. Eq a => a -> a -> Bool
/= a -> A a
forall a. a -> A a
A a
forall a. C a => a
AlgAdd.zero) [A a]
coeffs
get :: A a -> a
get :: A a -> a
get (A a
x) = a
x
terms :: [Spray a]
terms = (Int -> Spray a) -> [Int] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> A a -> a
get ([A a]
coeffs[A a] -> Int -> A a
forall a. HasCallStack => [a] -> Int -> a
!!Int
i) a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ (Int -> Spray a
forall a. C a => Int -> Spray a
lone Int
1 Spray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^ Int
i)) [Int]
indices
qPolynomialToQSpray :: QPolynomial -> QSpray
qPolynomialToQSpray :: QPolynomial -> Spray Rational
qPolynomialToQSpray QPolynomial
pol = [Spray Rational] -> Spray Rational
forall a. C a => [a] -> a
AlgAdd.sum [Spray Rational]
terms
where
coeffs :: [A Rational']
coeffs = QPolynomial -> [A Rational']
forall a. T a -> [a]
MathPol.coeffs QPolynomial
pol
indices :: [Int]
indices = (A Rational' -> Bool) -> [A Rational'] -> [Int]
forall a. (a -> Bool) -> [a] -> [Int]
findIndices (A Rational' -> A Rational' -> Bool
forall a. Eq a => a -> a -> Bool
/= Rational' -> A Rational'
forall a. a -> A a
A Rational'
0) [A Rational']
coeffs
get :: A Rational' -> Rational
get :: A Rational' -> Rational
get (A Rational'
x) = Rational' -> Integer
forall a. T a -> a
NumberRatio.numerator Rational'
x Integer -> Integer -> Rational
forall a. a -> a -> Ratio a
DR.:% Rational' -> Integer
forall a. T a -> a
NumberRatio.denominator Rational'
x
terms :: [Spray Rational]
terms = (Int -> Spray Rational) -> [Int] -> [Spray Rational]
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> A Rational' -> Rational
get ([A Rational']
coeffs[A Rational'] -> Int -> A Rational'
forall a. HasCallStack => [a] -> Int -> a
!!Int
i) Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ (Int -> Spray Rational
qlone Int
1 Spray Rational -> Int -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^ Int
i)) [Int]
indices
bracify :: (String, String) -> String -> String
bracify :: (String, String) -> ShowS
bracify (String
lbrace, String
rbrace) String
x = String
lbrace String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
x String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
rbrace
showRatioOfPolynomials :: forall a. (Eq a, AlgField.C a)
=> (Spray a -> String) -> RatioOfPolynomials a -> String
showRatioOfPolynomials :: forall a.
(Eq a, C a) =>
(Spray a -> String) -> RatioOfPolynomials a -> String
showRatioOfPolynomials Spray a -> String
sprayShower RatioOfPolynomials a
polysRatio =
String
numeratorString String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
denominatorString
where
numerator :: Polynomial a
numerator = RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.numerator RatioOfPolynomials a
polysRatio
denominator :: Polynomial a
denominator = RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.denominator RatioOfPolynomials a
polysRatio
brackets :: Bool
brackets = Polynomial a
denominator Polynomial a -> Polynomial a -> Bool
forall a. Eq a => a -> a -> Bool
/= A a -> Polynomial a
forall a. a -> T a
MathPol.const (a -> A a
forall a. a -> A a
A a
forall a. C a => a
AlgRing.one)
enclose :: ShowS
enclose = (String, String) -> ShowS
bracify (String
"[ ", String
" ]")
numeratorString :: String
numeratorString = if Bool
brackets
then ShowS
enclose (Spray a -> String
sprayShower (Polynomial a -> Spray a
forall a. (Eq a, C a) => Polynomial a -> Spray a
polynomialToSpray Polynomial a
numerator))
else Spray a -> String
sprayShower (Polynomial a -> Spray a
forall a. (Eq a, C a) => Polynomial a -> Spray a
polynomialToSpray Polynomial a
numerator)
denominatorString :: String
denominatorString = if Bool -> Bool
not Bool
brackets
then String
""
else String
" %//% " String -> ShowS
forall a. [a] -> [a] -> [a]
++ ShowS
enclose (Spray a -> String
sprayShower (Polynomial a -> Spray a
forall a. (Eq a, C a) => Polynomial a -> Spray a
polynomialToSpray Polynomial a
denominator))
prettyRatioOfQPolynomials ::
String
-> RatioOfQPolynomials
-> String
prettyRatioOfQPolynomials :: String -> RatioOfQPolynomials -> String
prettyRatioOfQPolynomials String
var = (Spray Rational' -> String) -> RatioOfQPolynomials -> String
forall a.
(Eq a, C a) =>
(Spray a -> String) -> RatioOfPolynomials a -> String
showRatioOfPolynomials ([String] -> Spray Rational' -> String
prettyQSprayXYZ' [String
var])
showQpol :: forall a. (Eq a, AlgField.C a)
=> Polynomial a -> String -> (a -> String) -> Bool -> String
showQpol :: forall a.
(Eq a, C a) =>
Polynomial a -> String -> (a -> String) -> Bool -> String
showQpol Polynomial a
pol String
variable a -> String
showCoeff Bool
brackets = if Bool
brackets
then String
"[ " String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
polyString String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
" ]"
else String
polyString
where
showCoeff' :: Int -> A a -> String
showCoeff' :: Int -> A a -> String
showCoeff' Int
i (A a
coeff) = case Int
i of
Int
0 -> ((String, String) -> ShowS
bracify (String
"(", String
")") ShowS -> (a -> String) -> a -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> String
showCoeff) a
coeff
Int
_ -> if a
coeff a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
forall a. C a => a
AlgRing.one
then String
""
else ((String, String) -> ShowS
bracify (String
"(", String
")") ShowS -> (a -> String) -> a -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> String
showCoeff) a
coeff
coeffs :: [A a]
coeffs = Polynomial a -> [A a]
forall a. T a -> [a]
MathPol.coeffs Polynomial a
pol
nonzeros :: [Int]
nonzeros = (A a -> Bool) -> [A a] -> [Int]
forall a. (a -> Bool) -> [a] -> [Int]
findIndices (A a -> A a -> Bool
forall a. Eq a => a -> a -> Bool
/= a -> A a
forall a. a -> A a
A a
forall a. C a => a
AlgAdd.zero) [A a]
coeffs
terms :: [Text]
terms = (Int -> Text) -> [Int] -> [Text]
forall a b. (a -> b) -> [a] -> [b]
map (String -> Text
pack (String -> Text) -> (Int -> String) -> Int -> Text
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> String
showTerm) [Int]
nonzeros
where
showTerm :: Int -> String
showTerm Int
i = case Int
i of
Int
0 -> Int -> A a -> String
showCoeff' Int
0 ([A a]
coeffs [A a] -> Int -> A a
forall a. HasCallStack => [a] -> Int -> a
!! Int
0)
Int
1 -> Int -> A a -> String
showCoeff' Int
1 ([A a]
coeffs [A a] -> Int -> A a
forall a. HasCallStack => [a] -> Int -> a
!! Int
1) String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
variable
Int
_ -> Int -> A a -> String
showCoeff' Int
i ([A a]
coeffs [A a] -> Int -> A a
forall a. HasCallStack => [a] -> Int -> a
!! Int
i) String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
variable String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"^" String -> ShowS
forall a. [a] -> [a] -> [a]
++ Int -> String
forall a. Show a => a -> String
show Int
i
polyString :: String
polyString = Text -> String
unpack (Text -> [Text] -> Text
intercalate (String -> Text
pack String
" + ") [Text]
terms)
showQpolysRatio :: forall a. (Eq a, AlgField.C a)
=> String -> (a -> String) -> RatioOfPolynomials a -> String
showQpolysRatio :: forall a.
(Eq a, C a) =>
String -> (a -> String) -> RatioOfPolynomials a -> String
showQpolysRatio String
var a -> String
showCoeff RatioOfPolynomials a
polysRatio = String
numeratorString String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
denominatorString
where
denominator :: Polynomial a
denominator = RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.denominator RatioOfPolynomials a
polysRatio
brackets :: Bool
brackets = Polynomial a
denominator Polynomial a -> Polynomial a -> Bool
forall a. Eq a => a -> a -> Bool
/= A a -> Polynomial a
forall a. a -> T a
MathPol.const (a -> A a
forall a. a -> A a
A a
forall a. C a => a
AlgRing.one)
numeratorString :: String
numeratorString =
Polynomial a -> String -> (a -> String) -> Bool -> String
forall a.
(Eq a, C a) =>
Polynomial a -> String -> (a -> String) -> Bool -> String
showQpol (RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.numerator RatioOfPolynomials a
polysRatio) String
var a -> String
showCoeff Bool
brackets
denominatorString :: String
denominatorString = if Bool -> Bool
not Bool
brackets
then String
""
else String
" %//% " String -> ShowS
forall a. [a] -> [a] -> [a]
++ Polynomial a -> String -> (a -> String) -> Bool -> String
forall a.
(Eq a, C a) =>
Polynomial a -> String -> (a -> String) -> Bool -> String
showQpol Polynomial a
denominator String
var a -> String
showCoeff Bool
True
prettyRatioOfPolynomials :: (Eq a, AlgField.C a, Show a)
=> String
-> RatioOfPolynomials a
-> String
prettyRatioOfPolynomials :: forall a.
(Eq a, C a, Show a) =>
String -> RatioOfPolynomials a -> String
prettyRatioOfPolynomials String
var = String -> (a -> String) -> RatioOfPolynomials a -> String
forall a.
(Eq a, C a) =>
String -> (a -> String) -> RatioOfPolynomials a -> String
showQpolysRatio String
var a -> String
forall a. Show a => a -> String
show
evalRatioOfPolynomials :: AlgField.C a
=> a
-> RatioOfPolynomials a
-> a
evalRatioOfPolynomials :: forall a. C a => a -> RatioOfPolynomials a -> a
evalRatioOfPolynomials a
value RatioOfPolynomials a
polysRatio =
a
resultNumerator a -> a -> a
forall a. C a => a -> a -> a
AlgField./ a
resultDenominator
where
A a
resultNumerator =
T (A a) -> A a -> A a
forall a. C a => T a -> a -> a
MathPol.evaluate (RatioOfPolynomials a -> T (A a)
forall a. T a -> a
NumberRatio.numerator RatioOfPolynomials a
polysRatio) (a -> A a
forall a. a -> A a
A a
value)
A a
resultDenominator =
T (A a) -> A a -> A a
forall a. C a => T a -> a -> a
MathPol.evaluate (RatioOfPolynomials a -> T (A a)
forall a. T a -> a
NumberRatio.denominator RatioOfPolynomials a
polysRatio) (a -> A a
forall a. a -> A a
A a
value)
type OneParameterSpray a = Spray (RatioOfPolynomials a)
type OneParameterQSpray = OneParameterSpray Rational'
prettyOneParameterSprayX1X2X3 ::
(Eq a, Show a, AlgField.C a)
=> String
-> String
-> OneParameterSpray a
-> String
prettyOneParameterSprayX1X2X3 :: forall a.
(Eq a, Show a, C a) =>
String -> String -> OneParameterSpray a -> String
prettyOneParameterSprayX1X2X3 String
a = (RatioOfPolynomials a -> String)
-> (String, String)
-> String
-> Spray (RatioOfPolynomials a)
-> String
forall a.
(a -> String) -> (String, String) -> String -> Spray a -> String
showSprayX1X2X3 (String -> RatioOfPolynomials a -> String
forall a.
(Eq a, C a, Show a) =>
String -> RatioOfPolynomials a -> String
prettyRatioOfPolynomials String
a) (String
"{ ", String
" }")
prettyOneParameterSprayXYZ ::
(Eq a, Show a, AlgField.C a)
=> String
-> [String]
-> OneParameterSpray a
-> String
prettyOneParameterSprayXYZ :: forall a.
(Eq a, Show a, C a) =>
String -> [String] -> OneParameterSpray a -> String
prettyOneParameterSprayXYZ String
a = (RatioOfPolynomials a -> String)
-> (String, String)
-> [String]
-> Spray (RatioOfPolynomials a)
-> String
forall a.
(a -> String) -> (String, String) -> [String] -> Spray a -> String
showSprayXYZ (String -> RatioOfPolynomials a -> String
forall a.
(Eq a, C a, Show a) =>
String -> RatioOfPolynomials a -> String
prettyRatioOfPolynomials String
a) (String
"{ ", String
" }")
prettyOneParameterSpray ::
(Eq a, Show a, AlgField.C a)
=> String
-> OneParameterSpray a
-> String
prettyOneParameterSpray :: forall a.
(Eq a, Show a, C a) =>
String -> OneParameterSpray a -> String
prettyOneParameterSpray String
a = String -> [String] -> OneParameterSpray a -> String
forall a.
(Eq a, Show a, C a) =>
String -> [String] -> OneParameterSpray a -> String
prettyOneParameterSprayXYZ String
a [String
"x", String
"y", String
"z"]
prettyOneParameterSpray' ::
(Eq a, Show a, AlgField.C a)
=> String
-> OneParameterSpray a
-> String
prettyOneParameterSpray' :: forall a.
(Eq a, Show a, C a) =>
String -> OneParameterSpray a -> String
prettyOneParameterSpray' String
a = String -> [String] -> OneParameterSpray a -> String
forall a.
(Eq a, Show a, C a) =>
String -> [String] -> OneParameterSpray a -> String
prettyOneParameterSprayXYZ String
a [String
"X", String
"Y", String
"Z"]
prettyOneParameterQSprayX1X2X3 ::
String
-> String
-> OneParameterQSpray
-> String
prettyOneParameterQSprayX1X2X3 :: String -> String -> OneParameterQSpray -> String
prettyOneParameterQSprayX1X2X3 String
a String
x =
(RatioOfQPolynomials -> String)
-> (String, String)
-> ([Seq Int] -> [String])
-> OneParameterQSpray
-> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray (String -> RatioOfQPolynomials -> String
prettyRatioOfQPolynomials String
a) (String
"{ ", String
" }") (String -> [Seq Int] -> [String]
showMonomialsX1X2X3 String
x)
prettyOneParameterQSprayXYZ ::
String
-> [String]
-> OneParameterQSpray
-> String
prettyOneParameterQSprayXYZ :: String -> [String] -> OneParameterQSpray -> String
prettyOneParameterQSprayXYZ String
a [String]
letters =
(RatioOfQPolynomials -> String)
-> (String, String)
-> ([Seq Int] -> [String])
-> OneParameterQSpray
-> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray (String -> RatioOfQPolynomials -> String
prettyRatioOfQPolynomials String
a) (String
"{ ", String
" }") ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String]
letters)
prettyOneParameterQSpray ::
String
-> OneParameterQSpray
-> String
prettyOneParameterQSpray :: String -> OneParameterQSpray -> String
prettyOneParameterQSpray String
a = String -> [String] -> OneParameterQSpray -> String
prettyOneParameterQSprayXYZ String
a [String
"x", String
"y", String
"z"]
prettyOneParameterQSpray' ::
String
-> OneParameterQSpray
-> String
prettyOneParameterQSpray' :: String -> OneParameterQSpray -> String
prettyOneParameterQSpray' String
a = String -> [String] -> OneParameterQSpray -> String
prettyOneParameterQSprayXYZ String
a [String
"X", String
"Y", String
"Z"]
evalOneParameterSpray ::
(Eq a, AlgField.C a) => OneParameterSpray a -> a -> Spray a
evalOneParameterSpray :: forall a. (Eq a, C a) => OneParameterSpray a -> a -> Spray a
evalOneParameterSpray OneParameterSpray a
spray a
x =
Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a
removeZeroTerms (Spray a -> Spray a) -> Spray a -> Spray a
forall a b. (a -> b) -> a -> b
$ (RatioOfPolynomials a -> a) -> OneParameterSpray a -> Spray a
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map (a -> RatioOfPolynomials a -> a
forall a. C a => a -> RatioOfPolynomials a -> a
evalRatioOfPolynomials a
x) OneParameterSpray a
spray
evalOneParameterSpray' :: (Eq a, AlgField.C a)
=> OneParameterSpray a
-> a
-> [a]
-> a
evalOneParameterSpray' :: forall a. (Eq a, C a) => OneParameterSpray a -> a -> [a] -> a
evalOneParameterSpray' OneParameterSpray a
spray a
x [a]
xs = if [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
xs Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= OneParameterSpray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables OneParameterSpray a
spray
then Spray a -> [a] -> a
forall a. (Eq a, C a) => Spray a -> [a] -> a
evalSpray (OneParameterSpray a -> a -> Spray a
forall a. (Eq a, C a) => OneParameterSpray a -> a -> Spray a
evalOneParameterSpray OneParameterSpray a
spray a
x) [a]
xs
else String -> a
forall a. HasCallStack => String -> a
error String
"evalOneParameterSpray': not enough values provided."
evalOneParameterMonomial :: (Eq a, AlgField.C a)
=> [a] -> Monomial (RatioOfPolynomials a) -> RatioOfPolynomials a
evalOneParameterMonomial :: forall a.
(Eq a, C a) =>
[a] -> Monomial (RatioOfPolynomials a) -> RatioOfPolynomials a
evalOneParameterMonomial [a]
xs (Powers
powers, RatioOfPolynomials a
coeff) =
[a] -> a
forall a. C a => [a] -> a
AlgRing.product ((a -> Integer -> a) -> [a] -> [Integer] -> [a]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith a -> Integer -> a
forall a. C a => a -> Integer -> a
(AlgRing.^) [a]
xs [Integer]
pows) a -> RatioOfPolynomials a -> RatioOfPolynomials a
forall a.
(Eq a, C a) =>
a -> RatioOfPolynomials a -> RatioOfPolynomials a
*. RatioOfPolynomials a
coeff
where
pows :: [Integer]
pows = Seq Integer -> [Integer]
forall a. Seq a -> [a]
forall (t :: * -> *) a. Foldable t => t a -> [a]
DF.toList (Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Seq Int -> Seq Integer
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Powers -> Seq Int
exponents Powers
powers)
evalOneParameterSpray'' ::
(Eq a, AlgField.C a) => OneParameterSpray a -> [a] -> RatioOfPolynomials a
evalOneParameterSpray'' :: forall a.
(Eq a, C a) =>
OneParameterSpray a -> [a] -> RatioOfPolynomials a
evalOneParameterSpray'' OneParameterSpray a
spray [a]
xs = if [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
xs Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= OneParameterSpray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables OneParameterSpray a
spray
then [RatioOfPolynomials a] -> RatioOfPolynomials a
forall a. C a => [a] -> a
AlgAdd.sum ([RatioOfPolynomials a] -> RatioOfPolynomials a)
-> [RatioOfPolynomials a] -> RatioOfPolynomials a
forall a b. (a -> b) -> a -> b
$ (Monomial (RatioOfPolynomials a) -> RatioOfPolynomials a)
-> [Monomial (RatioOfPolynomials a)] -> [RatioOfPolynomials a]
forall a b. (a -> b) -> [a] -> [b]
map ([a] -> Monomial (RatioOfPolynomials a) -> RatioOfPolynomials a
forall a.
(Eq a, C a) =>
[a] -> Monomial (RatioOfPolynomials a) -> RatioOfPolynomials a
evalOneParameterMonomial [a]
xs) (OneParameterSpray a -> [Monomial (RatioOfPolynomials a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList OneParameterSpray a
spray)
else String -> RatioOfPolynomials a
forall a. HasCallStack => String -> a
error String
"evalOneParameterSpray'': not enough values provided."
data Powers = Powers
{ Powers -> Seq Int
exponents :: Seq Int
, Powers -> Int
nvariables :: Int
}
deriving Int -> Powers -> ShowS
[Powers] -> ShowS
Powers -> String
(Int -> Powers -> ShowS)
-> (Powers -> String) -> ([Powers] -> ShowS) -> Show Powers
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
$cshowsPrec :: Int -> Powers -> ShowS
showsPrec :: Int -> Powers -> ShowS
$cshow :: Powers -> String
show :: Powers -> String
$cshowList :: [Powers] -> ShowS
showList :: [Powers] -> ShowS
Show
instance Eq Powers where
(==) :: Powers -> Powers -> Bool
Powers
pows1 == :: Powers -> Powers -> Bool
== Powers
pows2 = Powers -> Seq Int
exponents Powers
pows1' Seq Int -> Seq Int -> Bool
forall a. Eq a => a -> a -> Bool
== Powers -> Seq Int
exponents Powers
pows2'
where
(Powers
pows1', Powers
pows2') = (Powers, Powers) -> (Powers, Powers)
harmonize (Powers
pows1, Powers
pows2)
instance Hashable Powers where
hashWithSalt :: Int -> Powers -> Int
hashWithSalt :: Int -> Powers -> Int
hashWithSalt Int
k Powers
pows = Int -> (Seq Int, Int) -> Int
forall a. Hashable a => Int -> a -> Int
hashWithSalt Int
k (Powers -> Seq Int
exponents Powers
pows, Powers -> Int
nvariables Powers
pows)
growSequence :: Seq Int -> Int -> Int -> Seq Int
growSequence :: Seq Int -> Int -> Int -> Seq Int
growSequence Seq Int
s Int
m Int
n = Seq Int
s Seq Int -> Seq Int -> Seq Int
forall a. Seq a -> Seq a -> Seq a
>< Seq Int
t where t :: Seq Int
t = Int -> Int -> Seq Int
forall a. Int -> a -> Seq a
S.replicate (Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
m) Int
0
growSequence' :: Int -> Seq Int -> Seq Int
growSequence' :: Int -> Seq Int -> Seq Int
growSequence' Int
n Seq Int
s = Seq Int -> Int -> Int -> Seq Int
growSequence Seq Int
s (Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
s) Int
n
harmonize :: (Powers, Powers) -> (Powers, Powers)
harmonize :: (Powers, Powers) -> (Powers, Powers)
harmonize (Powers
pows1, Powers
pows2) = (Seq Int -> Int -> Powers
Powers Seq Int
e1' Int
n, Seq Int -> Int -> Powers
Powers Seq Int
e2' Int
n)
where
e1 :: Seq Int
e1 = Powers -> Seq Int
exponents Powers
pows1
e2 :: Seq Int
e2 = Powers -> Seq Int
exponents Powers
pows2
n1 :: Int
n1 = Powers -> Int
nvariables Powers
pows1
n2 :: Int
n2 = Powers -> Int
nvariables Powers
pows2
(Seq Int
e1', Seq Int
e2', Int
n) = if Int
n1 Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
n2
then (Seq Int -> Int -> Int -> Seq Int
growSequence Seq Int
e1 Int
n1 Int
n2, Seq Int
e2, Int
n2)
else (Seq Int
e1, Seq Int -> Int -> Int -> Seq Int
growSequence Seq Int
e2 Int
n2 Int
n1, Int
n1)
simplifyPowers :: Powers -> Powers
simplifyPowers :: Powers -> Powers
simplifyPowers Powers
pows = Seq Int -> Int -> Powers
Powers Seq Int
s (Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
s)
where
s :: Seq Int
s = (Int -> Bool) -> Seq Int -> Seq Int
forall a. (a -> Bool) -> Seq a -> Seq a
dropWhileR (Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0) (Powers -> Seq Int
exponents Powers
pows)
type Monomial a = (Powers, a)
type Spray a = HashMap Powers a
type QSpray = Spray Rational
type QSpray' = Spray Rational'
instance (AlgRing.C a, Eq a) => HasVariables (Spray a) where
type BaseRing (Spray a) = a
type VariablesType (Spray a) = Spray a
evaluate :: Spray a -> [a] -> a
evaluate :: Spray a -> [a] -> a
evaluate Spray a
spray [a]
xyz = if [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
xyz Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray
then [a] -> Spray a -> a
forall a. C a => [a] -> Spray a -> a
evalSprayHelper [a]
xyz Spray a
spray
else String -> a
forall a. HasCallStack => String -> a
error String
"evaluate: not enough values provided."
substitute :: [Maybe a] -> Spray a -> Spray a
substitute :: [Maybe a] -> Spray a -> Spray a
substitute [Maybe a]
subs Spray a
spray = if [Maybe a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Maybe a]
subs Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
n
then Spray a
spray'
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"substitute: incorrect length of the substitutions list."
where
n :: Int
n = Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray
monomials :: [(Powers, a)]
monomials = Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray
spray' :: Spray a
spray' =
(Spray a -> Spray a -> Spray a) -> [Spray a] -> Spray a
forall a. HasCallStack => (a -> a -> a) -> [a] -> a
foldl1' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^+^) (((Powers, a) -> Spray a) -> [(Powers, a)] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map ((Powers, a) -> Spray a
forall a. Monomial a -> Spray a
fromMonomial ((Powers, a) -> Spray a)
-> ((Powers, a) -> (Powers, a)) -> (Powers, a) -> Spray a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Powers, a) -> (Powers, a)
substituteMonomial) [(Powers, a)]
monomials)
substituteMonomial :: Monomial a -> Monomial a
substituteMonomial :: (Powers, a) -> (Powers, a)
substituteMonomial (Powers
powers, a
coeff) = (Powers
powers'', a
coeff')
where
pows :: Seq Int
pows = Powers -> Seq Int
exponents Powers
powers
nv :: Int
nv = Powers -> Int
nvariables Powers
powers
indices :: [Int]
indices = (Maybe a -> Bool) -> [Maybe a] -> [Int]
forall a. (a -> Bool) -> [a] -> [Int]
findIndices Maybe a -> Bool
forall a. Maybe a -> Bool
isJust (Int -> [Maybe a] -> [Maybe a]
forall a. Int -> [a] -> [a]
take Int
nv [Maybe a]
subs)
pows' :: [Integer]
pows' = [Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Seq Int
pows Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
i) | Int
i <- [Int]
indices]
xyz :: [a]
xyz = [Maybe a -> a
forall a. HasCallStack => Maybe a -> a
fromJust ([Maybe a]
subs [Maybe a] -> Int -> Maybe a
forall a. HasCallStack => [a] -> Int -> a
!! Int
i) | Int
i <- [Int]
indices]
coeff' :: a
coeff' = a
coeff a -> a -> a
forall a. C a => a -> a -> a
AlgRing.* [a] -> a
forall a. C a => [a] -> a
AlgRing.product ((a -> Integer -> a) -> [a] -> [Integer] -> [a]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith a -> Integer -> a
forall a. C a => a -> Integer -> a
(AlgRing.^) [a]
xyz [Integer]
pows')
f :: Int -> Int -> Int
f Int
i Int
a = if Int
i Int -> [Int] -> Bool
forall a. Eq a => a -> [a] -> Bool
forall (t :: * -> *) a. (Foldable t, Eq a) => a -> t a -> Bool
`elem` [Int]
indices then Int
0 else Int
a
pows'' :: Seq Int
pows'' = (Int -> Int -> Int) -> Seq Int -> Seq Int
forall a b. (Int -> a -> b) -> Seq a -> Seq b
S.mapWithIndex Int -> Int -> Int
f Seq Int
pows
powers'' :: Powers
powers'' = Powers -> Powers
simplifyPowers (Powers -> Powers) -> Powers -> Powers
forall a b. (a -> b) -> a -> b
$ Seq Int -> Int -> Powers
Powers Seq Int
pows'' Int
nv
changeVariables :: Spray a -> [Spray a] -> Spray a
changeVariables :: Spray a -> [Spray a] -> Spray a
changeVariables = Spray a -> [Spray a] -> Spray a
forall a. (C a, Eq a) => Spray a -> [Spray a] -> Spray a
composeSpray
numberOfVariables :: Spray a -> Int
numberOfVariables :: Spray a -> Int
numberOfVariables Spray a
spray =
if [Powers] -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [Powers]
powers then Int
0 else [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum ((Powers -> Int) -> [Powers] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Int
nvariables [Powers]
powers)
where
powers :: [Powers]
powers = Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
spray
permuteVariables :: [Int] -> Spray a -> Spray a
permuteVariables :: [Int] -> Spray a -> Spray a
permuteVariables [Int]
permutation Spray a
spray =
if Int
n' Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
n Bool -> Bool -> Bool
&& [Int] -> Bool
isPermutation [Int]
permutation
then Spray a
spray'
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"permuteVariables: invalid permutation."
where
n :: Int
n = Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray
n' :: Int
n' = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
permutation
isPermutation :: [Int] -> Bool
isPermutation [Int]
pmtn = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
minimum [Int]
pmtn Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 Bool -> Bool -> Bool
&& [Int] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length ([Int] -> [Int]
forall a. Eq a => [a] -> [a]
nub [Int]
pmtn) Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
n'
intmap :: IntMap Int
intmap = [(Int, Int)] -> IntMap Int
forall a. [(Int, a)] -> IntMap a
IM.fromList ([Int] -> [Int] -> [(Int, Int)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Int]
permutation [Int
1 .. Int
n'])
invpermutation :: [Int]
invpermutation = [IntMap Int
intmap IntMap Int -> Int -> Int
forall a. IntMap a -> Int -> a
IM.! Int
i | Int
i <- [Int
1 .. Int
n']]
permuteSeq :: Seq Int -> Seq Int
permuteSeq Seq Int
x =
(Int -> Int -> Int) -> Seq Int -> Seq Int
forall a b. (Int -> a -> b) -> Seq a -> Seq b
S.mapWithIndex (\Int
i Int
_ -> Seq Int
x Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` ([Int]
invpermutation [Int] -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!! Int
i Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1)) Seq Int
x
([Powers]
powers, [a]
coeffs) = [(Powers, a)] -> ([Powers], [a])
forall a b. [(a, b)] -> ([a], [b])
unzip (Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray)
expnts :: [Seq Int]
expnts = (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
powers
expnts' :: [Seq Int]
expnts' = (Seq Int -> Seq Int) -> [Seq Int] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Seq Int
permuteSeq (Seq Int -> Seq Int) -> (Seq Int -> Seq Int) -> Seq Int -> Seq Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> Seq Int -> Seq Int
growSequence' Int
n') [Seq Int]
expnts
powers' :: [Powers]
powers' = (Seq Int -> Powers) -> [Seq Int] -> [Powers]
forall a b. (a -> b) -> [a] -> [b]
map (\Seq Int
exps -> Powers -> Powers
simplifyPowers (Seq Int -> Int -> Powers
Powers Seq Int
exps Int
n')) [Seq Int]
expnts'
spray' :: Spray a
spray' = [(Powers, a)] -> Spray a
forall k v. (Eq k, Hashable k) => [(k, v)] -> HashMap k v
HM.fromList ([Powers] -> [a] -> [(Powers, a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Powers]
powers' [a]
coeffs)
swapVariables :: (Int, Int) -> Spray a -> Spray a
swapVariables :: (Int, Int) -> Spray a -> Spray a
swapVariables (Int
i, Int
j) Spray a
spray =
if Int
iInt -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>=Int
1 Bool -> Bool -> Bool
&& Int
jInt -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>=Int
1
then Spray a
spray'
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"swapVariables: invalid indices."
where
n :: Int
n = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray, Int
i, Int
j]
f :: Int -> Int
f Int
k | Int
k Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
i = Int
j
| Int
k Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
j = Int
i
| Bool
otherwise = Int
k
transposition :: [Int]
transposition = (Int -> Int) -> [Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map Int -> Int
f [Int
1 .. Int
n]
permuteSeq :: Seq Int -> Seq Int
permuteSeq Seq Int
x =
(Int -> Int -> Int) -> Seq Int -> Seq Int
forall a b. (Int -> a -> b) -> Seq a -> Seq b
S.mapWithIndex (\Int
ii Int
_ -> Seq Int
x Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` ([Int]
transposition [Int] -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!! Int
ii Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1)) Seq Int
x
([Powers]
powers, [a]
coeffs) = [(Powers, a)] -> ([Powers], [a])
forall a b. [(a, b)] -> ([a], [b])
unzip (Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray)
expnts :: [Seq Int]
expnts = (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
powers
expnts' :: [Seq Int]
expnts' = (Seq Int -> Seq Int) -> [Seq Int] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Seq Int
permuteSeq (Seq Int -> Seq Int) -> (Seq Int -> Seq Int) -> Seq Int -> Seq Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> Seq Int -> Seq Int
growSequence' Int
n) [Seq Int]
expnts
powers' :: [Powers]
powers' = (Seq Int -> Powers) -> [Seq Int] -> [Powers]
forall a b. (a -> b) -> [a] -> [b]
map (\Seq Int
exps -> Powers -> Powers
simplifyPowers (Seq Int -> Int -> Powers
Powers Seq Int
exps Int
n)) [Seq Int]
expnts'
spray' :: Spray a
spray' = [(Powers, a)] -> Spray a
forall k v. (Eq k, Hashable k) => [(k, v)] -> HashMap k v
HM.fromList ([Powers] -> [a] -> [(Powers, a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Powers]
powers' [a]
coeffs)
derivative :: Int -> Spray a -> Spray a
derivative :: Int -> Spray a -> Spray a
derivative Int
i Spray a
p = if Int
i Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
1
then Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a
cleanSpray (Spray a -> Spray a) -> Spray a -> Spray a
forall a b. (a -> b) -> a -> b
$ (a -> a -> a) -> [(Powers, a)] -> Spray a
forall k v.
(Eq k, Hashable k) =>
(v -> v -> v) -> [(k, v)] -> HashMap k v
HM.fromListWith a -> a -> a
forall a. C a => a -> a -> a
(AlgAdd.+) [(Powers, a)]
monomials
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"derivative: invalid index."
where
p' :: [(Powers, a)]
p' = Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
p
monomials :: [(Powers, a)]
monomials = [ (Powers, a) -> (Powers, a)
derivMonomial (Powers, a)
mp | (Powers, a)
mp <- [(Powers, a)]
p' ]
derivMonomial :: Monomial a -> Monomial a
derivMonomial :: (Powers, a) -> (Powers, a)
derivMonomial (Powers
pows, a
coef) = if Int
i' Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
expts
then (Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0, a
forall a. C a => a
AlgAdd.zero)
else (Powers
pows', a
coef')
where
i' :: Int
i' = Int
i Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1
expts :: Seq Int
expts = Powers -> Seq Int
exponents Powers
pows
expt_i :: Int
expt_i = Seq Int
expts Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
i'
expts' :: Seq Int
expts' = (Int -> Int) -> Int -> Seq Int -> Seq Int
forall a. (a -> a) -> Int -> Seq a -> Seq a
adjust (Int -> Int -> Int
forall a. Num a => a -> a -> a
subtract Int
1) Int
i' Seq Int
expts
coef' :: a
coef' = Int
expt_i Int -> a -> a
forall a. (C a, Eq a) => Int -> a -> a
.^ a
coef
pows' :: Powers
pows' = Seq Int -> Int -> Powers
Powers Seq Int
expts' (Powers -> Int
nvariables Powers
pows)
addSprays :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a
addSprays :: forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
addSprays Spray a
p Spray a
q = Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a
cleanSpray (Spray a -> Spray a) -> Spray a -> Spray a
forall a b. (a -> b) -> a -> b
$ (Spray a -> Powers -> a -> Spray a)
-> Spray a -> Spray a -> Spray a
forall a k v. (a -> k -> v -> a) -> a -> HashMap k v -> a
HM.foldlWithKey' Spray a -> Powers -> a -> Spray a
forall {k} {v}.
(Hashable k, C v) =>
HashMap k v -> k -> v -> HashMap k v
f Spray a
p Spray a
q
where
f :: HashMap k v -> k -> v -> HashMap k v
f HashMap k v
s k
powers v
coef = (v -> v -> v) -> k -> v -> HashMap k v -> HashMap k v
forall k v.
(Eq k, Hashable k) =>
(v -> v -> v) -> k -> v -> HashMap k v -> HashMap k v
HM.insertWith v -> v -> v
forall a. C a => a -> a -> a
(AlgAdd.+) k
powers v
coef HashMap k v
s
negateSpray :: AlgAdd.C a => Spray a -> Spray a
negateSpray :: forall a. C a => Spray a -> Spray a
negateSpray = (a -> a) -> HashMap Powers a -> HashMap Powers a
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map a -> a
forall a. C a => a -> a
AlgAdd.negate
scaleSpray :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a
scaleSpray :: forall a. (C a, Eq a) => a -> Spray a -> Spray a
scaleSpray a
lambda Spray a
p = Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a
cleanSpray (Spray a -> Spray a) -> Spray a -> Spray a
forall a b. (a -> b) -> a -> b
$ (a -> a) -> Spray a -> Spray a
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map (a
lambda a -> a -> a
forall a. C a => a -> a -> a
AlgRing.*) Spray a
p
multMonomial :: AlgRing.C a => Monomial a -> Monomial a -> Monomial a
multMonomial :: forall a. C a => Monomial a -> Monomial a -> Monomial a
multMonomial (Powers
pows1, a
coef1) (Powers
pows2, a
coef2) = (Powers
pows, a
coef1 a -> a -> a
forall a. C a => a -> a -> a
AlgRing.* a
coef2)
where
(Powers
pows1', Powers
pows2') = (Powers, Powers) -> (Powers, Powers)
harmonize (Powers
pows1, Powers
pows2)
expts :: Seq Int
expts = (Int -> Int -> Int) -> Seq Int -> Seq Int -> Seq Int
forall a b c. (a -> b -> c) -> Seq a -> Seq b -> Seq c
S.zipWith Int -> Int -> Int
forall a. Num a => a -> a -> a
(+) (Powers -> Seq Int
exponents Powers
pows1') (Powers -> Seq Int
exponents Powers
pows2')
pows :: Powers
pows = Seq Int -> Int -> Powers
Powers Seq Int
expts (Powers -> Int
nvariables Powers
pows1')
multSprays :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a
multSprays :: forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
multSprays Spray a
p Spray a
q = Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a
cleanSpray (Spray a -> Spray a) -> Spray a -> Spray a
forall a b. (a -> b) -> a -> b
$ (a -> a -> a) -> [(Powers, a)] -> Spray a
forall k v.
(Eq k, Hashable k) =>
(v -> v -> v) -> [(k, v)] -> HashMap k v
HM.fromListWith a -> a -> a
forall a. C a => a -> a -> a
(AlgAdd.+) [(Powers, a)]
prods
where
p' :: [(Powers, a)]
p' = Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
p
q' :: [(Powers, a)]
q' = Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
q
prods :: [(Powers, a)]
prods = [ (Powers, a) -> (Powers, a) -> (Powers, a)
forall a. C a => Monomial a -> Monomial a -> Monomial a
multMonomial (Powers, a)
mp (Powers, a)
mq | (Powers, a)
mp <- [(Powers, a)]
p', (Powers, a)
mq <- [(Powers, a)]
q' ]
instance (AlgAdd.C a, Eq a) => AlgAdd.C (Spray a) where
(+) :: Spray a -> Spray a -> Spray a
Spray a
p + :: Spray a -> Spray a -> Spray a
+ Spray a
q = Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
addSprays Spray a
p Spray a
q
zero :: Spray a
zero :: Spray a
zero = Spray a
forall k v. HashMap k v
HM.empty
negate :: Spray a -> Spray a
negate :: Spray a -> Spray a
negate = Spray a -> Spray a
forall a. C a => Spray a -> Spray a
negateSpray
instance (AlgRing.C a, Eq a) => AlgMod.C a (Spray a) where
(*>) :: a -> Spray a -> Spray a
a
lambda *> :: a -> Spray a -> Spray a
*> Spray a
p = a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
scaleSpray a
lambda Spray a
p
instance (AlgRing.C a, Eq a) => AlgRightMod.C a (Spray a) where
(<*) :: Spray a -> a -> Spray a
Spray a
p <* :: Spray a -> a -> Spray a
<* a
lambda = a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
scaleSpray a
lambda Spray a
p
instance (AlgRing.C a, Eq a) => AlgRing.C (Spray a) where
(*) :: Spray a -> Spray a -> Spray a
Spray a
p * :: Spray a -> Spray a -> Spray a
* Spray a
q = Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
multSprays Spray a
p Spray a
q
one :: Spray a
one :: Spray a
one = Int -> Spray a
forall a. C a => Int -> Spray a
lone Int
0
infixl 6 ^+^
(^+^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ :: forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^+^) Spray a
p Spray a
q = Spray a
p Spray a -> Spray a -> Spray a
forall a. C a => a -> a -> a
AlgAdd.+ Spray a
q
infixl 6 ^-^
(^-^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ :: forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^-^) Spray a
p Spray a
q = Spray a
p Spray a -> Spray a -> Spray a
forall a. C a => a -> a -> a
AlgAdd.- Spray a
q
infixl 7 ^*^
(^*^) :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ :: forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^*^) Spray a
p Spray a
q = Spray a
p Spray a -> Spray a -> Spray a
forall a. C a => a -> a -> a
AlgRing.* Spray a
q
infixr 8 ^**^
(^**^) :: (AlgRing.C a, Eq a) => Spray a -> Int -> Spray a
^**^ :: forall a. (C a, Eq a) => Spray a -> Int -> Spray a
(^**^) Spray a
p Int
n = if Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
0
then Spray a
p Spray a -> Integer -> Spray a
forall a. C a => a -> Integer -> a
AlgRing.^ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
n
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"(^**^): negative power of a spray is not allowed."
infixr 7 *^
(*^) :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a
*^ :: forall a. (C a, Eq a) => a -> Spray a -> Spray a
(*^) a
lambda Spray a
pol = a
lambda a -> Spray a -> Spray a
forall a v. C a v => a -> v -> v
AlgMod.*> Spray a
pol
infixr 7 /^
(/^) :: (AlgField.C a, Eq a) => Spray a -> a -> Spray a
/^ :: forall a. (C a, Eq a) => Spray a -> a -> Spray a
(/^) Spray a
spray a
lambda = a -> a
forall a. C a => a -> a
AlgField.recip a
lambda a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ Spray a
spray
simplifySpray :: Spray a -> Spray a
simplifySpray :: forall a. Spray a -> Spray a
simplifySpray = (Powers -> Powers) -> HashMap Powers a -> HashMap Powers a
forall k2 k1 v.
(Eq k2, Hashable k2) =>
(k1 -> k2) -> HashMap k1 v -> HashMap k2 v
HM.mapKeys Powers -> Powers
simplifyPowers
removeZeroTerms :: (AlgAdd.C a, Eq a) => Spray a -> Spray a
removeZeroTerms :: forall a. (C a, Eq a) => Spray a -> Spray a
removeZeroTerms = (a -> Bool) -> HashMap Powers a -> HashMap Powers a
forall v k. (v -> Bool) -> HashMap k v -> HashMap k v
HM.filter (a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
forall a. C a => a
AlgAdd.zero)
cleanSpray :: (AlgAdd.C a, Eq a) => Spray a -> Spray a
cleanSpray :: forall a. (C a, Eq a) => Spray a -> Spray a
cleanSpray Spray a
p = Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a
removeZeroTerms (Spray a -> Spray a
forall a. Spray a -> Spray a
simplifySpray Spray a
p)
lone :: AlgRing.C a => Int -> Spray a
lone :: forall a. C a => Int -> Spray a
lone Int
n = if Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
0
then Powers -> a -> HashMap Powers a
forall k v. Hashable k => k -> v -> HashMap k v
HM.singleton Powers
pows a
forall a. C a => a
AlgRing.one
else String -> HashMap Powers a
forall a. HasCallStack => String -> a
error String
"lone: invalid index."
where
pows :: Powers
pows = if Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0
then Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0
else Seq Int -> Int -> Powers
Powers (Int -> Int -> Seq Int
forall a. Int -> a -> Seq a
S.replicate (Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1) Int
forall a. C a => a
AlgAdd.zero Seq Int -> Int -> Seq Int
forall a. Seq a -> a -> Seq a
|> Int
forall a. C a => a
AlgRing.one) Int
n
qlone :: Int -> QSpray
qlone :: Int -> Spray Rational
qlone = Int -> Spray Rational
forall a. C a => Int -> Spray a
lone
unitSpray :: AlgRing.C a => Spray a
unitSpray :: forall a. C a => Spray a
unitSpray = Int -> Spray a
forall a. C a => Int -> Spray a
lone Int
0
zeroSpray :: (Eq a, AlgAdd.C a) => Spray a
zeroSpray :: forall a. (Eq a, C a) => Spray a
zeroSpray = Spray a
forall a. C a => a
AlgAdd.zero
isZeroSpray :: Spray a -> Bool
isZeroSpray :: forall a. Spray a -> Bool
isZeroSpray = HashMap Powers a -> Bool
forall k v. HashMap k v -> Bool
HM.null
constantSpray :: (AlgRing.C a, Eq a) => a -> Spray a
constantSpray :: forall a. (C a, Eq a) => a -> Spray a
constantSpray a
c = a
c a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ Int -> Spray a
forall a. C a => Int -> Spray a
lone Int
0
getCoefficient :: AlgAdd.C a => [Int] -> Spray a -> a
getCoefficient :: forall a. C a => [Int] -> Spray a -> a
getCoefficient [Int]
expnts Spray a
spray = a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup Powers
powers Spray a
spray)
where
expnts' :: Seq Int
expnts' = (Int -> Bool) -> Seq Int -> Seq Int
forall a. (a -> Bool) -> Seq a -> Seq a
S.dropWhileR (Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0) ([Int] -> Seq Int
forall a. [a] -> Seq a
S.fromList [Int]
expnts)
powers :: Powers
powers = Seq Int -> Int -> Powers
Powers Seq Int
expnts' (Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
expnts')
getConstantTerm :: AlgAdd.C a => Spray a -> a
getConstantTerm :: forall a. C a => Spray a -> a
getConstantTerm Spray a
spray = a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup Powers
powers Spray a
spray)
where
powers :: Powers
powers = Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0
isConstantSpray :: (Eq a, AlgRing.C a) => Spray a -> Bool
isConstantSpray :: forall a. (Eq a, C a) => Spray a -> Bool
isConstantSpray = Spray a -> Bool
forall b. HasVariables b => b -> Bool
isConstant
evalSprayHelper :: forall a. AlgRing.C a => [a] -> Spray a -> a
evalSprayHelper :: forall a. C a => [a] -> Spray a -> a
evalSprayHelper [a]
xyz Spray a
spray =
[a] -> a
forall a. C a => [a] -> a
AlgAdd.sum ([a] -> a) -> [a] -> a
forall a b. (a -> b) -> a -> b
$ (Monomial a -> a) -> [Monomial a] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map Monomial a -> a
evalMonomial (Spray a -> [Monomial a]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray)
where
evalMonomial :: Monomial a -> a
evalMonomial :: Monomial a -> a
evalMonomial (Powers
powers, a
coeff) =
a
coeff a -> a -> a
forall a. C a => a -> a -> a
AlgRing.* [a] -> a
forall a. C a => [a] -> a
AlgRing.product ((a -> Integer -> a) -> [a] -> [Integer] -> [a]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith a -> Integer -> a
forall a. C a => a -> Integer -> a
(AlgRing.^) [a]
xyz [Integer]
pows)
where
pows :: [Integer]
pows = Seq Integer -> [Integer]
forall a. Seq a -> [a]
forall (t :: * -> *) a. Foldable t => t a -> [a]
DF.toList (Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Seq Int -> Seq Integer
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Powers -> Seq Int
exponents Powers
powers)
evalSpray :: (Eq a, AlgRing.C a) => Spray a -> [a] -> a
evalSpray :: forall a. (Eq a, C a) => Spray a -> [a] -> a
evalSpray = Spray a -> [a] -> a
Spray a -> [BaseRing (Spray a)] -> BaseRing (Spray a)
forall b. HasVariables b => b -> [BaseRing b] -> BaseRing b
evaluate
evalSpraySpray :: (Eq a, AlgRing.C a) => Spray (Spray a) -> [a] -> Spray a
evalSpraySpray :: forall a. (Eq a, C a) => Spray (Spray a) -> [a] -> Spray a
evalSpraySpray Spray (Spray a)
spray [a]
xyz = if [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
xyz Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
n
then (Spray a -> a) -> Spray (Spray a) -> Spray a
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map ([a] -> Spray a -> a
forall a. C a => [a] -> Spray a -> a
evalSprayHelper [a]
xyz) Spray (Spray a)
spray
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"evalSpraySpray: not enough values provided."
where
n :: Int
n = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum (HashMap Powers Int -> [Int]
forall k v. HashMap k v -> [v]
HM.elems (HashMap Powers Int -> [Int]) -> HashMap Powers Int -> [Int]
forall a b. (a -> b) -> a -> b
$ (Spray a -> Int) -> Spray (Spray a) -> HashMap Powers Int
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray (Spray a)
spray)
fromMonomial :: Monomial a -> Spray a
fromMonomial :: forall a. Monomial a -> Spray a
fromMonomial (Powers
pows, a
coeff) = Powers -> a -> HashMap Powers a
forall k v. Hashable k => k -> v -> HashMap k v
HM.singleton Powers
pows a
coeff
substituteSpray :: (Eq a, AlgRing.C a) => [Maybe a] -> Spray a -> Spray a
substituteSpray :: forall a. (Eq a, C a) => [Maybe a] -> Spray a -> Spray a
substituteSpray = [Maybe a] -> Spray a -> Spray a
[Maybe (BaseRing (Spray a))] -> Spray a -> Spray a
forall b. HasVariables b => [Maybe (BaseRing b)] -> b -> b
substitute
fromRationalSpray :: Spray Rational -> Spray Double
fromRationalSpray :: Spray Rational -> Spray Double
fromRationalSpray = (Rational -> Double) -> Spray Rational -> Spray Double
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map Rational -> Double
forall a. Fractional a => Rational -> a
fromRational
composeSpray ::
forall a. (AlgRing.C a, Eq a) => Spray a -> [Spray a] -> Spray a
composeSpray :: forall a. (C a, Eq a) => Spray a -> [Spray a] -> Spray a
composeSpray Spray a
p = Spray (Spray a) -> [Spray a] -> Spray a
forall a. (Eq a, C a) => Spray a -> [a] -> a
evalSpray (Spray a -> Spray (Spray a)
identify Spray a
p)
where
identify :: Spray a -> Spray (Spray a)
identify :: Spray a -> Spray (Spray a)
identify = (a -> Spray a) -> Spray a -> Spray (Spray a)
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray
fromList :: (AlgRing.C a, Eq a) => [([Int], a)] -> Spray a
fromList :: forall a. (C a, Eq a) => [([Int], a)] -> Spray a
fromList [([Int], a)]
x = Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a
cleanSpray (Spray a -> Spray a) -> Spray a -> Spray a
forall a b. (a -> b) -> a -> b
$ [(Powers, a)] -> Spray a
forall k v. (Eq k, Hashable k) => [(k, v)] -> HashMap k v
HM.fromList ([(Powers, a)] -> Spray a) -> [(Powers, a)] -> Spray a
forall a b. (a -> b) -> a -> b
$ (([Int], a) -> (Powers, a)) -> [([Int], a)] -> [(Powers, a)]
forall a b. (a -> b) -> [a] -> [b]
map
(\([Int]
expts, a
coef) -> (Seq Int -> Int -> Powers
Powers ([Int] -> Seq Int
forall a. [a] -> Seq a
S.fromList [Int]
expts) ([Int] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Int]
expts), a
coef)) [([Int], a)]
x
showSpray ::
(a -> String)
-> (String, String)
-> ([Seq Int] -> [String])
-> Spray a
-> String
showSpray :: forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray a -> String
showCoef (String, String)
braces [Seq Int] -> [String]
showMonomials Spray a
spray =
if Spray a -> Bool
forall a. Spray a -> Bool
isZeroSpray Spray a
spray
then String
"0"
else Text -> String
unpack (Text -> String) -> Text -> String
forall a b. (a -> b) -> a -> b
$ Text -> [Text] -> Text
intercalate (String -> Text
pack String
" + ") [Text]
stringTerms
where
terms :: [(Powers, a)]
terms = ((Powers, a) -> (Powers, a) -> Ordering)
-> [(Powers, a)] -> [(Powers, a)]
forall a. (a -> a -> Ordering) -> [a] -> [a]
sortBy ((Seq Int -> Seq Int -> Ordering) -> Seq Int -> Seq Int -> Ordering
forall a b c. (a -> b -> c) -> b -> a -> c
flip Seq Int -> Seq Int -> Ordering
forall a. Ord a => a -> a -> Ordering
compare (Seq Int -> Seq Int -> Ordering)
-> ((Powers, a) -> Seq Int)
-> (Powers, a)
-> (Powers, a)
-> Ordering
forall b c a. (b -> b -> c) -> (a -> b) -> a -> a -> c
`on` (Powers, a) -> Seq Int
forall {b}. (Powers, b) -> Seq Int
fexpts) (Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray)
fexpts :: (Powers, b) -> Seq Int
fexpts (Powers, b)
term = Powers -> Seq Int
exponents (Powers -> Seq Int) -> Powers -> Seq Int
forall a b. (a -> b) -> a -> b
$ (Powers, b) -> Powers
forall a b. (a, b) -> a
fst (Powers, b)
term
coeffs :: [a]
coeffs = ((Powers, a) -> a) -> [(Powers, a)] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map (Powers, a) -> a
forall a b. (a, b) -> b
snd [(Powers, a)]
terms
powers :: [Seq Int]
powers = ((Powers, a) -> Seq Int) -> [(Powers, a)] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Powers -> Seq Int
exponents (Powers -> Seq Int)
-> ((Powers, a) -> Powers) -> (Powers, a) -> Seq Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Powers, a) -> Powers
forall a b. (a, b) -> a
fst) [(Powers, a)]
terms
stringMonomials :: [String]
stringMonomials = [Seq Int] -> [String]
showMonomials [Seq Int]
powers
stringTerms :: [Text]
stringTerms = (a -> String -> Text) -> [a] -> [String] -> [Text]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith a -> String -> Text
f [a]
coeffs [String]
stringMonomials
f :: a -> String -> Text
f a
coeff String
smonomial
| String
smonomial String -> String -> Bool
forall a. Eq a => a -> a -> Bool
== String
"" = String -> Text
pack String
scoeff'
| String
scoeff String -> String -> Bool
forall a. Eq a => a -> a -> Bool
== String
"" = String -> Text
pack String
smonomial
| Bool
otherwise = String -> Text
pack (String -> Text) -> String -> Text
forall a b. (a -> b) -> a -> b
$ String
scoeff' String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"*" String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
smonomial
where
scoeff :: String
scoeff = a -> String
showCoef a
coeff
scoeff' :: String
scoeff' = (String, String) -> ShowS
bracify (String, String)
braces String
scoeff
showSprayXYZ ::
(a -> String)
-> (String, String)
-> [String]
-> Spray a
-> String
showSprayXYZ :: forall a.
(a -> String) -> (String, String) -> [String] -> Spray a -> String
showSprayXYZ a -> String
showCoef (String, String)
braces [String]
letters Spray a
spray =
if [String] -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [String]
letters
then ShowS
forall a. HasCallStack => String -> a
error String
"showSprayXYZ: empty list of strings."
else (a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray a -> String
showCoef (String, String)
braces ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String]
letters) Spray a
spray
showSprayXYZ' ::
(a -> String)
-> [String]
-> Spray a
-> String
showSprayXYZ' :: forall a. (a -> String) -> [String] -> Spray a -> String
showSprayXYZ' a -> String
showCoef = (a -> String) -> (String, String) -> [String] -> Spray a -> String
forall a.
(a -> String) -> (String, String) -> [String] -> Spray a -> String
showSprayXYZ a -> String
showCoef (String
"(", String
")")
prettySprayXYZ ::
(Show a)
=> [String]
-> Spray a
-> String
prettySprayXYZ :: forall a. Show a => [String] -> Spray a -> String
prettySprayXYZ = (a -> String) -> [String] -> Spray a -> String
forall a. (a -> String) -> [String] -> Spray a -> String
showSprayXYZ' a -> String
forall a. Show a => a -> String
show
showSprayX1X2X3 ::
(a -> String)
-> (String, String)
-> String
-> Spray a
-> String
showSprayX1X2X3 :: forall a.
(a -> String) -> (String, String) -> String -> Spray a -> String
showSprayX1X2X3 a -> String
showCoef (String, String)
braces String
letter =
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray a -> String
showCoef (String, String)
braces (String -> [Seq Int] -> [String]
showMonomialsX1X2X3 String
letter)
showSprayX1X2X3' ::
(a -> String)
-> String
-> Spray a
-> String
showSprayX1X2X3' :: forall a. (a -> String) -> String -> Spray a -> String
showSprayX1X2X3' a -> String
showCoef = (a -> String) -> (String, String) -> String -> Spray a -> String
forall a.
(a -> String) -> (String, String) -> String -> Spray a -> String
showSprayX1X2X3 a -> String
showCoef (String
"(", String
")")
prettySprayX1X2X3 ::
Show a
=> String
-> Spray a
-> String
prettySprayX1X2X3 :: forall a. Show a => String -> Spray a -> String
prettySprayX1X2X3 = (a -> String) -> String -> Spray a -> String
forall a. (a -> String) -> String -> Spray a -> String
showSprayX1X2X3' a -> String
forall a. Show a => a -> String
show
prettySpray :: (Show a) => Spray a -> String
prettySpray :: forall a. Show a => Spray a -> String
prettySpray = [String] -> Spray a -> String
forall a. Show a => [String] -> Spray a -> String
prettySprayXYZ [String
"x", String
"y", String
"z"]
prettySpray' :: Show a => Spray a -> String
prettySpray' :: forall a. Show a => Spray a -> String
prettySpray' = String -> Spray a -> String
forall a. Show a => String -> Spray a -> String
prettySprayX1X2X3 String
"x"
showMonomialsOld :: String -> [Seq Int] -> [String]
showMonomialsOld :: String -> [Seq Int] -> [String]
showMonomialsOld String
var = (Seq Int -> String) -> [Seq Int] -> [String]
forall a b. (a -> b) -> [a] -> [b]
map (String -> Seq Int -> String
showMonomialOld String
var)
where
showMonomialOld :: String -> Seq Int -> String
showMonomialOld :: String -> Seq Int -> String
showMonomialOld String
a Seq Int
pows =
Text -> String
unpack (Text -> String) -> Text -> String
forall a b. (a -> b) -> a -> b
$ Text -> Text -> Text
append (String -> Text
pack String
x) (Char -> Text -> Text
cons Char
'(' (Text -> Text) -> Text -> Text
forall a b. (a -> b) -> a -> b
$ Text -> Char -> Text
snoc Text
string Char
')')
where
x :: String
x = String
a String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"^"
string :: Text
string = Text -> [Text] -> Text
intercalate (String -> Text
pack String
", ") ((Int -> Text) -> [Int] -> [Text]
forall a b. (a -> b) -> [a] -> [b]
map (String -> Text
pack (String -> Text) -> (Int -> String) -> Int -> Text
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> String
forall a. Show a => a -> String
show) (Seq Int -> [Int]
forall a. Seq a -> [a]
forall (t :: * -> *) a. Foldable t => t a -> [a]
DF.toList Seq Int
pows))
prettySpray'' ::
Show a
=> String
-> Spray a
-> String
prettySpray'' :: forall a. Show a => String -> Spray a -> String
prettySpray'' String
var = (a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray a -> String
forall a. Show a => a -> String
show (String
"(", String
")") (String -> [Seq Int] -> [String]
showMonomialsOld String
var)
showNumSpray ::
(Num a, Ord a)
=> ([Seq Int] -> [String])
-> (a -> String)
-> Spray a
-> String
showNumSpray :: forall a.
(Num a, Ord a) =>
([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
showNumSpray [Seq Int] -> [String]
showMonomials a -> String
showCoeff Spray a
spray =
if Spray a -> Bool
forall a. Spray a -> Bool
isZeroSpray Spray a
spray
then String
"0"
else [String] -> String
forall (t :: * -> *) a. Foldable t => t [a] -> [a]
concat ([String] -> String) -> [String] -> String
forall a b. (a -> b) -> a -> b
$ (String -> ShowS) -> [String] -> [String] -> [String]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith String -> ShowS
forall a. [a] -> [a] -> [a]
(++) [String]
stringSigns [String]
stringTerms
where
terms :: [(Powers, a)]
terms = ((Powers, a) -> (Powers, a) -> Ordering)
-> [(Powers, a)] -> [(Powers, a)]
forall a. (a -> a -> Ordering) -> [a] -> [a]
sortBy ((Seq Int -> Seq Int -> Ordering) -> Seq Int -> Seq Int -> Ordering
forall a b c. (a -> b -> c) -> b -> a -> c
flip Seq Int -> Seq Int -> Ordering
forall a. Ord a => a -> a -> Ordering
compare (Seq Int -> Seq Int -> Ordering)
-> ((Powers, a) -> Seq Int)
-> (Powers, a)
-> (Powers, a)
-> Ordering
forall b c a. (b -> b -> c) -> (a -> b) -> a -> a -> c
`on` (Powers -> Seq Int
exponents (Powers -> Seq Int)
-> ((Powers, a) -> Powers) -> (Powers, a) -> Seq Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Powers, a) -> Powers
forall a b. (a, b) -> a
fst)) (Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray)
coeffs :: [a]
coeffs = ((Powers, a) -> a) -> [(Powers, a)] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map (Powers, a) -> a
forall a b. (a, b) -> b
snd [(Powers, a)]
terms
(a
firstCoeff, [a]
otherCoeffs) = Maybe (a, [a]) -> (a, [a])
forall a. HasCallStack => Maybe a -> a
fromJust ([a] -> Maybe (a, [a])
forall a. [a] -> Maybe (a, [a])
uncons [a]
coeffs)
firstSign :: String
firstSign = if a
firstCoeff a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> a
0 then String
"" else String
"-"
otherSigns :: [String]
otherSigns = (a -> String) -> [a] -> [String]
forall a b. (a -> b) -> [a] -> [b]
map (\a
x -> if a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> a
0 then String
" + " else String
" - ") [a]
otherCoeffs
stringSigns :: [String]
stringSigns = String
firstSign String -> [String] -> [String]
forall a. a -> [a] -> [a]
: [String]
otherSigns
absCoeffs :: [a]
absCoeffs = (a -> a) -> [a] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map a -> a
forall a. Num a => a -> a
abs [a]
coeffs
powers :: [Seq Int]
powers = ((Powers, a) -> Seq Int) -> [(Powers, a)] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Powers -> Seq Int
exponents (Powers -> Seq Int)
-> ((Powers, a) -> Powers) -> (Powers, a) -> Seq Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Powers, a) -> Powers
forall a b. (a, b) -> a
fst) [(Powers, a)]
terms
stringMonomials :: [String]
stringMonomials = [Seq Int] -> [String]
showMonomials [Seq Int]
powers
stringTerms :: [String]
stringTerms = (a -> ShowS) -> [a] -> [String] -> [String]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith a -> ShowS
f [a]
absCoeffs [String]
stringMonomials
f :: a -> ShowS
f a
acoeff String
smonomial
| String
smonomial String -> String -> Bool
forall a. Eq a => a -> a -> Bool
== String
"" = a -> String
showCoeff a
acoeff
| String
scoeff String -> String -> Bool
forall a. Eq a => a -> a -> Bool
== String
"" = String
smonomial
| Bool
otherwise = String
scoeff String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"*" String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
smonomial
where
scoeff :: String
scoeff = if a
acoeff a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
1 then String
"" else a -> String
showCoeff a
acoeff
showMonomialX1X2X3 :: String -> Seq Int -> Text
showMonomialX1X2X3 :: String -> Seq Int -> Text
showMonomialX1X2X3 String
x Seq Int
pows = Text
x1x2x3
where
f :: Int -> Int -> Text
f Int
i Int
p
| Int
p Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = String -> Text
pack String
""
| Int
p Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 = String -> Text
pack (String -> Text) -> String -> Text
forall a b. (a -> b) -> a -> b
$ String
x String -> ShowS
forall a. [a] -> [a] -> [a]
++ Int -> String
forall a. Show a => a -> String
show Int
i
| Bool
otherwise = String -> Text
pack (String -> Text) -> String -> Text
forall a b. (a -> b) -> a -> b
$ String
x String -> ShowS
forall a. [a] -> [a] -> [a]
++ Int -> String
forall a. Show a => a -> String
show Int
i String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"^" String -> ShowS
forall a. [a] -> [a] -> [a]
++ Int -> String
forall a. Show a => a -> String
show Int
p
indices :: [Int]
indices = (Int -> Bool) -> Seq Int -> [Int]
forall a. (a -> Bool) -> Seq a -> [Int]
S.findIndicesL (Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
/= Int
0) Seq Int
pows
x1x2x3 :: Text
x1x2x3 =
Text -> [Text] -> Text
intercalate (String -> Text
pack String
".") ((Int -> Text) -> [Int] -> [Text]
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> Int -> Int -> Text
f (Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) (Seq Int
pows Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
i)) [Int]
indices)
showMonomialsX1X2X3 :: String -> [Seq Int] -> [String]
showMonomialsX1X2X3 :: String -> [Seq Int] -> [String]
showMonomialsX1X2X3 String
x = (Seq Int -> String) -> [Seq Int] -> [String]
forall a b. (a -> b) -> [a] -> [b]
map (Text -> String
unpack (Text -> String) -> (Seq Int -> Text) -> Seq Int -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. String -> Seq Int -> Text
showMonomialX1X2X3 String
x)
showMonomialXYZ :: [String] -> Int -> Seq Int -> Text
showMonomialXYZ :: [String] -> Int -> Seq Int -> Text
showMonomialXYZ [String]
letters Int
n Seq Int
pows = if Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= [String] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [String]
letters
then Text
xyz
else String -> Seq Int -> Text
showMonomialX1X2X3 ([String]
letters [String] -> Int -> String
forall a. HasCallStack => [a] -> Int -> a
!! Int
0) Seq Int
pows
where
f :: String -> a -> Text
f String
letter a
p
| a
p a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = String -> Text
pack String
""
| a
p a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
1 = String -> Text
pack String
letter
| Bool
otherwise = String -> Text
pack (String -> Text) -> String -> Text
forall a b. (a -> b) -> a -> b
$ String
letter String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"^" String -> ShowS
forall a. [a] -> [a] -> [a]
++ a -> String
forall a. Show a => a -> String
show a
p
indices :: [Int]
indices = (Int -> Bool) -> Seq Int -> [Int]
forall a. (a -> Bool) -> Seq a -> [Int]
S.findIndicesL (Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
/= Int
0) Seq Int
pows
xyz :: Text
xyz = Text -> [Text] -> Text
intercalate (String -> Text
pack String
".")
((Int -> Text) -> [Int] -> [Text]
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> String -> Int -> Text
forall {a}. (Eq a, Num a, Show a) => String -> a -> Text
f ([String]
letters[String] -> Int -> String
forall a. HasCallStack => [a] -> Int -> a
!!Int
i) (Seq Int
pows Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
i)) [Int]
indices)
showMonomialsXYZ :: [String] -> [Seq Int] -> [String]
showMonomialsXYZ :: [String] -> [Seq Int] -> [String]
showMonomialsXYZ [String]
letters [Seq Int]
powers = (Seq Int -> String) -> [Seq Int] -> [String]
forall a b. (a -> b) -> [a] -> [b]
map (Text -> String
unpack (Text -> String) -> (Seq Int -> Text) -> Seq Int -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [String] -> Int -> Seq Int -> Text
showMonomialXYZ [String]
letters Int
n) [Seq Int]
powers
where
n :: Int
n = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum ((Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map Seq Int -> Int
forall a. Seq a -> Int
S.length [Seq Int]
powers)
prettyNumSprayX1X2X3 :: (Num a, Ord a, Show a)
=> String
-> Spray a
-> String
prettyNumSprayX1X2X3 :: forall a. (Num a, Ord a, Show a) => String -> Spray a -> String
prettyNumSprayX1X2X3 String
x = ([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
forall a.
(Num a, Ord a) =>
([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
showNumSpray (String -> [Seq Int] -> [String]
showMonomialsX1X2X3 String
x) a -> String
forall a. Show a => a -> String
show
prettyNumSprayXYZ :: (Num a, Ord a, Show a)
=> [String]
-> Spray a
-> String
prettyNumSprayXYZ :: forall a. (Num a, Ord a, Show a) => [String] -> Spray a -> String
prettyNumSprayXYZ [String]
letters = ([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
forall a.
(Num a, Ord a) =>
([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
showNumSpray ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String]
letters) a -> String
forall a. Show a => a -> String
show
showRatio :: Rational -> String
showRatio :: Rational -> String
showRatio Rational
q = if Integer
d Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
1
then Integer -> String
forall a. Show a => a -> String
show Integer
n
else String
"(" String -> ShowS
forall a. [a] -> [a] -> [a]
++ Integer -> String
forall a. Show a => a -> String
show Integer
n String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"/" String -> ShowS
forall a. [a] -> [a] -> [a]
++ Integer -> String
forall a. Show a => a -> String
show Integer
d String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
")"
where
n :: Integer
n = Rational -> Integer
forall a. Ratio a -> a
DR.numerator Rational
q
d :: Integer
d = Rational -> Integer
forall a. Ratio a -> a
DR.denominator Rational
q
showRatio' :: (Eq a, Num a, Show a) => NumberRatio.T a -> String
showRatio' :: forall a. (Eq a, Num a, Show a) => T a -> String
showRatio' T a
q = if a
d a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
1
then a -> String
forall a. Show a => a -> String
show a
n
else String
"(" String -> ShowS
forall a. [a] -> [a] -> [a]
++ a -> String
forall a. Show a => a -> String
show a
n String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"/" String -> ShowS
forall a. [a] -> [a] -> [a]
++ a -> String
forall a. Show a => a -> String
show a
d String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
")"
where
n :: a
n = T a -> a
forall a. T a -> a
NumberRatio.numerator T a
q
d :: a
d = T a -> a
forall a. T a -> a
NumberRatio.denominator T a
q
showQSpray ::
([Seq Int] -> [String])
-> QSpray
-> String
showQSpray :: ([Seq Int] -> [String]) -> Spray Rational -> String
showQSpray [Seq Int] -> [String]
showMonomials = ([Seq Int] -> [String])
-> (Rational -> String) -> Spray Rational -> String
forall a.
(Num a, Ord a) =>
([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
showNumSpray [Seq Int] -> [String]
showMonomials Rational -> String
showRatio
showQSpray' ::
([Seq Int] -> [String])
-> QSpray'
-> String
showQSpray' :: ([Seq Int] -> [String]) -> Spray Rational' -> String
showQSpray' [Seq Int] -> [String]
showMonomials = ([Seq Int] -> [String])
-> (Rational' -> String) -> Spray Rational' -> String
forall a.
(Num a, Ord a) =>
([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
showNumSpray [Seq Int] -> [String]
showMonomials Rational' -> String
forall a. (Eq a, Num a, Show a) => T a -> String
showRatio'
prettyQSprayX1X2X3 ::
String
-> QSpray
-> String
prettyQSprayX1X2X3 :: String -> Spray Rational -> String
prettyQSprayX1X2X3 String
x = ([Seq Int] -> [String]) -> Spray Rational -> String
showQSpray (String -> [Seq Int] -> [String]
showMonomialsX1X2X3 String
x)
prettyQSprayX1X2X3' ::
String
-> QSpray'
-> String
prettyQSprayX1X2X3' :: String -> Spray Rational' -> String
prettyQSprayX1X2X3' String
x = ([Seq Int] -> [String]) -> Spray Rational' -> String
showQSpray' (String -> [Seq Int] -> [String]
showMonomialsX1X2X3 String
x)
prettyQSprayXYZ ::
[String]
-> QSpray
-> String
prettyQSprayXYZ :: [String] -> Spray Rational -> String
prettyQSprayXYZ [String]
letters = ([Seq Int] -> [String]) -> Spray Rational -> String
showQSpray ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String]
letters)
prettyQSprayXYZ' ::
[String]
-> QSpray'
-> String
prettyQSprayXYZ' :: [String] -> Spray Rational' -> String
prettyQSprayXYZ' [String]
letters = ([Seq Int] -> [String]) -> Spray Rational' -> String
showQSpray' ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String]
letters)
prettyQSpray :: QSpray -> String
prettyQSpray :: Spray Rational -> String
prettyQSpray = [String] -> Spray Rational -> String
prettyQSprayXYZ [String
"x", String
"y", String
"z"]
prettyQSpray'' :: QSpray -> String
prettyQSpray'' :: Spray Rational -> String
prettyQSpray'' = [String] -> Spray Rational -> String
prettyQSprayXYZ [String
"X", String
"Y", String
"Z"]
prettyQSpray' :: QSpray' -> String
prettyQSpray' :: Spray Rational' -> String
prettyQSpray' = [String] -> Spray Rational' -> String
prettyQSprayXYZ' [String
"x", String
"y", String
"z"]
prettyQSpray''' :: QSpray' -> String
prettyQSpray''' :: Spray Rational' -> String
prettyQSpray''' = [String] -> Spray Rational' -> String
prettyQSprayXYZ' [String
"X", String
"Y", String
"Z"]
prettyNumSpray :: (Num a, Ord a, Show a) => Spray a -> String
prettyNumSpray :: forall a. (Num a, Ord a, Show a) => Spray a -> String
prettyNumSpray = [String] -> Spray a -> String
forall a. (Num a, Ord a, Show a) => [String] -> Spray a -> String
prettyNumSprayXYZ [String
"x", String
"y", String
"z"]
prettyNumSpray' :: (Num a, Ord a, Show a) => Spray a -> String
prettyNumSpray' :: forall a. (Num a, Ord a, Show a) => Spray a -> String
prettyNumSpray' = [String] -> Spray a -> String
forall a. (Num a, Ord a, Show a) => [String] -> Spray a -> String
prettyNumSprayXYZ [String
"X", String
"Y", String
"Z"]
sprayTerms :: Spray a -> HashMap (Seq Int) a
sprayTerms :: forall a. Spray a -> HashMap (Seq Int) a
sprayTerms = (Powers -> Seq Int) -> HashMap Powers a -> HashMap (Seq Int) a
forall k2 k1 v.
(Eq k2, Hashable k2) =>
(k1 -> k2) -> HashMap k1 v -> HashMap k2 v
HM.mapKeys Powers -> Seq Int
exponents
toList :: Spray a -> [([Int], a)]
toList :: forall a. Spray a -> [([Int], a)]
toList Spray a
p = HashMap [Int] a -> [([Int], a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList (HashMap [Int] a -> [([Int], a)])
-> HashMap [Int] a -> [([Int], a)]
forall a b. (a -> b) -> a -> b
$ (Powers -> [Int]) -> Spray a -> HashMap [Int] a
forall k2 k1 v.
(Eq k2, Hashable k2) =>
(k1 -> k2) -> HashMap k1 v -> HashMap k2 v
HM.mapKeys (Seq Int -> [Int]
forall a. Seq a -> [a]
forall (t :: * -> *) a. Foldable t => t a -> [a]
DF.toList (Seq Int -> [Int]) -> (Powers -> Seq Int) -> Powers -> [Int]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Powers -> Seq Int
exponents) Spray a
p
bombieriSpray :: AlgAdd.C a => Spray a -> Spray a
bombieriSpray :: forall a. C a => Spray a -> Spray a
bombieriSpray = (Powers -> a -> a) -> HashMap Powers a -> HashMap Powers a
forall k v1 v2. (k -> v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.mapWithKey Powers -> a -> a
forall {a}. C a => Powers -> a -> a
f
where
f :: Powers -> a -> a
f Powers
pows = Int -> a -> a
forall {a}. C a => Int -> a -> a
times (Seq Int -> Int
forall {a}. (Num a, Enum a, Eq a) => Seq a -> a
pfactorial (Seq Int -> Int) -> Seq Int -> Int
forall a b. (a -> b) -> a -> b
$ Powers -> Seq Int
exponents Powers
pows)
pfactorial :: Seq a -> a
pfactorial Seq a
pows = [a] -> a
forall a. Num a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
product ([a] -> a) -> [a] -> a
forall a b. (a -> b) -> a -> b
$ Seq a -> [a]
forall a. Seq a -> [a]
forall (t :: * -> *) a. Foldable t => t a -> [a]
DF.toList (Seq a -> [a]) -> Seq a -> [a]
forall a b. (a -> b) -> a -> b
$ a -> a
forall {a}. (Num a, Enum a) => a -> a
factorial (a -> a) -> Seq a -> Seq a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (a -> Bool) -> Seq a -> Seq a
forall a. (a -> Bool) -> Seq a -> Seq a
S.filter (a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
0) Seq a
pows
factorial :: a -> a
factorial a
n = [a] -> a
forall a. Num a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
product [a
1 .. a
n]
times :: Int -> a -> a
times Int
k a
x = [a] -> a
forall a. C a => [a] -> a
AlgAdd.sum (Int -> a -> [a]
forall a. Int -> a -> [a]
replicate Int
k a
x)
collinearSprays :: (Eq a, AlgField.C a) => Spray a -> Spray a -> Bool
collinearSprays :: forall a. (Eq a, C a) => Spray a -> Spray a -> Bool
collinearSprays Spray a
spray1 Spray a
spray2 = a
r a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ Spray a
spray2 Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
spray1
where
r :: a
r = (Powers, a) -> a
forall a b. (a, b) -> b
snd (Spray a -> (Powers, a)
forall a. Spray a -> Monomial a
leadingTerm Spray a
spray1) a -> a -> a
forall a. C a => a -> a -> a
AlgField./ (Powers, a) -> a
forall a b. (a, b) -> b
snd (Spray a -> (Powers, a)
forall a. Spray a -> Monomial a
leadingTerm Spray a
spray2)
maxIndex :: Ord a => [a] -> Int
maxIndex :: forall a. Ord a => [a] -> Int
maxIndex = (Int, a) -> Int
forall a b. (a, b) -> a
fst ((Int, a) -> Int) -> ([a] -> (Int, a)) -> [a] -> Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. ((Int, a) -> (Int, a) -> Ordering) -> [(Int, a)] -> (Int, a)
forall (t :: * -> *) a.
Foldable t =>
(a -> a -> Ordering) -> t a -> a
maximumBy (((Int, a) -> a) -> (Int, a) -> (Int, a) -> Ordering
forall a b. Ord a => (b -> a) -> b -> b -> Ordering
comparing (Int, a) -> a
forall a b. (a, b) -> b
snd) ([(Int, a)] -> (Int, a)) -> ([a] -> [(Int, a)]) -> [a] -> (Int, a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [Int] -> [a] -> [(Int, a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Int
0 .. ]
leadingTerm :: Spray a -> Monomial a
leadingTerm :: forall a. Spray a -> Monomial a
leadingTerm Spray a
p = (Powers
biggest, Spray a
p Spray a -> Powers -> a
forall k v.
(Eq k, Hashable k, HasCallStack) =>
HashMap k v -> k -> v
HM.! Powers
biggest)
where
powers :: [Powers]
powers = Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
p
i :: Int
i = [Seq Int] -> Int
forall a. Ord a => [a] -> Int
maxIndex ([Seq Int] -> Int) -> [Seq Int] -> Int
forall a b. (a -> b) -> a -> b
$ (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
powers
biggest :: Powers
biggest = [Powers]
powers [Powers] -> Int -> Powers
forall a. HasCallStack => [a] -> Int -> a
!! Int
i
divides :: Monomial a -> Monomial a -> Bool
divides :: forall a. Monomial a -> Monomial a -> Bool
divides (Powers
powsP, a
_) (Powers
powsQ, a
_) = Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
expntsP Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
expntsQ Bool -> Bool -> Bool
&& Bool
lower
where
expntsP :: Seq Int
expntsP = Powers -> Seq Int
exponents Powers
powsP
expntsQ :: Seq Int
expntsQ = Powers -> Seq Int
exponents Powers
powsQ
lower :: Bool
lower = ((Int, Int) -> Bool) -> Seq (Int, Int) -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
DF.all ((Int -> Int -> Bool) -> (Int, Int) -> Bool
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
(<=)) (Seq Int -> Seq Int -> Seq (Int, Int)
forall a b. Seq a -> Seq b -> Seq (a, b)
S.zip Seq Int
expntsP Seq Int
expntsQ)
quotient :: AlgField.C a => Monomial a -> Monomial a -> Monomial a
quotient :: forall a. C a => Monomial a -> Monomial a -> Monomial a
quotient (Powers
powsQ, a
coeffQ) (Powers
powsP, a
coeffP) = (Powers
pows, a
coeff)
where
(Powers
powsP', Powers
powsQ') = (Powers, Powers) -> (Powers, Powers)
harmonize (Powers
powsP, Powers
powsQ)
expntsP :: Seq Int
expntsP = Powers -> Seq Int
exponents Powers
powsP'
expntsQ :: Seq Int
expntsQ = Powers -> Seq Int
exponents Powers
powsQ'
expnts :: Seq Int
expnts = (Int -> Int -> Int) -> Seq Int -> Seq Int -> Seq Int
forall a b c. (a -> b -> c) -> Seq a -> Seq b -> Seq c
S.zipWith (-) Seq Int
expntsQ Seq Int
expntsP
n :: Int
n = Powers -> Int
nvariables Powers
powsP'
pows :: Powers
pows = Seq Int -> Int -> Powers
Powers Seq Int
expnts Int
n
coeff :: a
coeff = a
coeffQ a -> a -> a
forall a. C a => a -> a -> a
AlgField./ a
coeffP
sprayDivisionRemainder :: forall a. (Eq a, AlgField.C a)
=> Spray a -> [Spray a] -> Spray a
sprayDivisionRemainder :: forall a. (Eq a, C a) => Spray a -> [Spray a] -> Spray a
sprayDivisionRemainder Spray a
p [Spray a]
qs =
if Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0
then String -> Spray a
forall a. HasCallStack => String -> a
error String
"sprayDivisionRemainder: the list of divisors is empty."
else (Spray a, Spray a) -> Spray a
forall a b. (a, b) -> b
snd ((Spray a, Spray a) -> Spray a) -> (Spray a, Spray a) -> Spray a
forall a b. (a -> b) -> a -> b
$ Spray a -> Spray a -> (Spray a, Spray a)
ogo Spray a
p Spray a
forall a. C a => a
AlgAdd.zero
where
n :: Int
n = [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
qs
qsltqs :: [(Spray a, Monomial a)]
qsltqs = [Spray a] -> [Monomial a] -> [(Spray a, Monomial a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Spray a]
qs ((Spray a -> Monomial a) -> [Spray a] -> [Monomial a]
forall a b. (a -> b) -> [a] -> [b]
map Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm [Spray a]
qs)
g :: Monomial a -> Spray a -> Spray a -> (Spray a, Spray a)
g :: Monomial a -> Spray a -> Spray a -> (Spray a, Spray a)
g Monomial a
lts Spray a
s Spray a
r = (Spray a
s Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
ltsspray, Spray a
r Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray a
ltsspray)
where
ltsspray :: Spray a
ltsspray = Monomial a -> Spray a
forall a. Monomial a -> Spray a
fromMonomial Monomial a
lts
go :: Monomial a -> Spray a -> Spray a -> Int -> Bool -> (Spray a, Spray a)
go :: Monomial a
-> Spray a -> Spray a -> Int -> Bool -> (Spray a, Spray a)
go Monomial a
lts !Spray a
s Spray a
r !Int
i !Bool
divoccured
| Bool
divoccured = (Spray a
s, Spray a
r)
| Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
n = Monomial a -> Spray a -> Spray a -> (Spray a, Spray a)
g Monomial a
lts Spray a
s Spray a
r
| Bool
otherwise = Monomial a
-> Spray a -> Spray a -> Int -> Bool -> (Spray a, Spray a)
go Monomial a
lts Spray a
news Spray a
r (Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) Bool
newdivoccured
where
(Spray a
q, Monomial a
ltq) = [(Spray a, Monomial a)]
qsltqs [(Spray a, Monomial a)] -> Int -> (Spray a, Monomial a)
forall a. HasCallStack => [a] -> Int -> a
!! Int
i
newdivoccured :: Bool
newdivoccured = Monomial a -> Monomial a -> Bool
forall a. Monomial a -> Monomial a -> Bool
divides Monomial a
ltq Monomial a
lts
news :: Spray a
news = if Bool
newdivoccured
then Spray a
s Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ (Monomial a -> Spray a
forall a. Monomial a -> Spray a
fromMonomial (Monomial a -> Monomial a -> Monomial a
forall a. C a => Monomial a -> Monomial a -> Monomial a
quotient Monomial a
lts Monomial a
ltq) Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q)
else Spray a
s
ogo :: Spray a -> Spray a -> (Spray a, Spray a)
ogo :: Spray a -> Spray a -> (Spray a, Spray a)
ogo !Spray a
s !Spray a
r
| Spray a
s Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. C a => a
AlgAdd.zero = (Spray a
s, Spray a
r)
| Bool
otherwise = Spray a -> Spray a -> (Spray a, Spray a)
ogo Spray a
s' Spray a
r'
where
(Spray a
s', Spray a
r') = Monomial a
-> Spray a -> Spray a -> Int -> Bool -> (Spray a, Spray a)
go (Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm Spray a
s) Spray a
s Spray a
r Int
0 Bool
False
sprayDivision :: forall a. (Eq a, AlgField.C a)
=> Spray a
-> Spray a
-> (Spray a, Spray a)
sprayDivision :: forall a. (Eq a, C a) => Spray a -> Spray a -> (Spray a, Spray a)
sprayDivision Spray a
sprayA Spray a
sprayB =
if Spray a -> Bool
forall b. HasVariables b => b -> Bool
isConstant Spray a
sprayB
then if Spray a -> Bool
forall a. Spray a -> Bool
isZeroSpray Spray a
sprayB
then
String -> (Spray a, Spray a)
forall a. HasCallStack => String -> a
error String
"sprayDivision: division by zero."
else
let c :: a
c = Spray a -> a
forall a. C a => Spray a -> a
getConstantTerm Spray a
sprayB in (Spray a
sprayA Spray a -> a -> Spray a
forall k a. (C k, C k a) => a -> k -> a
/> a
c, Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray)
else Spray a -> Spray a -> Spray a -> (Spray a, Spray a)
ogo Spray a
sprayA Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray
where
go :: Monomial a -> Spray a -> Spray a -> Spray a -> Int -> Bool
-> (Spray a, Spray a, Spray a)
go :: Monomial a
-> Spray a
-> Spray a
-> Spray a
-> Int
-> Bool
-> (Spray a, Spray a, Spray a)
go Monomial a
ltp !Spray a
p !Spray a
q Spray a
r !Int
i !Bool
divoccured
| Bool
divoccured = (Spray a
p, Spray a
q, Spray a
r)
| Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 = (Spray a
p Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
ltpspray, Spray a
q, Spray a
r Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray a
ltpspray)
| Bool
otherwise = Monomial a
-> Spray a
-> Spray a
-> Spray a
-> Int
-> Bool
-> (Spray a, Spray a, Spray a)
go Monomial a
ltp Spray a
newp Spray a
newq Spray a
r Int
1 Bool
newdivoccured
where
ltpspray :: Spray a
ltpspray = Monomial a -> Spray a
forall a. Monomial a -> Spray a
fromMonomial Monomial a
ltp
ltB :: Monomial a
ltB = Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm Spray a
sprayB
newdivoccured :: Bool
newdivoccured = Monomial a -> Monomial a -> Bool
forall a. Monomial a -> Monomial a -> Bool
divides Monomial a
ltB Monomial a
ltp
(Spray a
newp, Spray a
newq) = if Bool
newdivoccured
then (Spray a
p Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ (Spray a
qtnt Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
sprayB), Spray a
q Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray a
qtnt)
else (Spray a
p, Spray a
q)
where
qtnt :: Spray a
qtnt = Monomial a -> Spray a
forall a. Monomial a -> Spray a
fromMonomial (Monomial a -> Spray a) -> Monomial a -> Spray a
forall a b. (a -> b) -> a -> b
$ Monomial a -> Monomial a -> Monomial a
forall a. C a => Monomial a -> Monomial a -> Monomial a
quotient Monomial a
ltp Monomial a
ltB
ogo :: Spray a -> Spray a -> Spray a -> (Spray a, Spray a)
ogo :: Spray a -> Spray a -> Spray a -> (Spray a, Spray a)
ogo !Spray a
p !Spray a
q !Spray a
r
| Spray a
p Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. C a => a
AlgAdd.zero = (Spray a
q, Spray a
r)
| Bool
otherwise = Spray a -> Spray a -> Spray a -> (Spray a, Spray a)
ogo Spray a
p' Spray a
q' Spray a
r'
where
(Spray a
p', Spray a
q', Spray a
r') = Monomial a
-> Spray a
-> Spray a
-> Spray a
-> Int
-> Bool
-> (Spray a, Spray a, Spray a)
go (Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm Spray a
p) Spray a
p Spray a
q Spray a
r Int
0 Bool
False
sprayDivisionRemainder' ::
forall a. (Eq a, AlgField.C a)
=> Spray a -> HashMap Int (Spray a, Monomial a) -> Spray a
sprayDivisionRemainder' :: forall a.
(Eq a, C a) =>
Spray a -> HashMap Int (Spray a, Monomial a) -> Spray a
sprayDivisionRemainder' Spray a
p HashMap Int (Spray a, Monomial a)
qsltqs = (Spray a, Spray a) -> Spray a
forall a b. (a, b) -> b
snd ((Spray a, Spray a) -> Spray a) -> (Spray a, Spray a) -> Spray a
forall a b. (a -> b) -> a -> b
$ Spray a -> Spray a -> (Spray a, Spray a)
ogo Spray a
p Spray a
forall a. C a => a
AlgAdd.zero
where
n :: Int
n = HashMap Int (Spray a, Monomial a) -> Int
forall k v. HashMap k v -> Int
HM.size HashMap Int (Spray a, Monomial a)
qsltqs
g :: Monomial a -> Spray a -> Spray a -> (Spray a, Spray a)
g :: Monomial a -> Spray a -> Spray a -> (Spray a, Spray a)
g Monomial a
lts Spray a
s Spray a
r = (Spray a
s Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
ltsspray, Spray a
r Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray a
ltsspray)
where
ltsspray :: Spray a
ltsspray = Monomial a -> Spray a
forall a. Monomial a -> Spray a
fromMonomial Monomial a
lts
go :: Monomial a -> Spray a -> Spray a -> Int -> Bool -> (Spray a, Spray a)
go :: Monomial a
-> Spray a -> Spray a -> Int -> Bool -> (Spray a, Spray a)
go Monomial a
lts !Spray a
s Spray a
r !Int
i !Bool
divoccured
| Bool
divoccured = (Spray a
s, Spray a
r)
| Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
n = Monomial a -> Spray a -> Spray a -> (Spray a, Spray a)
g Monomial a
lts Spray a
s Spray a
r
| Bool
otherwise = Monomial a
-> Spray a -> Spray a -> Int -> Bool -> (Spray a, Spray a)
go Monomial a
lts Spray a
news Spray a
r (Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) Bool
newdivoccured
where
(Spray a
q, Monomial a
ltq) = HashMap Int (Spray a, Monomial a)
qsltqs HashMap Int (Spray a, Monomial a) -> Int -> (Spray a, Monomial a)
forall k v.
(Eq k, Hashable k, HasCallStack) =>
HashMap k v -> k -> v
HM.! Int
i
newdivoccured :: Bool
newdivoccured = Monomial a -> Monomial a -> Bool
forall a. Monomial a -> Monomial a -> Bool
divides Monomial a
ltq Monomial a
lts
news :: Spray a
news = if Bool
newdivoccured
then Spray a
s Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ (Monomial a -> Spray a
forall a. Monomial a -> Spray a
fromMonomial (Monomial a -> Monomial a -> Monomial a
forall a. C a => Monomial a -> Monomial a -> Monomial a
quotient Monomial a
lts Monomial a
ltq) Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q)
else Spray a
s
ogo :: Spray a -> Spray a -> (Spray a, Spray a)
ogo :: Spray a -> Spray a -> (Spray a, Spray a)
ogo !Spray a
s !Spray a
r
| Spray a
s Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. C a => a
AlgAdd.zero = (Spray a
s, Spray a
r)
| Bool
otherwise = Spray a -> Spray a -> (Spray a, Spray a)
ogo Spray a
s' Spray a
r'
where
(Spray a
s', Spray a
r') = Monomial a
-> Spray a -> Spray a -> Int -> Bool -> (Spray a, Spray a)
go (Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm Spray a
s) Spray a
s Spray a
r Int
0 Bool
False
combn2 :: Int -> Int -> HashMap Int (Int, Int)
combn2 :: Int -> Int -> HashMap Int (Int, Int)
combn2 Int
n Int
s = [(Int, (Int, Int))] -> HashMap Int (Int, Int)
forall k v. (Eq k, Hashable k) => [(k, v)] -> HashMap k v
HM.fromList ([Int] -> [(Int, Int)] -> [(Int, (Int, Int))]
forall a b. [a] -> [b] -> [(a, b)]
zip [Int]
range0 ([Int] -> [Int] -> [(Int, Int)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Int]
row1 [Int]
row2))
where
range0 :: [Int]
range0 = [Int
0 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
2]
range1 :: [Int]
range1 = [Int
1 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1]
row1 :: [Int]
row1 = Int -> [Int] -> [Int]
forall a. Int -> [a] -> [a]
drop Int
s ([Int] -> [Int]) -> [Int] -> [Int]
forall a b. (a -> b) -> a -> b
$ (Int -> [Int]) -> [Int] -> [Int]
forall (t :: * -> *) a b. Foldable t => (a -> [b]) -> t a -> [b]
concatMap (\Int
i -> [Int
0 .. Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1]) [Int]
range1
row2 :: [Int]
row2 = Int -> [Int] -> [Int]
forall a. Int -> [a] -> [a]
drop Int
s ([Int] -> [Int]) -> [Int] -> [Int]
forall a b. (a -> b) -> a -> b
$ (Int -> [Int]) -> [Int] -> [Int]
forall (t :: * -> *) a b. Foldable t => (a -> [b]) -> t a -> [b]
concatMap (\Int
i -> Int -> Int -> [Int]
forall a. Int -> a -> [a]
replicate Int
i Int
i) [Int]
range1
sPolynomial :: (Eq a, AlgField.C a)
=> (Spray a, Monomial a) -> (Spray a, Monomial a) -> Spray a
sPolynomial :: forall a.
(Eq a, C a) =>
(Spray a, Monomial a) -> (Spray a, Monomial a) -> Spray a
sPolynomial (Spray a, Monomial a)
pltp (Spray a, Monomial a)
qltq = Spray a
wp Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
p Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
wq Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q
where
p :: Spray a
p = (Spray a, Monomial a) -> Spray a
forall a b. (a, b) -> a
fst (Spray a, Monomial a)
pltp
q :: Spray a
q = (Spray a, Monomial a) -> Spray a
forall a b. (a, b) -> a
fst (Spray a, Monomial a)
qltq
(Powers
lpowsP, a
lcoefP) = (Spray a, Monomial a) -> Monomial a
forall a b. (a, b) -> b
snd (Spray a, Monomial a)
pltp
(Powers
lpowsQ, a
lcoefQ) = (Spray a, Monomial a) -> Monomial a
forall a b. (a, b) -> b
snd (Spray a, Monomial a)
qltq
(Powers
lpowsP', Powers
lpowsQ') = (Powers, Powers) -> (Powers, Powers)
harmonize (Powers
lpowsP, Powers
lpowsQ)
lexpntsP :: Seq Int
lexpntsP = Powers -> Seq Int
exponents Powers
lpowsP'
lexpntsQ :: Seq Int
lexpntsQ = Powers -> Seq Int
exponents Powers
lpowsQ'
gamma :: Seq Int
gamma = (Int -> Int -> Int) -> Seq Int -> Seq Int -> Seq Int
forall a b c. (a -> b -> c) -> Seq a -> Seq b -> Seq c
S.zipWith Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Seq Int
lexpntsP Seq Int
lexpntsQ
betaP :: Seq Int
betaP = (Int -> Int -> Int) -> Seq Int -> Seq Int -> Seq Int
forall a b c. (a -> b -> c) -> Seq a -> Seq b -> Seq c
S.zipWith (-) Seq Int
gamma Seq Int
lexpntsP
betaQ :: Seq Int
betaQ = (Int -> Int -> Int) -> Seq Int -> Seq Int -> Seq Int
forall a b c. (a -> b -> c) -> Seq a -> Seq b -> Seq c
S.zipWith (-) Seq Int
gamma Seq Int
lexpntsQ
n :: Int
n = Powers -> Int
nvariables Powers
lpowsP'
wp :: Spray a
wp = Monomial a -> Spray a
forall a. Monomial a -> Spray a
fromMonomial (Seq Int -> Int -> Powers
Powers Seq Int
betaP Int
n, a -> a
forall a. C a => a -> a
AlgField.recip a
lcoefP)
wq :: Spray a
wq = Monomial a -> Spray a
forall a. Monomial a -> Spray a
fromMonomial (Seq Int -> Int -> Powers
Powers Seq Int
betaQ Int
n, a -> a
forall a. C a => a -> a
AlgField.recip a
lcoefQ)
groebner00 :: forall a. (Eq a, AlgField.C a) => [Spray a] -> [Spray a]
groebner00 :: forall a. (Eq a, C a) => [Spray a] -> [Spray a]
groebner00 [Spray a]
sprays = Int
-> Int
-> HashMap Int (Int, Int)
-> HashMap Int (Spray a, Monomial a)
-> [Spray a]
go Int
0 Int
j0 HashMap Int (Int, Int)
combins0 HashMap Int (Spray a, Monomial a)
spraysMap
where
j0 :: Int
j0 = [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
sprays
combins0 :: HashMap Int (Int, Int)
combins0 = Int -> Int -> HashMap Int (Int, Int)
combn2 Int
j0 Int
0
ltsprays :: [Monomial a]
ltsprays = (Spray a -> Monomial a) -> [Spray a] -> [Monomial a]
forall a b. (a -> b) -> [a] -> [b]
map Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm [Spray a]
sprays
spraysltsprays :: [(Spray a, Monomial a)]
spraysltsprays = [Spray a] -> [Monomial a] -> [(Spray a, Monomial a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Spray a]
sprays [Monomial a]
ltsprays
spraysMap :: HashMap Int (Spray a, Monomial a)
spraysMap = [(Int, (Spray a, Monomial a))] -> HashMap Int (Spray a, Monomial a)
forall k v. (Eq k, Hashable k) => [(k, v)] -> HashMap k v
HM.fromList ([Int] -> [(Spray a, Monomial a)] -> [(Int, (Spray a, Monomial a))]
forall a b. [a] -> [b] -> [(a, b)]
zip [Int
0 .. Int
j0Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1] [(Spray a, Monomial a)]
spraysltsprays)
go :: Int -> Int -> HashMap Int (Int, Int)
-> HashMap Int (Spray a, Monomial a) -> [Spray a]
go :: Int
-> Int
-> HashMap Int (Int, Int)
-> HashMap Int (Spray a, Monomial a)
-> [Spray a]
go !Int
i !Int
j !HashMap Int (Int, Int)
combins !HashMap Int (Spray a, Monomial a)
gpolysMap
| Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== HashMap Int (Int, Int) -> Int
forall a. HashMap Int a -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length HashMap Int (Int, Int)
combins = ((Spray a, Monomial a) -> Spray a)
-> [(Spray a, Monomial a)] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map (Spray a, Monomial a) -> Spray a
forall a b. (a, b) -> a
fst (HashMap Int (Spray a, Monomial a) -> [(Spray a, Monomial a)]
forall k v. HashMap k v -> [v]
HM.elems HashMap Int (Spray a, Monomial a)
gpolysMap)
| Bool
otherwise = Int
-> Int
-> HashMap Int (Int, Int)
-> HashMap Int (Spray a, Monomial a)
-> [Spray a]
go Int
i' Int
j' HashMap Int (Int, Int)
combins' HashMap Int (Spray a, Monomial a)
gpolysMap'
where
(Int
k, Int
l) = HashMap Int (Int, Int)
combins HashMap Int (Int, Int) -> Int -> (Int, Int)
forall k v.
(Eq k, Hashable k, HasCallStack) =>
HashMap k v -> k -> v
HM.! Int
i
sfg :: Spray a
sfg = (Spray a, Monomial a) -> (Spray a, Monomial a) -> Spray a
forall a.
(Eq a, C a) =>
(Spray a, Monomial a) -> (Spray a, Monomial a) -> Spray a
sPolynomial (HashMap Int (Spray a, Monomial a)
gpolysMap HashMap Int (Spray a, Monomial a) -> Int -> (Spray a, Monomial a)
forall k v.
(Eq k, Hashable k, HasCallStack) =>
HashMap k v -> k -> v
HM.! Int
k) (HashMap Int (Spray a, Monomial a)
gpolysMap HashMap Int (Spray a, Monomial a) -> Int -> (Spray a, Monomial a)
forall k v.
(Eq k, Hashable k, HasCallStack) =>
HashMap k v -> k -> v
HM.! Int
l)
sbarfg :: Spray a
sbarfg = Spray a -> HashMap Int (Spray a, Monomial a) -> Spray a
forall a.
(Eq a, C a) =>
Spray a -> HashMap Int (Spray a, Monomial a) -> Spray a
sprayDivisionRemainder' Spray a
sfg HashMap Int (Spray a, Monomial a)
gpolysMap
ltsbarfg :: Monomial a
ltsbarfg = Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm Spray a
sbarfg
(Int
i', Int
j', HashMap Int (Spray a, Monomial a)
gpolysMap', HashMap Int (Int, Int)
combins') = if Spray a
sbarfg Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. C a => a
AlgAdd.zero
then
(Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1, Int
j, HashMap Int (Spray a, Monomial a)
gpolysMap, HashMap Int (Int, Int)
combins)
else
( Int
0
, Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1
, Int
-> (Spray a, Monomial a)
-> HashMap Int (Spray a, Monomial a)
-> HashMap Int (Spray a, Monomial a)
forall k v.
(Eq k, Hashable k) =>
k -> v -> HashMap k v -> HashMap k v
HM.insert Int
j (Spray a
sbarfg, Monomial a
ltsbarfg) HashMap Int (Spray a, Monomial a)
gpolysMap
, Int -> Int -> HashMap Int (Int, Int)
combn2 (Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) (Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
)
groebner0 :: forall a. (Eq a, AlgField.C a) => [Spray a] -> [Spray a]
groebner0 :: forall a. (Eq a, C a) => [Spray a] -> [Spray a]
groebner0 [Spray a]
sprays =
if Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
1 then [Spray a]
sprays else [[Spray a]
basis00 [Spray a] -> Int -> Spray a
forall a. HasCallStack => [a] -> Int -> a
!! Int
k | Int
k <- [Int
0 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1] [Int] -> [Int] -> [Int]
forall a. Eq a => [a] -> [a] -> [a]
\\ [Int]
discard]
where
n :: Int
n = [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
basis00
basis00 :: [Spray a]
basis00 = [Spray a] -> [Spray a]
forall a. (Eq a, C a) => [Spray a] -> [Spray a]
groebner00 [Spray a]
sprays
go :: Int -> [Int] -> [Int]
go :: Int -> [Int] -> [Int]
go !Int
i [Int]
toRemove
| Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
n = [Int]
toRemove
| Bool
otherwise = Int -> [Int] -> [Int]
go (Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) [Int]
toRemove'
where
ltf :: Monomial a
ltf = Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm ([Spray a]
basis00 [Spray a] -> Int -> Spray a
forall a. HasCallStack => [a] -> Int -> a
!! Int
i)
toDrop :: [Int]
toDrop = [Int]
toRemove [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
i]
igo :: Int -> Bool
igo :: Int -> Bool
igo !Int
j
| Int
j Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
n = Bool
False
| Int
j Int -> [Int] -> Bool
forall a. Eq a => a -> [a] -> Bool
forall (t :: * -> *) a. (Foldable t, Eq a) => a -> t a -> Bool
`elem` [Int]
toDrop = Int -> Bool
igo (Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
| Bool
otherwise = Bool
ok Bool -> Bool -> Bool
|| Int -> Bool
igo (Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1)
where
ok :: Bool
ok = Monomial a -> Monomial a -> Bool
forall a. Monomial a -> Monomial a -> Bool
divides (Spray a -> Monomial a
forall a. Spray a -> Monomial a
leadingTerm ([Spray a]
basis00 [Spray a] -> Int -> Spray a
forall a. HasCallStack => [a] -> Int -> a
!! Int
j)) Monomial a
ltf
toRemove' :: [Int]
toRemove' = if Int -> Bool
igo Int
0 then [Int]
toDrop else [Int]
toRemove
discard :: [Int]
discard = Int -> [Int] -> [Int]
go Int
0 []
reduceGroebnerBasis :: forall a. (Eq a, AlgField.C a) => [Spray a] -> [Spray a]
reduceGroebnerBasis :: forall a. (Eq a, C a) => [Spray a] -> [Spray a]
reduceGroebnerBasis [Spray a]
gbasis =
if [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
gbasis Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
2
then (Int -> Spray a) -> [Int] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map Int -> Spray a
reduction [Int
0 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1]
else [Spray a]
ngbasis
where
normalize :: Spray a -> Spray a
normalize :: Spray a -> Spray a
normalize Spray a
spray = a -> a
forall a. C a => a -> a
AlgField.recip a
coef a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ Spray a
spray
where
(Powers
_, a
coef) = Spray a -> (Powers, a)
forall a. Spray a -> Monomial a
leadingTerm Spray a
spray
ngbasis :: [Spray a]
ngbasis = (Spray a -> Spray a) -> [Spray a] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map Spray a -> Spray a
normalize [Spray a]
gbasis
n :: Int
n = [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
ngbasis
reduction :: Int -> Spray a
reduction :: Int -> Spray a
reduction Int
i = Spray a -> [Spray a] -> Spray a
forall a. (Eq a, C a) => Spray a -> [Spray a] -> Spray a
sprayDivisionRemainder ([Spray a]
ngbasis [Spray a] -> Int -> Spray a
forall a. HasCallStack => [a] -> Int -> a
!! Int
i) [Spray a]
rest
where
rest :: [Spray a]
rest = [[Spray a]
ngbasis [Spray a] -> Int -> Spray a
forall a. HasCallStack => [a] -> Int -> a
!! Int
k | Int
k <- [Int
0 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1] [Int] -> [Int] -> [Int]
forall a. Eq a => [a] -> [a] -> [a]
\\ [Int
i]]
groebner ::
forall a. (Eq a, AlgField.C a)
=> [Spray a]
-> Bool
-> [Spray a]
groebner :: forall a. (Eq a, C a) => [Spray a] -> Bool -> [Spray a]
groebner [Spray a]
sprays Bool
reduced =
if Bool
reduced then [Spray a] -> [Spray a]
forall a. (Eq a, C a) => [Spray a] -> [Spray a]
reduceGroebnerBasis [Spray a]
gbasis0 else (Spray a -> Spray a) -> [Spray a] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map Spray a -> Spray a
normalize [Spray a]
gbasis0
where
gbasis0 :: [Spray a]
gbasis0 = [Spray a] -> [Spray a]
forall a. (Eq a, C a) => [Spray a] -> [Spray a]
groebner0 [Spray a]
sprays
normalize :: Spray a -> Spray a
normalize :: Spray a -> Spray a
normalize Spray a
spray = a -> a
forall a. C a => a -> a
AlgField.recip a
coef a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ Spray a
spray
where
(Powers
_, a
coef) = Spray a -> (Powers, a)
forall a. Spray a -> Monomial a
leadingTerm Spray a
spray
combinationsOf :: Int -> [a] -> [[a]]
combinationsOf :: forall a. Int -> [a] -> [[a]]
combinationsOf Int
_ [] = String -> [[a]]
forall a. HasCallStack => String -> a
error String
"combinationsOf: should not happen."
combinationsOf Int
1 [a]
as = (a -> [a]) -> [a] -> [[a]]
forall a b. (a -> b) -> [a] -> [b]
map a -> [a]
forall a. a -> [a]
forall (f :: * -> *) a. Applicative f => a -> f a
pure [a]
as
combinationsOf Int
k as :: [a]
as@(a
_:[a]
xs) =
Int -> Int -> [a] -> [[a]] -> [[a]]
forall a. Int -> Int -> [a] -> [[a]] -> [[a]]
run (Int
lInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) (Int
kInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) [a]
as ([[a]] -> [[a]]) -> [[a]] -> [[a]]
forall a b. (a -> b) -> a -> b
$ Int -> [a] -> [[a]]
forall a. Int -> [a] -> [[a]]
combinationsOf (Int
kInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) [a]
xs
where
l :: Int
l = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
as
run :: Int -> Int -> [a] -> [[a]] -> [[a]]
run :: forall a. Int -> Int -> [a] -> [[a]] -> [[a]]
run Int
n Int
i [a]
ys [[a]]
cs
| Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
i = ([a] -> [a]) -> [[a]] -> [[a]]
forall a b. (a -> b) -> [a] -> [b]
map ([a]
ys [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++) [[a]]
cs
| Bool
otherwise = ([a] -> [a]) -> [[a]] -> [[a]]
forall a b. (a -> b) -> [a] -> [b]
map (a
qa -> [a] -> [a]
forall a. a -> [a] -> [a]
:) [[a]]
cs [[a]] -> [[a]] -> [[a]]
forall a. [a] -> [a] -> [a]
++ Int -> Int -> [a] -> [[a]] -> [[a]]
forall a. Int -> Int -> [a] -> [[a]] -> [[a]]
run (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) Int
i [a]
qs (Int -> [[a]] -> [[a]]
forall a. Int -> [a] -> [a]
drop Int
dc [[a]]
cs)
where
f :: [a] -> (a, [a])
f :: forall a. [a] -> (a, [a])
f [] = String -> (a, [a])
forall a. HasCallStack => String -> a
error String
"combinationsOf: should not happen."
f (a
b:[a]
bs) = (a
b, [a]
bs)
(a
q, [a]
qs) = [a] -> (a, [a])
forall a. [a] -> (a, [a])
f (Int -> [a] -> [a]
forall a. Int -> [a] -> [a]
take (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) [a]
ys)
dc :: Int
dc = [Int] -> Int
forall a. Num a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
product [(Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
kInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1) .. (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1)] Int -> Int -> Int
forall a. Integral a => a -> a -> a
`div` [Int] -> Int
forall a. Num a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Num a) => t a -> a
product [Int
1 .. Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1]
permutationsBinarySequence :: Int -> Int -> [Seq Int]
permutationsBinarySequence :: Int -> Int -> [Seq Int]
permutationsBinarySequence Int
nzeros Int
nones =
let n :: Int
n = Int
nzeros Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
nones in
([Int] -> Seq Int) -> [[Int]] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Int -> [Int] -> Seq Int
binarySequence Int
n) (Int -> [Int] -> [[Int]]
forall a. Int -> [a] -> [[a]]
combinationsOf Int
nones [Int
0 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1])
where
binarySequence :: Int -> [Int] -> Seq Int
binarySequence :: Int -> [Int] -> Seq Int
binarySequence Int
n [Int]
combo = Int -> (Int -> Int) -> Seq Int
forall a. Int -> (Int -> a) -> Seq a
fromFunction Int
n Int -> Int
f
where
f :: Int -> Int
f :: Int -> Int
f Int
i = Bool -> Int
forall a. Enum a => a -> Int
fromEnum (Int
i Int -> [Int] -> Bool
forall a. Eq a => a -> [a] -> Bool
forall (t :: * -> *) a. (Foldable t, Eq a) => a -> t a -> Bool
`elem` [Int]
combo)
esPolynomial ::
(AlgRing.C a, Eq a)
=> Int
-> Int
-> Spray a
esPolynomial :: forall a. (C a, Eq a) => Int -> Int -> Spray a
esPolynomial Int
n Int
k
| Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
0 Bool -> Bool -> Bool
|| Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
0
= String -> Spray a
forall a. HasCallStack => String -> a
error String
"esPolynomial: both arguments must be positive integers."
| Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
n = Spray a
forall a. C a => a
AlgAdd.zero
| Int
k Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = Spray a
forall a. C a => Spray a
unitSpray
| Bool
otherwise = Spray a -> Spray a
forall a. Spray a -> Spray a
simplifySpray Spray a
spray
where
perms :: [Seq Int]
perms = Int -> Int -> [Seq Int]
permutationsBinarySequence (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
k) Int
k
spray :: Spray a
spray = [(Powers, a)] -> Spray a
forall k v. (Eq k, Hashable k) => [(k, v)] -> HashMap k v
HM.fromList ([(Powers, a)] -> Spray a) -> [(Powers, a)] -> Spray a
forall a b. (a -> b) -> a -> b
$ (Seq Int -> (Powers, a)) -> [Seq Int] -> [(Powers, a)]
forall a b. (a -> b) -> [a] -> [b]
map (\Seq Int
expts -> (Seq Int -> Int -> Powers
Powers Seq Int
expts Int
n, a
forall a. C a => a
AlgRing.one)) [Seq Int]
perms
psPolynomial ::
forall a. (AlgRing.C a, Eq a)
=> Int
-> Int
-> Spray a
psPolynomial :: forall a. (C a, Eq a) => Int -> Int -> Spray a
psPolynomial Int
n Int
k
| Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
0 Bool -> Bool -> Bool
|| Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
0
= String -> Spray a
forall a. HasCallStack => String -> a
error String
"psPolynomial: both arguments must be positive integers."
| Int
k Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
n = Spray a
forall a. C a => a
AlgAdd.zero
| Int
k Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = Int
n Int -> Spray a -> Spray a
forall a. (C a, Eq a) => Int -> a -> a
.^ Spray a
forall a. C a => Spray a
unitSpray
| Bool
otherwise = Spray a
spray
where
spray :: Spray a
spray = [(Powers, a)] -> Spray a
forall k v. (Eq k, Hashable k) => [(k, v)] -> HashMap k v
HM.fromList ([(Powers, a)] -> Spray a) -> [(Powers, a)] -> Spray a
forall a b. (a -> b) -> a -> b
$ (Int -> (Powers, a)) -> [Int] -> [(Powers, a)]
forall a b. (a -> b) -> [a] -> [b]
map Int -> (Powers, a)
f [Int
1 .. Int
n]
f :: Int -> (Powers, a)
f :: Int -> (Powers, a)
f Int
j = (Seq Int -> Int -> Powers
Powers Seq Int
expts Int
j, a
forall a. C a => a
AlgRing.one)
where
expts :: Seq Int
expts = Int -> Int -> Seq Int
forall a. Int -> a -> Seq a
S.replicate (Int
jInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) Int
0 Seq Int -> Int -> Seq Int
forall a. Seq a -> a -> Seq a
|> Int
k
isSymmetricSpray :: forall a. (AlgField.C a, Eq a) => Spray a -> Bool
isSymmetricSpray :: forall a. (C a, Eq a) => Spray a -> Bool
isSymmetricSpray Spray a
spray = Bool
check1 Bool -> Bool -> Bool
&& Bool
check2
where
n :: Int
n = Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray
indices :: [Int]
indices = [Int
1 .. Int
n]
gPolys :: [Spray a]
gPolys = (Int -> Spray a) -> [Int] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> Int -> Int -> Spray a
forall a. (C a, Eq a) => Int -> Int -> Spray a
esPolynomial Int
n Int
i Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Int -> Spray a
forall a. C a => Int -> Spray a
lone (Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
i)) [Int]
indices
gbasis :: [Spray a]
gbasis = [Spray a] -> [Spray a]
forall a. (Eq a, C a) => [Spray a] -> [Spray a]
groebner0 [Spray a]
gPolys
spray' :: Spray a
spray' = Spray a
spray Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray (Spray a -> a
forall a. C a => Spray a -> a
getConstantTerm Spray a
spray)
g :: Spray a
g = Spray a -> [Spray a] -> Spray a
forall a. (Eq a, C a) => Spray a -> [Spray a] -> Spray a
sprayDivisionRemainder Spray a
spray' [Spray a]
gbasis
gpowers :: [Powers]
gpowers = Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
g
check1 :: Bool
check1 = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
minimum ((Powers -> Int) -> [Powers] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Int
nvariables [Powers]
gpowers) Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
n
expnts :: [Seq Int]
expnts = (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
gpowers
check2 :: Bool
check2 = (Seq Int -> Bool) -> [Seq Int] -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
DF.all ((Int -> Bool) -> Seq Int -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
DF.all (Int
0 Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
==)) ((Seq Int -> Seq Int) -> [Seq Int] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Int -> Seq Int -> Seq Int
forall a. Int -> Seq a -> Seq a
S.take Int
n) [Seq Int]
expnts)
isPolynomialOf :: forall a. (AlgField.C a, Eq a)
=> Spray a -> [Spray a] -> (Bool, Maybe (Spray a))
isPolynomialOf :: forall a.
(C a, Eq a) =>
Spray a -> [Spray a] -> (Bool, Maybe (Spray a))
isPolynomialOf Spray a
spray [Spray a]
sprays = (Bool, Maybe (Spray a))
result
where
nov :: Int
nov = Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray
n :: Int
n = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum ([Int] -> Int) -> [Int] -> Int
forall a b. (a -> b) -> a -> b
$ (Spray a -> Int) -> [Spray a] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables [Spray a]
sprays
result :: (Bool, Maybe (Spray a))
result
| Int
nov Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
n = (Bool
False, Maybe (Spray a)
forall a. Maybe a
Nothing)
| Bool
otherwise = (Bool
checks, Maybe (Spray a)
poly)
where
m :: Int
m = [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
sprays
yPolys :: [Spray a]
yPolys = (Int -> Spray a) -> [Int] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> Int -> Spray a
forall a. C a => Int -> Spray a
lone (Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
i) :: Spray a) [Int
1 .. Int
m]
gPolys :: [Spray a]
gPolys = (Spray a -> Spray a -> Spray a)
-> [Spray a] -> [Spray a] -> [Spray a]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^-^) [Spray a]
sprays [Spray a]
yPolys
gbasis0 :: [Spray a]
gbasis0 = [Spray a] -> [Spray a]
forall a. (Eq a, C a) => [Spray a] -> [Spray a]
groebner0 [Spray a]
gPolys
constantTerm :: Spray a
constantTerm = a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray (Spray a -> a
forall a. C a => Spray a -> a
getConstantTerm Spray a
spray)
spray' :: Spray a
spray' = Spray a
spray Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
constantTerm
g :: Spray a
g = Spray a -> [Spray a] -> Spray a
forall a. (Eq a, C a) => Spray a -> [Spray a] -> Spray a
sprayDivisionRemainder Spray a
spray' [Spray a]
gbasis0
gpowers :: [Powers]
gpowers = Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
g
check1 :: Bool
check1 = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
minimum ((Powers -> Int) -> [Powers] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Int
nvariables [Powers]
gpowers) Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
n
expnts :: [Seq Int]
expnts = (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
gpowers
check2 :: Bool
check2 = (Seq Int -> Bool) -> [Seq Int] -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
DF.all ((Int -> Bool) -> Seq Int -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
DF.all (Int
0 Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
==)) ((Seq Int -> Seq Int) -> [Seq Int] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Int -> Seq Int -> Seq Int
forall a. Int -> Seq a -> Seq a
S.take Int
n) [Seq Int]
expnts)
checks :: Bool
checks = Bool
check1 Bool -> Bool -> Bool
&& Bool
check2
poly :: Maybe (Spray a)
poly = if Bool
checks
then Spray a -> Maybe (Spray a)
forall a. a -> Maybe a
Just (Spray a -> Maybe (Spray a)) -> Spray a -> Maybe (Spray a)
forall a b. (a -> b) -> a -> b
$ Spray a -> Spray a
dropXis Spray a
g Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray a
constantTerm
else Maybe (Spray a)
forall a. Maybe a
Nothing
dropXis :: Spray a -> Spray a
dropXis = (Powers -> Powers) -> Spray a -> Spray a
forall k2 k1 v.
(Eq k2, Hashable k2) =>
(k1 -> k2) -> HashMap k1 v -> HashMap k2 v
HM.mapKeys Powers -> Powers
f
f :: Powers -> Powers
f (Powers Seq Int
expnnts Int
_) = Seq Int -> Int -> Powers
Powers (Int -> Seq Int -> Seq Int
forall a. Int -> Seq a -> Seq a
S.drop Int
n Seq Int
expnnts) Int
n
sylvesterMatrix :: AlgAdd.C a => [a] -> [a] -> Matrix a
sylvesterMatrix :: forall a. C a => [a] -> [a] -> Matrix a
sylvesterMatrix [a]
x [a]
y = [[a]] -> Matrix a
forall a. [[a]] -> Matrix a
fromLists ([[a]]
xrows [[a]] -> [[a]] -> [[a]]
forall a. [a] -> [a] -> [a]
++ [[a]]
yrows)
where
m :: Int
m = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
x Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1
n :: Int
n = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
y Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1
xrows :: [[a]]
xrows = [Int -> a -> [a]
forall a. Int -> a -> [a]
replicate Int
i a
forall a. C a => a
AlgAdd.zero [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ [a]
x [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ Int -> a -> [a]
forall a. Int -> a -> [a]
replicate (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) a
forall a. C a => a
AlgAdd.zero
| Int
i <- [Int
0 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1]]
yrows :: [[a]]
yrows = [Int -> a -> [a]
forall a. Int -> a -> [a]
replicate Int
i a
forall a. C a => a
AlgAdd.zero [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ [a]
y [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ Int -> a -> [a]
forall a. Int -> a -> [a]
replicate (Int
mInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) a
forall a. C a => a
AlgAdd.zero
| Int
i <- [Int
0 .. Int
mInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1]]
sylvesterMatrix' :: AlgRing.C a => [a] -> [a] -> Int -> Matrix a
sylvesterMatrix' :: forall a. C a => [a] -> [a] -> Int -> Matrix a
sylvesterMatrix' [a]
x [a]
y Int
k = if Int
s Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0
then [[a]] -> Matrix a
forall a. [[a]] -> Matrix a
fromLists [[a
forall a. C a => a
AlgRing.one]]
else Int -> Int -> Int -> Int -> Matrix a -> Matrix a
forall a. Int -> Int -> Int -> Int -> Matrix a -> Matrix a
submatrix Int
1 Int
s Int
1 Int
s (Matrix a -> Matrix a) -> Matrix a -> Matrix a
forall a b. (a -> b) -> a -> b
$ [[a]] -> Matrix a
forall a. [[a]] -> Matrix a
fromLists ([[a]]
xrows [[a]] -> [[a]] -> [[a]]
forall a. [a] -> [a] -> [a]
++ [[a]]
yrows)
where
m :: Int
m = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
x Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1
n :: Int
n = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
y Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1
s :: Int
s = Int
m Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
2Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
k
xrows :: [[a]]
xrows = [Int -> a -> [a]
forall a. Int -> a -> [a]
replicate Int
i a
forall a. C a => a
AlgAdd.zero [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ [a]
x [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ Int -> a -> [a]
forall a. Int -> a -> [a]
replicate (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) a
forall a. C a => a
AlgAdd.zero
| Int
i <- [Int
0 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
k]]
yrows :: [[a]]
yrows = [Int -> a -> [a]
forall a. Int -> a -> [a]
replicate Int
i a
forall a. C a => a
AlgAdd.zero [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ [a]
y [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ Int -> a -> [a]
forall a. Int -> a -> [a]
replicate (Int
mInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
iInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) a
forall a. C a => a
AlgAdd.zero
| Int
i <- [Int
0 .. Int
mInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
k]]
sprayCoefficients :: (Eq a, AlgRing.C a) => Spray a -> [Spray a]
sprayCoefficients :: forall a. (Eq a, C a) => Spray a -> [Spray a]
sprayCoefficients Spray a
spray =
if Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0
then [Spray a
constantTerm]
else [Spray a] -> [Spray a]
forall a. [a] -> [a]
reverse [Spray a]
sprays
where
n :: Int
n = Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray
([Powers]
powers, [a]
coeffs) = [(Powers, a)] -> ([Powers], [a])
forall a b. [(a, b)] -> ([a], [b])
unzip (Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray)
expnts :: [Seq Int]
expnts = (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
powers
constantTerm :: Spray a
constantTerm =
a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray (a -> Spray a) -> a -> Spray a
forall a b. (a -> b) -> a -> b
$ a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0) Spray a
spray)
([Seq Int]
expnts', [a]
coeffs') =
[(Seq Int, a)] -> ([Seq Int], [a])
forall a b. [(a, b)] -> ([a], [b])
unzip ([(Seq Int, a)] -> ([Seq Int], [a]))
-> [(Seq Int, a)] -> ([Seq Int], [a])
forall a b. (a -> b) -> a -> b
$ ((Seq Int, a) -> Bool) -> [(Seq Int, a)] -> [(Seq Int, a)]
forall a. (a -> Bool) -> [a] -> [a]
filter (\(Seq Int
s,a
_) -> Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
s Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0) ([Seq Int] -> [a] -> [(Seq Int, a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Seq Int]
expnts [a]
coeffs)
xpows :: [Int]
xpows = (Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) [Seq Int]
expnts'
expnts'' :: [Seq Int]
expnts'' = (Seq Int -> Seq Int) -> [Seq Int] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Int -> Seq Int -> Seq Int
forall a. Int -> Seq a -> Seq a
S.deleteAt Int
0) [Seq Int]
expnts'
powers'' :: [Powers]
powers'' = (Seq Int -> Powers) -> [Seq Int] -> [Powers]
forall a b. (a -> b) -> [a] -> [b]
map (\Seq Int
s -> Seq Int -> Int -> Powers
Powers Seq Int
s (Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
s)) [Seq Int]
expnts''
sprays'' :: [Spray a]
sprays'' = (Powers -> a -> Spray a) -> [Powers] -> [a] -> [Spray a]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith (((Powers, a) -> Spray a) -> Powers -> a -> Spray a
forall a b c. ((a, b) -> c) -> a -> b -> c
curry (Powers, a) -> Spray a
forall a. Monomial a -> Spray a
fromMonomial) [Powers]
powers'' [a]
coeffs'
imap :: IntMap (Spray a)
imap = (Spray a -> Spray a -> Spray a)
-> [(Int, Spray a)] -> IntMap (Spray a)
forall a. (a -> a -> a) -> [(Int, a)] -> IntMap a
IM.fromListWith Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^+^) ([Int] -> [Spray a] -> [(Int, Spray a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Int]
xpows [Spray a]
sprays'')
imap' :: IntMap (Spray a)
imap' = (Spray a -> Spray a -> Spray a)
-> Int -> Spray a -> IntMap (Spray a) -> IntMap (Spray a)
forall a. (a -> a -> a) -> Int -> a -> IntMap a -> IntMap a
IM.insertWith Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^+^) Int
0 Spray a
constantTerm IntMap (Spray a)
imap
permutation :: [Int]
permutation = [Int
2 .. Int
n] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
1]
sprays :: [Spray a]
sprays = [
[Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation (Spray a -> Maybe (Spray a) -> Spray a
forall a. a -> Maybe a -> a
fromMaybe Spray a
forall a. C a => a
AlgAdd.zero (Int -> IntMap (Spray a) -> Maybe (Spray a)
forall a. Int -> IntMap a -> Maybe a
IM.lookup Int
i IntMap (Spray a)
imap'))
| Int
i <- [Int
0 .. [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
xpows]
]
resultant1 :: (Eq a, AlgRing.C a) => Spray a -> Spray a -> a
resultant1 :: forall a. (Eq a, C a) => Spray a -> Spray a -> a
resultant1 Spray a
p Spray a
q =
if Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
1
then Matrix a -> a
forall a. (Eq a, C a) => Matrix a -> a
detLaplace (Matrix a -> a) -> Matrix a -> a
forall a b. (a -> b) -> a -> b
$ [a] -> [a] -> Matrix a
forall a. C a => [a] -> [a] -> Matrix a
sylvesterMatrix [a]
pcoeffs [a]
qcoeffs
else String -> a
forall a. HasCallStack => String -> a
error String
"resultant1: the two sprays must be univariate."
where
n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
p) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
q)
pexpnts :: [Int]
pexpnts =
(Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) ([Seq Int] -> [Int]) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> a -> b
$ (Seq Int -> Bool) -> [Seq Int] -> [Seq Int]
forall a. (a -> Bool) -> [a] -> [a]
filter (Bool -> Bool
not (Bool -> Bool) -> (Seq Int -> Bool) -> Seq Int -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Seq Int -> Bool
forall a. Seq a -> Bool
S.null) ((Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents (Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
p))
qexpnts :: [Int]
qexpnts =
(Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) ([Seq Int] -> [Int]) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> a -> b
$ (Seq Int -> Bool) -> [Seq Int] -> [Seq Int]
forall a. (a -> Bool) -> [a] -> [a]
filter (Bool -> Bool
not (Bool -> Bool) -> (Seq Int -> Bool) -> Seq Int -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Seq Int -> Bool
forall a. Seq a -> Bool
S.null) ((Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents (Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
q))
p0 :: a
p0 = a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0) Spray a
p)
q0 :: a
q0 = a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0) Spray a
q)
pcoeffs :: [a]
pcoeffs = if [Int] -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [Int]
pexpnts
then [a
p0]
else [a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers (Int -> Seq Int
forall a. a -> Seq a
S.singleton Int
i) Int
1) Spray a
p)
| Int
i <- [Int
maxp, Int
maxpInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1 .. Int
1]] [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ [a
p0]
where
maxp :: Int
maxp = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
pexpnts
qcoeffs :: [a]
qcoeffs = if [Int] -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [Int]
qexpnts
then [a
q0]
else [a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers (Int -> Seq Int
forall a. a -> Seq a
S.singleton Int
i) Int
1) Spray a
q)
| Int
i <- [Int
maxq, Int
maxqInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1 .. Int
1]] [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ [a
q0]
where
maxq :: Int
maxq = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
qexpnts
subresultants1 :: (Eq a, AlgRing.C a) => Spray a -> Spray a -> [a]
subresultants1 :: forall a. (Eq a, C a) => Spray a -> Spray a -> [a]
subresultants1 Spray a
p Spray a
q = if Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
1
then (Int -> a) -> [Int] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map (Matrix a -> a
forall a. (Eq a, C a) => Matrix a -> a
detLaplace (Matrix a -> a) -> (Int -> Matrix a) -> Int -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [a] -> [a] -> Int -> Matrix a
forall a. C a => [a] -> [a] -> Int -> Matrix a
sylvesterMatrix' [a]
pcoeffs [a]
qcoeffs) [Int
0 .. Int -> Int -> Int
forall a. Ord a => a -> a -> a
min Int
d Int
e Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1]
else String -> [a]
forall a. HasCallStack => String -> a
error String
"subresultants1: the two sprays must be univariate."
where
n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
p) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
q)
pexpnts :: [Int]
pexpnts =
(Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) ([Seq Int] -> [Int]) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> a -> b
$ (Seq Int -> Bool) -> [Seq Int] -> [Seq Int]
forall a. (a -> Bool) -> [a] -> [a]
filter (Bool -> Bool
not (Bool -> Bool) -> (Seq Int -> Bool) -> Seq Int -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Seq Int -> Bool
forall a. Seq a -> Bool
S.null) ((Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents (Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
p))
qexpnts :: [Int]
qexpnts =
(Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) ([Seq Int] -> [Int]) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> a -> b
$ (Seq Int -> Bool) -> [Seq Int] -> [Seq Int]
forall a. (a -> Bool) -> [a] -> [a]
filter (Bool -> Bool
not (Bool -> Bool) -> (Seq Int -> Bool) -> Seq Int -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Seq Int -> Bool
forall a. Seq a -> Bool
S.null) ((Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents (Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
q))
p0 :: a
p0 = a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0) Spray a
p)
q0 :: a
q0 = a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0) Spray a
q)
pcoeffs :: [a]
pcoeffs = if [Int] -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [Int]
pexpnts
then [a
p0]
else [a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers (Int -> Seq Int
forall a. a -> Seq a
S.singleton Int
i) Int
1) Spray a
p)
| Int
i <- [Int
maxp, Int
maxpInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1 .. Int
1]] [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ [a
p0]
where
maxp :: Int
maxp = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
pexpnts
qcoeffs :: [a]
qcoeffs = if [Int] -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [Int]
qexpnts
then [a
q0]
else [a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers (Int -> Seq Int
forall a. a -> Seq a
S.singleton Int
i) Int
1) Spray a
q)
| Int
i <- [Int
maxq, Int
maxqInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1 .. Int
1]] [a] -> [a] -> [a]
forall a. [a] -> [a] -> [a]
++ [a
q0]
where
maxq :: Int
maxq = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
qexpnts
d :: Int
d = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
pcoeffs
e :: Int
e = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
qcoeffs
resultant :: (Eq a, AlgRing.C a)
=> Int
-> Spray a
-> Spray a
-> Spray a
resultant :: forall a. (Eq a, C a) => Int -> Spray a -> Spray a -> Spray a
resultant Int
var Spray a
p Spray a
q =
if Int
var Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
1 Bool -> Bool -> Bool
&& Int
var Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
n
then [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation' Spray a
det
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"resultant: invalid variable index."
where
n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
p) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
q)
permutation :: [Int]
permutation = [Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
varInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
2 .. Int
n] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
1 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
varInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1]
permutation' :: [Int]
permutation' = [Int
var .. Int
n] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
1 .. Int
varInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1]
p' :: Spray a
p' = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
p
q' :: Spray a
q' = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
q
det :: Spray a
det = Matrix (Spray a) -> Spray a
forall a. (Eq a, C a) => Matrix a -> a
detLaplace (Matrix (Spray a) -> Spray a) -> Matrix (Spray a) -> Spray a
forall a b. (a -> b) -> a -> b
$
[Spray a] -> [Spray a] -> Matrix (Spray a)
forall a. C a => [a] -> [a] -> Matrix a
sylvesterMatrix (Spray a -> [Spray a]
forall a. (Eq a, C a) => Spray a -> [Spray a]
sprayCoefficients Spray a
p') (Spray a -> [Spray a]
forall a. (Eq a, C a) => Spray a -> [Spray a]
sprayCoefficients Spray a
q')
subresultants :: (Eq a, AlgRing.C a)
=> Int
-> Spray a
-> Spray a
-> [Spray a]
subresultants :: forall a. (Eq a, C a) => Int -> Spray a -> Spray a -> [Spray a]
subresultants Int
var Spray a
p Spray a
q
| Int
var Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
1 = String -> [Spray a]
forall a. HasCallStack => String -> a
error String
"subresultants: invalid variable index."
| Int
var Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
n = String -> [Spray a]
forall a. HasCallStack => String -> a
error String
"subresultants: too large variable index."
| Bool
otherwise = (Int -> Spray a) -> [Int] -> [Spray a]
forall a b. (a -> b) -> [a] -> [b]
map (Spray a -> Spray a
permute' (Spray a -> Spray a) -> (Int -> Spray a) -> Int -> Spray a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Matrix (Spray a) -> Spray a
forall a. (Eq a, C a) => Matrix a -> a
detLaplace (Matrix (Spray a) -> Spray a)
-> (Int -> Matrix (Spray a)) -> Int -> Spray a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [Spray a] -> [Spray a] -> Int -> Matrix (Spray a)
forall a. C a => [a] -> [a] -> Int -> Matrix a
sylvesterMatrix' [Spray a]
pcoeffs [Spray a]
qcoeffs)
[Int
0 .. Int -> Int -> Int
forall a. Ord a => a -> a -> a
min Int
d Int
e Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1]
where
pcoeffs :: [Spray a]
pcoeffs = Spray a -> [Spray a]
forall a. (Eq a, C a) => Spray a -> [Spray a]
sprayCoefficients Spray a
p'
qcoeffs :: [Spray a]
qcoeffs = Spray a -> [Spray a]
forall a. (Eq a, C a) => Spray a -> [Spray a]
sprayCoefficients Spray a
q'
d :: Int
d = [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
pcoeffs
e :: Int
e = [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
qcoeffs
n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
p) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
q)
permutation :: [Int]
permutation = Int
var Int -> [Int] -> [Int]
forall a. a -> [a] -> [a]
: [Int
1 .. Int
varInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
varInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1 .. Int
n]
permute :: Spray a -> Spray a
permute = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation
p' :: Spray a
p' = Spray a -> Spray a
permute Spray a
p
q' :: Spray a
q' = Spray a -> Spray a
permute Spray a
q
permutation' :: [Int]
permutation' = [Int
2 .. Int
var] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ (Int
1 Int -> [Int] -> [Int]
forall a. a -> [a] -> [a]
: [Int
varInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1 .. Int
n])
permute' :: Spray a -> Spray a
permute' = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation'
resultant' :: forall a. (Eq a, AlgField.C a)
=> Int
-> Spray a
-> Spray a
-> Spray a
resultant' :: forall a. (Eq a, C a) => Int -> Spray a -> Spray a -> Spray a
resultant' Int
var Spray a
sprayA Spray a
sprayB
| Int
var Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
1 Bool -> Bool -> Bool
|| Int
var Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
n
= String -> Spray a
forall a. HasCallStack => String -> a
error String
"resultant': invalid variable index."
| Spray a
sprayA Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray Bool -> Bool -> Bool
|| Spray a
sprayB Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray
= Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray
| Bool
otherwise
= [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation' (Spray a -> Spray a) -> Spray a -> Spray a
forall a b. (a -> b) -> a -> b
$ Spray a -> Spray a -> Spray a -> Spray a -> Spray a -> Spray a
go Spray a
forall a. C a => Spray a
unitSpray Spray a
forall a. C a => Spray a
unitSpray Spray a
s0 Spray a
p0 Spray a
q0
where
n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
sprayA) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
sprayB)
permutation :: [Int]
permutation = [Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
varInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1 .. Int
n] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
1 .. Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
var]
permutation' :: [Int]
permutation' = [Int
varInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1 .. Int
n] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
1 .. Int
var]
sprayA' :: Spray a
sprayA' = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
sprayA
sprayB' :: Spray a
sprayB' = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
sprayB
degA :: Int
degA = Int -> Spray a -> Int
forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n Spray a
sprayA'
degB :: Int
degB = Int -> Spray a -> Int
forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n Spray a
sprayB'
content :: Spray a -> Spray a
content :: Spray a -> Spray a
content Spray a
spray = (Spray a -> Spray a -> Spray a) -> [Spray a] -> Spray a
forall a. HasCallStack => (a -> a -> a) -> [a] -> a
foldl1' Spray a -> Spray a -> Spray a
forall a. (Eq a, C a) => Spray a -> Spray a -> Spray a
gcdSpray (Int -> Spray a -> [Spray a]
forall a. (Eq a, C a) => Int -> Spray a -> [Spray a]
sprayCoefficients' Int
n Spray a
spray)
exactDivisionBy :: Spray a -> Spray a -> Spray a
exactDivisionBy :: Spray a -> Spray a -> Spray a
exactDivisionBy Spray a
b Spray a
a =
if (Spray a, Spray a) -> Spray a
forall a b. (a, b) -> b
snd (Spray a, Spray a)
division Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray
then (Spray a, Spray a) -> Spray a
forall a b. (a, b) -> a
fst (Spray a, Spray a)
division
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"exactDivisionBy: should not happen."
where
division :: (Spray a, Spray a)
division = Spray a -> Spray a -> (Spray a, Spray a)
forall a. (Eq a, C a) => Spray a -> Spray a -> (Spray a, Spray a)
sprayDivision Spray a
a Spray a
b
contA :: Spray a
contA = Spray a -> Spray a
content Spray a
sprayA'
contB :: Spray a
contB = Spray a -> Spray a
content Spray a
sprayB'
sprayA'' :: Spray a
sprayA'' = Spray a -> Spray a -> Spray a
exactDivisionBy Spray a
contA Spray a
sprayA'
sprayB'' :: Spray a
sprayB'' = Spray a -> Spray a -> Spray a
exactDivisionBy Spray a
contB Spray a
sprayB'
t :: Spray a
t = Spray a
contASpray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
degB Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
contBSpray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
degA
s0 :: Spray a
s0 = if Int
degA Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
degB Bool -> Bool -> Bool
&& Int -> Bool
forall a. Integral a => a -> Bool
odd Int
degA Bool -> Bool -> Bool
&& Int -> Bool
forall a. Integral a => a -> Bool
odd Int
degB
then Spray a -> Spray a
forall a. C a => a -> a
AlgAdd.negate Spray a
forall a. C a => Spray a
unitSpray :: Spray a
else Spray a
forall a. C a => Spray a
unitSpray
(Spray a
p0, Spray a
q0) = if Int
degA Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Int
degB
then (Spray a
sprayA'', Spray a
sprayB'')
else (Spray a
sprayB'', Spray a
sprayA'')
go :: Spray a -> Spray a -> Spray a -> Spray a -> Spray a -> Spray a
go :: Spray a -> Spray a -> Spray a -> Spray a -> Spray a -> Spray a
go Spray a
g Spray a
h Spray a
s Spray a
p Spray a
q =
if Int
degq' Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0
then Spray a
s' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
t Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
h''
else Spray a -> Spray a -> Spray a -> Spray a -> Spray a -> Spray a
go Spray a
g' Spray a
h' Spray a
s' Spray a
p' Spray a
q'
where
degp :: Int
degp = Int -> Spray a -> Int
forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n Spray a
p
degq :: Int
degq = Int -> Spray a -> Int
forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n Spray a
q
delta :: Int
delta = Int
degp Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
degq
s' :: Spray a
s' = if Int -> Bool
forall a. Integral a => a -> Bool
odd Int
degp Bool -> Bool -> Bool
&& Int -> Bool
forall a. Integral a => a -> Bool
odd Int
degq
then Spray a -> Spray a
forall a. C a => a -> a
AlgAdd.negate Spray a
s
else Spray a
s
(Spray a
_, (Spray a
_, Spray a
r)) = Int -> Spray a -> Spray a -> (Spray a, (Spray a, Spray a))
forall a.
(Eq a, C a) =>
Int -> Spray a -> Spray a -> (Spray a, (Spray a, Spray a))
pseudoDivision Int
n Spray a
p Spray a
q
p' :: Spray a
p' = Spray a
q
q' :: Spray a
q' = Spray a -> Spray a -> Spray a
exactDivisionBy (Spray a
g Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
hSpray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
delta) Spray a
r
(Int
degp', Spray a
ellp') = Int -> Spray a -> (Int, Spray a)
forall a. (Eq a, C a) => Int -> Spray a -> (Int, Spray a)
degreeAndLeadingCoefficient Int
n Spray a
p'
(Int
degq', Spray a
ellq') = Int -> Spray a -> (Int, Spray a)
forall a. (Eq a, C a) => Int -> Spray a -> (Int, Spray a)
degreeAndLeadingCoefficient Int
n Spray a
q'
g' :: Spray a
g' = Spray a
ellp'
h' :: Spray a
h' = Spray a -> Spray a -> Spray a
exactDivisionBy (Spray a
hSpray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
delta) (Spray a
h Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
g'Spray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
delta)
h'' :: Spray a
h'' = Spray a -> Spray a -> Spray a
exactDivisionBy (Spray a
h'Spray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
degp') (Spray a
h' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
ellq'Spray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
degp')
sprayCoefficients' :: (Eq a, AlgRing.C a) => Int -> Spray a -> [Spray a]
sprayCoefficients' :: forall a. (Eq a, C a) => Int -> Spray a -> [Spray a]
sprayCoefficients' Int
n Spray a
spray
| Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
/= Int
n = [Spray a
spray]
| Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = [a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray a
constantTerm]
| Bool
otherwise = [Spray a]
sprays
where
permutation :: [Int]
permutation = [Int
2 .. Int
n] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
1]
spray' :: Spray a
spray' = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
spray
([Powers]
powers, [a]
coeffs) = [(Powers, a)] -> ([Powers], [a])
forall a b. [(a, b)] -> ([a], [b])
unzip (Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray')
expnts :: [Seq Int]
expnts = (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
powers
constantTerm :: a
constantTerm = a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0) Spray a
spray')
([Seq Int]
expnts', [a]
coeffs') =
[(Seq Int, a)] -> ([Seq Int], [a])
forall a b. [(a, b)] -> ([a], [b])
unzip ([(Seq Int, a)] -> ([Seq Int], [a]))
-> [(Seq Int, a)] -> ([Seq Int], [a])
forall a b. (a -> b) -> a -> b
$ ((Seq Int, a) -> Bool) -> [(Seq Int, a)] -> [(Seq Int, a)]
forall a. (a -> Bool) -> [a] -> [a]
filter (\(Seq Int
s,a
_) -> (Bool -> Bool
not (Bool -> Bool) -> (Seq Int -> Bool) -> Seq Int -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Seq Int -> Bool
forall a. Seq a -> Bool
S.null) Seq Int
s) ([Seq Int] -> [a] -> [(Seq Int, a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Seq Int]
expnts [a]
coeffs)
xpows :: [Int]
xpows = (Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) [Seq Int]
expnts'
expnts'' :: [Seq Int]
expnts'' = (Seq Int -> Seq Int) -> [Seq Int] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map (Int -> Seq Int -> Seq Int
forall a. Int -> Seq a -> Seq a
S.deleteAt Int
0) [Seq Int]
expnts'
powers'' :: [Powers]
powers'' = (Seq Int -> Powers) -> [Seq Int] -> [Powers]
forall a b. (a -> b) -> [a] -> [b]
map (\Seq Int
s -> Seq Int -> Int -> Powers
Powers Seq Int
s (Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
s)) [Seq Int]
expnts''
sprays'' :: [Spray a]
sprays'' = (Powers -> a -> Spray a) -> [Powers] -> [a] -> [Spray a]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith (((Powers, a) -> Spray a) -> Powers -> a -> Spray a
forall a b c. ((a, b) -> c) -> a -> b -> c
curry (Powers, a) -> Spray a
forall a. Monomial a -> Spray a
fromMonomial) [Powers]
powers'' [a]
coeffs'
imap :: IntMap (Spray a)
imap = (Spray a -> Spray a -> Spray a)
-> [(Int, Spray a)] -> IntMap (Spray a)
forall a. (a -> a -> a) -> [(Int, a)] -> IntMap a
IM.fromListWith Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^+^) ([Int] -> [Spray a] -> [(Int, Spray a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Int]
xpows [Spray a]
sprays'')
imap' :: IntMap (Spray a)
imap' = (Spray a -> Spray a -> Spray a)
-> Int -> Spray a -> IntMap (Spray a) -> IntMap (Spray a)
forall a. (a -> a -> a) -> Int -> a -> IntMap a -> IntMap a
IM.insertWith Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^+^) Int
0 (a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray a
constantTerm) IntMap (Spray a)
imap
deg :: Int
deg = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
xpows
sprays :: [Spray a]
sprays = [
Spray a -> Maybe (Spray a) -> Spray a
forall a. a -> Maybe a -> a
fromMaybe Spray a
forall a. C a => a
AlgAdd.zero (Int -> IntMap (Spray a) -> Maybe (Spray a)
forall a. Int -> IntMap a -> Maybe a
IM.lookup Int
i IntMap (Spray a)
imap')
| Int
i <- [Int
deg, Int
degInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1 .. Int
0]
]
degree :: (Eq a, AlgRing.C a) => Int -> Spray a -> Int
degree :: forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n Spray a
spray
| Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 =
if Spray a
spray Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray
then Int
forall a. Bounded a => a
minBound
else Int
0
| Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
/= Int
n = Int
0
| Bool
otherwise = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
xpows
where
permutation :: [Int]
permutation = [Int
2 .. Int
n] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
1]
spray' :: Spray a
spray' = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
spray
expnts :: [Seq Int]
expnts = (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents ([Powers] -> [Seq Int]) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> a -> b
$ Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
spray'
expnts' :: [Seq Int]
expnts' = (Seq Int -> Bool) -> [Seq Int] -> [Seq Int]
forall a. (a -> Bool) -> [a] -> [a]
filter (Bool -> Bool
not (Bool -> Bool) -> (Seq Int -> Bool) -> Seq Int -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Seq Int -> Bool
forall a. Seq a -> Bool
S.null) [Seq Int]
expnts
xpows :: [Int]
xpows = (Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) [Seq Int]
expnts'
degreeAndLeadingCoefficient :: (Eq a, AlgRing.C a)
=> Int -> Spray a -> (Int, Spray a)
degreeAndLeadingCoefficient :: forall a. (Eq a, C a) => Int -> Spray a -> (Int, Spray a)
degreeAndLeadingCoefficient Int
n Spray a
spray
| Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = (
if a
constantTerm a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
forall a. C a => a
AlgAdd.zero
then Int
forall a. Bounded a => a
minBound
else Int
0,
a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray a
constantTerm
)
| Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
/= Int
n = (Int
0, Spray a
spray)
| Bool
otherwise = (Int
deg, Spray a
leadingCoeff)
where
permutation :: [Int]
permutation = [Int
2 .. Int
n] [Int] -> [Int] -> [Int]
forall a. [a] -> [a] -> [a]
++ [Int
1]
spray' :: Spray a
spray' = [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
spray
([Powers]
powers, [a]
coeffs) = [(Powers, a)] -> ([Powers], [a])
forall a b. [(a, b)] -> ([a], [b])
unzip (Spray a -> [(Powers, a)]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray a
spray')
expnts :: [Seq Int]
expnts = (Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
powers
constantTerm :: a
constantTerm = a -> Maybe a -> a
forall a. a -> Maybe a -> a
fromMaybe a
forall a. C a => a
AlgAdd.zero (Powers -> Spray a -> Maybe a
forall k v. (Eq k, Hashable k) => k -> HashMap k v -> Maybe v
HM.lookup (Seq Int -> Int -> Powers
Powers Seq Int
forall a. Seq a
S.empty Int
0) Spray a
spray')
([Seq Int]
expnts', [a]
coeffs') =
[(Seq Int, a)] -> ([Seq Int], [a])
forall a b. [(a, b)] -> ([a], [b])
unzip ([(Seq Int, a)] -> ([Seq Int], [a]))
-> [(Seq Int, a)] -> ([Seq Int], [a])
forall a b. (a -> b) -> a -> b
$ ((Seq Int, a) -> Bool) -> [(Seq Int, a)] -> [(Seq Int, a)]
forall a. (a -> Bool) -> [a] -> [a]
filter (\(Seq Int
s,a
_) -> Bool -> Bool
not (Bool -> Bool) -> Bool -> Bool
forall a b. (a -> b) -> a -> b
$ Seq Int -> Bool
forall a. Seq a -> Bool
S.null Seq Int
s) ([Seq Int] -> [a] -> [(Seq Int, a)]
forall a b. [a] -> [b] -> [(a, b)]
zip [Seq Int]
expnts [a]
coeffs)
xpows :: [Int]
xpows = (Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) [Seq Int]
expnts'
deg :: Int
deg = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum [Int]
xpows
is :: [Int]
is = Int -> [Int] -> [Int]
forall a. Eq a => a -> [a] -> [Int]
elemIndices Int
deg [Int]
xpows
expnts'' :: [Seq Int]
expnts'' = [Int -> Seq Int -> Seq Int
forall a. Int -> Seq a -> Seq a
S.deleteAt Int
0 ([Seq Int]
expnts' [Seq Int] -> Int -> Seq Int
forall a. HasCallStack => [a] -> Int -> a
!! Int
i) | Int
i <- [Int]
is]
powers'' :: [Powers]
powers'' = (Seq Int -> Powers) -> [Seq Int] -> [Powers]
forall a b. (a -> b) -> [a] -> [b]
map (\Seq Int
s -> Seq Int -> Int -> Powers
Powers Seq Int
s (Seq Int -> Int
forall a. Seq a -> Int
S.length Seq Int
s)) [Seq Int]
expnts''
coeffs'' :: [a]
coeffs'' = [[a]
coeffs' [a] -> Int -> a
forall a. HasCallStack => [a] -> Int -> a
!! Int
i | Int
i <- [Int]
is]
leadingCoeff :: Spray a
leadingCoeff =
(Spray a -> Spray a -> Spray a) -> [Spray a] -> Spray a
forall a. HasCallStack => (a -> a -> a) -> [a] -> a
foldl1' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
(^+^) ((Powers -> a -> Spray a) -> [Powers] -> [a] -> [Spray a]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith (((Powers, a) -> Spray a) -> Powers -> a -> Spray a
forall a b c. ((a, b) -> c) -> a -> b -> c
curry (Powers, a) -> Spray a
forall a. Monomial a -> Spray a
fromMonomial) [Powers]
powers'' [a]
coeffs'')
pseudoDivision :: (Eq a, AlgRing.C a)
=> Int
-> Spray a
-> Spray a
-> (Spray a, (Spray a, Spray a))
pseudoDivision :: forall a.
(Eq a, C a) =>
Int -> Spray a -> Spray a -> (Spray a, (Spray a, Spray a))
pseudoDivision Int
n Spray a
sprayA Spray a
sprayB
| Int
degB Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
forall a. Bounded a => a
minBound = String -> (Spray a, (Spray a, Spray a))
forall a. HasCallStack => String -> a
error String
"pseudoDivision: pseudo-division by 0."
| Int
degA Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
degB = String -> (Spray a, (Spray a, Spray a))
forall a. HasCallStack => String -> a
error String
"pseudoDivision: degree(A) < degree(B)."
| Bool
otherwise = (Spray a
ellB Spray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^ Int
delta , Spray a -> Spray a -> Int -> (Spray a, Spray a)
go Spray a
sprayA Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray Int
delta)
where
degA :: Int
degA = Int -> Spray a -> Int
forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n Spray a
sprayA
(Int
degB, Spray a
ellB) = Int -> Spray a -> (Int, Spray a)
forall a. (Eq a, C a) => Int -> Spray a -> (Int, Spray a)
degreeAndLeadingCoefficient Int
n Spray a
sprayB
delta :: Int
delta = Int
degA Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
degB Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
1
go :: Spray a -> Spray a -> Int -> (Spray a, Spray a)
go Spray a
sprayR Spray a
sprayQ Int
e =
if Int
degR Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
degB Bool -> Bool -> Bool
|| Spray a
sprayR Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray
then (Spray a
q Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
sprayQ, Spray a
q Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
sprayR)
else Spray a -> Spray a -> Int -> (Spray a, Spray a)
go (Spray a
ellB Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
sprayR Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
sprayS Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
sprayB)
(Spray a
ellB Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
sprayQ Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray a
sprayS)
(Int
e Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1)
where
(Int
degR, Spray a
ellR) = Int -> Spray a -> (Int, Spray a)
forall a. (Eq a, C a) => Int -> Spray a -> (Int, Spray a)
degreeAndLeadingCoefficient Int
n Spray a
sprayR
q :: Spray a
q = Spray a
ellB Spray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^ Int
e
sprayXn :: Spray a
sprayXn = Int -> Spray a
forall a. C a => Int -> Spray a
lone Int
n
sprayS :: Spray a
sprayS = Spray a
ellR Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
sprayXn Spray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^ (Int
degR Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
degB)
gcdKX1dotsXn :: forall a. (Eq a, AlgField.C a)
=> Int -> Spray a -> Spray a -> Spray a
gcdKX1dotsXn :: forall a. (Eq a, C a) => Int -> Spray a -> Spray a -> Spray a
gcdKX1dotsXn Int
n Spray a
sprayA Spray a
sprayB
| Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray (a -> Spray a) -> a -> Spray a
forall a b. (a -> b) -> a -> b
$ Spray a -> Spray a -> a
gcdKX0 Spray a
sprayA Spray a
sprayB
| Int
degB Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
degA = Int -> Spray a -> Spray a -> Spray a
forall a. (Eq a, C a) => Int -> Spray a -> Spray a -> Spray a
gcdKX1dotsXn Int
n Spray a
sprayB Spray a
sprayA
| Spray a
sprayB Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray = Spray a
sprayA
| Bool
otherwise = Spray a -> Spray a -> Spray a -> Spray a -> Spray a
go Spray a
sprayA' Spray a
sprayB' Spray a
forall a. C a => Spray a
unitSpray Spray a
forall a. C a => Spray a
unitSpray
where
gcdKX0 :: Spray a -> Spray a -> a
gcdKX0 :: Spray a -> Spray a -> a
gcdKX0 = (Spray a -> a) -> Spray a -> Spray a -> a
forall a b. a -> b -> a
const ((Spray a -> a) -> Spray a -> Spray a -> a)
-> (Spray a -> a) -> Spray a -> Spray a -> a
forall a b. (a -> b) -> a -> b
$ a -> Spray a -> a
forall a b. a -> b -> a
const a
forall a. C a => a
AlgRing.one
n' :: Int
n' = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
sprayA) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
sprayB)
degA :: Int
degA = Int -> Spray a -> Int
forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n' Spray a
sprayA
degB :: Int
degB = Int -> Spray a -> Int
forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n' Spray a
sprayB
gcdKX1dotsXm :: Spray a -> Spray a -> Spray a
gcdKX1dotsXm = Int -> Spray a -> Spray a -> Spray a
forall a. (Eq a, C a) => Int -> Spray a -> Spray a -> Spray a
gcdKX1dotsXn (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1)
content :: Spray a -> Spray a
content :: Spray a -> Spray a
content Spray a
spray = (Spray a -> Spray a -> Spray a) -> [Spray a] -> Spray a
forall a. HasCallStack => (a -> a -> a) -> [a] -> a
foldl1' Spray a -> Spray a -> Spray a
gcdKX1dotsXm (Int -> Spray a -> [Spray a]
forall a. (Eq a, C a) => Int -> Spray a -> [Spray a]
sprayCoefficients' Int
n' Spray a
spray)
exactDivisionBy :: Spray a -> Spray a -> Spray a
exactDivisionBy :: Spray a -> Spray a -> Spray a
exactDivisionBy Spray a
b Spray a
a =
if (Spray a, Spray a) -> Spray a
forall a b. (a, b) -> b
snd (Spray a, Spray a)
division Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray
then (Spray a, Spray a) -> Spray a
forall a b. (a, b) -> a
fst (Spray a, Spray a)
division
else String -> Spray a
forall a. HasCallStack => String -> a
error String
"exactDivisionBy: should not happen."
where
division :: (Spray a, Spray a)
division = Spray a -> Spray a -> (Spray a, Spray a)
forall a. (Eq a, C a) => Spray a -> Spray a -> (Spray a, Spray a)
sprayDivision Spray a
a Spray a
b
reduceSpray :: Spray a -> Spray a
reduceSpray :: Spray a -> Spray a
reduceSpray Spray a
spray = Spray a -> Spray a -> Spray a
exactDivisionBy Spray a
cntnt Spray a
spray
where
coeffs :: [Spray a]
coeffs = Int -> Spray a -> [Spray a]
forall a. (Eq a, C a) => Int -> Spray a -> [Spray a]
sprayCoefficients' Int
n' Spray a
spray
cntnt :: Spray a
cntnt = (Spray a -> Spray a -> Spray a) -> [Spray a] -> Spray a
forall a. HasCallStack => (a -> a -> a) -> [a] -> a
foldl1' Spray a -> Spray a -> Spray a
gcdKX1dotsXm [Spray a]
coeffs
contA :: Spray a
contA = Spray a -> Spray a
content Spray a
sprayA
contB :: Spray a
contB = Spray a -> Spray a
content Spray a
sprayB
d :: Spray a
d = Spray a -> Spray a -> Spray a
gcdKX1dotsXm Spray a
contA Spray a
contB
sprayA' :: Spray a
sprayA' = Spray a -> Spray a -> Spray a
exactDivisionBy Spray a
contA Spray a
sprayA
sprayB' :: Spray a
sprayB' = Spray a -> Spray a -> Spray a
exactDivisionBy Spray a
contB Spray a
sprayB
go :: Spray a -> Spray a -> Spray a -> Spray a -> Spray a
go :: Spray a -> Spray a -> Spray a -> Spray a -> Spray a
go Spray a
sprayA'' Spray a
sprayB'' Spray a
g Spray a
h
| Spray a
sprayR Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray = Spray a
d Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a -> Spray a
reduceSpray Spray a
sprayB''
| Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
sprayR Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = Spray a
d
| Bool
otherwise = Spray a -> Spray a -> Spray a -> Spray a -> Spray a
go Spray a
sprayB''
(Spray a -> Spray a -> Spray a
exactDivisionBy (Spray a
g Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
hSpray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
delta) Spray a
sprayR)
Spray a
ellA''
(Spray a -> Spray a -> Spray a
exactDivisionBy (Spray a
hSpray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
delta) (Spray a
h Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
gSpray a -> Int -> Spray a
forall a. (C a, Eq a) => Spray a -> Int -> Spray a
^**^Int
delta))
where
(Spray a
_, (Spray a
_, Spray a
sprayR)) = Int -> Spray a -> Spray a -> (Spray a, (Spray a, Spray a))
forall a.
(Eq a, C a) =>
Int -> Spray a -> Spray a -> (Spray a, (Spray a, Spray a))
pseudoDivision Int
n' Spray a
sprayA'' Spray a
sprayB''
(Int
degA'', Spray a
ellA'') = Int -> Spray a -> (Int, Spray a)
forall a. (Eq a, C a) => Int -> Spray a -> (Int, Spray a)
degreeAndLeadingCoefficient Int
n' Spray a
sprayA''
degB'' :: Int
degB'' = Int -> Spray a -> Int
forall a. (Eq a, C a) => Int -> Spray a -> Int
degree Int
n' Spray a
sprayB''
delta :: Int
delta = Int
degA'' Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
degB''
gcdSpray :: forall a. (Eq a, AlgField.C a) => Spray a -> Spray a -> Spray a
gcdSpray :: forall a. (Eq a, C a) => Spray a -> Spray a -> Spray a
gcdSpray Spray a
sprayA Spray a
sprayB = Int -> Spray a -> Spray a -> Spray a
forall a. (Eq a, C a) => Int -> Spray a -> Spray a -> Spray a
gcdKX1dotsXn Int
n Spray a
sprayA Spray a
sprayB
where
n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
sprayA) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
sprayB)
detLaplace :: forall a. (Eq a, AlgRing.C a) => Matrix a -> a
detLaplace :: forall a. (Eq a, C a) => Matrix a -> a
detLaplace Matrix a
b =
if Matrix a -> Int
forall a. Matrix a -> Int
nrows Matrix a
b Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Matrix a -> Int
forall a. Matrix a -> Int
ncols Matrix a
b
then Matrix a -> a
detUnsafe Matrix a
b
else String -> a
forall a. HasCallStack => String -> a
error String
"detLaplace: the matrix is not square."
where
detUnsafe :: Matrix a -> a
detUnsafe Matrix a
m = if Matrix a -> Int
forall a. Matrix a -> Int
nrows Matrix a
m Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1
then
Matrix a
m Matrix a -> (Int, Int) -> a
forall a. Matrix a -> (Int, Int) -> a
DM.! (Int
1,Int
1)
else
[a] -> a
suml1
[Int -> a -> a
forall {a} {a}. (Integral a, C a) => a -> a -> a
negateIf Int
i (a -> a -> a
times (Matrix a
m Matrix a -> (Int, Int) -> a
forall a. Matrix a -> (Int, Int) -> a
DM.! (Int
i,Int
1)) (Matrix a -> a
detUnsafe (Int -> Int -> Matrix a -> Matrix a
forall a. Int -> Int -> Matrix a -> Matrix a
minorMatrix Int
i Int
1 Matrix a
m)))
| Int
i <- [Int
1 .. Matrix a -> Int
forall a. Matrix a -> Int
nrows Matrix a
m]]
suml1 :: [a] -> a
suml1 = (a -> a -> a) -> [a] -> a
forall a. HasCallStack => (a -> a -> a) -> [a] -> a
foldl1' a -> a -> a
forall a. C a => a -> a -> a
(AlgAdd.+)
negateIf :: a -> a -> a
negateIf a
i = if a -> Bool
forall a. Integral a => a -> Bool
even a
i then a -> a
forall a. C a => a -> a
AlgAdd.negate else a -> a
forall a. a -> a
id
times :: a -> a -> a
times :: a -> a -> a
times a
x a
y = if a
x a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
forall a. C a => a
AlgAdd.zero then a
forall a. C a => a
AlgAdd.zero else a
x a -> a -> a
forall a. C a => a -> a -> a
AlgRing.* a
y
detLaplace' :: forall a. (Eq a, AlgRing.C a) => MathMatrix.T a -> a
detLaplace' :: forall a. (Eq a, C a) => T a -> a
detLaplace' T a
m = Matrix a -> a
forall a. (Eq a, C a) => Matrix a -> a
detLaplace ([[a]] -> Matrix a
forall a. [[a]] -> Matrix a
DM.fromLists ([[a]] -> Matrix a) -> [[a]] -> Matrix a
forall a b. (a -> b) -> a -> b
$ T a -> [[a]]
forall a. T a -> [[a]]
MathMatrix.rows T a
m)
characteristicPolynomial :: (Eq a, AlgRing.C a) => Matrix a -> Spray a
characteristicPolynomial :: forall a. (Eq a, C a) => Matrix a -> Spray a
characteristicPolynomial Matrix a
m =
if Matrix a -> Int
forall a. Matrix a -> Int
nrows Matrix a
m Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
/= Matrix a -> Int
forall a. Matrix a -> Int
ncols Matrix a
m
then String -> Spray a
forall a. HasCallStack => String -> a
error String
"characteristicPolynomial: the matrix is not square."
else Matrix (Spray a) -> Spray a
forall a. (Eq a, C a) => Matrix a -> a
detLaplace Matrix (Spray a)
m'
where
m' :: Matrix (Spray a)
m' = ((Int, Int) -> a -> Spray a) -> Matrix a -> Matrix (Spray a)
forall a b. ((Int, Int) -> a -> b) -> Matrix a -> Matrix b
DM.mapPos (Int, Int) -> a -> Spray a
f Matrix a
m
f :: (Int, Int) -> a -> Spray a
f (Int
i, Int
j) a
mij = if Int
i Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
j
then a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray a
mij Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
x
else a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray a
mij
x :: Spray a
x = Int -> Spray a
forall a. C a => Int -> Spray a
lone Int
1
data RatioOfSprays a = RatioOfSprays
{ forall a. RatioOfSprays a -> Spray a
_numerator :: Spray a
, forall a. RatioOfSprays a -> Spray a
_denominator :: Spray a
}
deriving Int -> RatioOfSprays a -> ShowS
[RatioOfSprays a] -> ShowS
RatioOfSprays a -> String
(Int -> RatioOfSprays a -> ShowS)
-> (RatioOfSprays a -> String)
-> ([RatioOfSprays a] -> ShowS)
-> Show (RatioOfSprays a)
forall a. Show a => Int -> RatioOfSprays a -> ShowS
forall a. Show a => [RatioOfSprays a] -> ShowS
forall a. Show a => RatioOfSprays a -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
$cshowsPrec :: forall a. Show a => Int -> RatioOfSprays a -> ShowS
showsPrec :: Int -> RatioOfSprays a -> ShowS
$cshow :: forall a. Show a => RatioOfSprays a -> String
show :: RatioOfSprays a -> String
$cshowList :: forall a. Show a => [RatioOfSprays a] -> ShowS
showList :: [RatioOfSprays a] -> ShowS
Show
type RatioOfQSprays = RatioOfSprays Rational
instance (Eq a, AlgField.C a) => HasVariables (RatioOfSprays a) where
type BaseRing (RatioOfSprays a) = a
type VariablesType (RatioOfSprays a) = Spray a
substitute :: [Maybe a] -> RatioOfSprays a -> RatioOfSprays a
substitute :: [Maybe a] -> RatioOfSprays a -> RatioOfSprays a
substitute [Maybe a]
subs (RatioOfSprays Spray a
p Spray a
q) =
[Maybe (BaseRing (Spray a))] -> Spray a -> Spray a
forall b. HasVariables b => [Maybe (BaseRing b)] -> b -> b
substitute [Maybe a]
[Maybe (BaseRing (Spray a))]
subs Spray a
p Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
%//% [Maybe (BaseRing (Spray a))] -> Spray a -> Spray a
forall b. HasVariables b => [Maybe (BaseRing b)] -> b -> b
substitute [Maybe a]
[Maybe (BaseRing (Spray a))]
subs Spray a
q
evaluate :: RatioOfSprays a -> [a] -> a
evaluate :: RatioOfSprays a -> [a] -> a
evaluate (RatioOfSprays Spray a
p Spray a
q) [a]
xyz = Spray a -> [BaseRing (Spray a)] -> BaseRing (Spray a)
forall b. HasVariables b => b -> [BaseRing b] -> BaseRing b
evaluate Spray a
p [a]
[BaseRing (Spray a)]
xyz a -> a -> a
forall a. C a => a -> a -> a
AlgField./ Spray a -> [BaseRing (Spray a)] -> BaseRing (Spray a)
forall b. HasVariables b => b -> [BaseRing b] -> BaseRing b
evaluate Spray a
q [a]
[BaseRing (Spray a)]
xyz
changeVariables :: RatioOfSprays a -> [Spray a] -> RatioOfSprays a
changeVariables :: RatioOfSprays a -> [Spray a] -> RatioOfSprays a
changeVariables RatioOfSprays a
rOS [Spray a]
newVariables =
if [Spray a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [Spray a]
newVariables Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< RatioOfSprays a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables RatioOfSprays a
rOS
then
String -> RatioOfSprays a
forall a. HasCallStack => String -> a
error String
"changeVariables: not enough new variables provided."
else
Spray a -> [VariablesType (Spray a)] -> Spray a
forall b. HasVariables b => b -> [VariablesType b] -> b
changeVariables (RatioOfSprays a -> Spray a
forall a. RatioOfSprays a -> Spray a
_numerator RatioOfSprays a
rOS) [Spray a]
[VariablesType (Spray a)]
newVariables
Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
%//% Spray a -> [VariablesType (Spray a)] -> Spray a
forall b. HasVariables b => b -> [VariablesType b] -> b
changeVariables (RatioOfSprays a -> Spray a
forall a. RatioOfSprays a -> Spray a
_denominator RatioOfSprays a
rOS) [Spray a]
[VariablesType (Spray a)]
newVariables
numberOfVariables :: RatioOfSprays a -> Int
numberOfVariables :: RatioOfSprays a -> Int
numberOfVariables (RatioOfSprays Spray a
p Spray a
q) =
Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
p) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
q)
permuteVariables :: [Int] -> RatioOfSprays a -> RatioOfSprays a
permuteVariables :: [Int] -> RatioOfSprays a -> RatioOfSprays a
permuteVariables [Int]
permutation (RatioOfSprays Spray a
p Spray a
q) =
[Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
p Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
%//% [Int] -> Spray a -> Spray a
forall b. HasVariables b => [Int] -> b -> b
permuteVariables [Int]
permutation Spray a
q
swapVariables :: (Int, Int) -> RatioOfSprays a -> RatioOfSprays a
swapVariables :: (Int, Int) -> RatioOfSprays a -> RatioOfSprays a
swapVariables (Int
i, Int
j) (RatioOfSprays Spray a
p Spray a
q) =
(Int, Int) -> Spray a -> Spray a
forall b. HasVariables b => (Int, Int) -> b -> b
swapVariables (Int
i, Int
j) Spray a
p Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
%//% (Int, Int) -> Spray a -> Spray a
forall b. HasVariables b => (Int, Int) -> b -> b
swapVariables (Int
i, Int
j) Spray a
q
derivative :: Int -> RatioOfSprays a -> RatioOfSprays a
derivative :: Int -> RatioOfSprays a -> RatioOfSprays a
derivative Int
i (RatioOfSprays Spray a
p Spray a
q) = (Spray a
p' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
p Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q') Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
%//% (Spray a
q Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q)
where
p' :: Spray a
p' = Int -> Spray a -> Spray a
forall b. HasVariables b => Int -> b -> b
derivative Int
i Spray a
p
q' :: Spray a
q' = Int -> Spray a -> Spray a
forall b. HasVariables b => Int -> b -> b
derivative Int
i Spray a
q
exactDivision :: (Eq a, AlgField.C a) => Spray a -> Spray a -> Spray a
exactDivision :: forall a. (Eq a, C a) => Spray a -> Spray a -> Spray a
exactDivision Spray a
p Spray a
q = (Spray a, Spray a) -> Spray a
forall a b. (a, b) -> a
fst (Spray a -> Spray a -> (Spray a, Spray a)
forall a. (Eq a, C a) => Spray a -> Spray a -> (Spray a, Spray a)
sprayDivision Spray a
p Spray a
q)
irreducibleFraction ::
(Eq a, AlgField.C a) => Spray a -> Spray a -> RatioOfSprays a
irreducibleFraction :: forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
irreducibleFraction Spray a
p Spray a
q = RatioOfSprays a -> RatioOfSprays a
forall a. (Eq a, C a) => RatioOfSprays a -> RatioOfSprays a
adjustFraction RatioOfSprays a
rOS
where
g :: Spray a
g = Spray a -> Spray a -> Spray a
forall a. (Eq a, C a) => Spray a -> Spray a -> Spray a
gcdSpray Spray a
p Spray a
q
a :: Spray a
a = Spray a -> Spray a -> Spray a
forall a. (Eq a, C a) => Spray a -> Spray a -> Spray a
exactDivision Spray a
p Spray a
g
b :: Spray a
b = Spray a -> Spray a -> Spray a
forall a. (Eq a, C a) => Spray a -> Spray a -> Spray a
exactDivision Spray a
q Spray a
g
rOS :: RatioOfSprays a
rOS = if Spray a -> Bool
forall b. HasVariables b => b -> Bool
isConstant Spray a
p Bool -> Bool -> Bool
|| Spray a -> Bool
forall b. HasVariables b => b -> Bool
isConstant Spray a
q
then Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays Spray a
p Spray a
q
else Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays Spray a
a Spray a
b
adjustFraction :: (Eq a, AlgField.C a) => RatioOfSprays a -> RatioOfSprays a
adjustFraction :: forall a. (Eq a, C a) => RatioOfSprays a -> RatioOfSprays a
adjustFraction (RatioOfSprays Spray a
p Spray a
q) = if Spray a -> Bool
forall b. HasVariables b => b -> Bool
isConstant Spray a
q
then Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays (Spray a
p Spray a -> a -> Spray a
forall a. (C a, Eq a) => Spray a -> a -> Spray a
/^ a
c) Spray a
forall a. C a => Spray a
unitSpray
else Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays Spray a
p Spray a
q
where
c :: a
c = Spray a -> a
forall a. C a => Spray a -> a
getConstantTerm Spray a
q
instance (AlgRing.C a, Eq a) => Eq (RatioOfSprays a) where
(==) :: RatioOfSprays a -> RatioOfSprays a -> Bool
== :: RatioOfSprays a -> RatioOfSprays a -> Bool
(==) (RatioOfSprays Spray a
p Spray a
q) (RatioOfSprays Spray a
p' Spray a
q') =
Spray a -> Bool
forall a. Spray a -> Bool
isZeroSpray (Spray a
p Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray a
p' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q)
instance (AlgField.C a, Eq a) => AlgAdd.C (RatioOfSprays a) where
(+) :: RatioOfSprays a -> RatioOfSprays a -> RatioOfSprays a
+ :: RatioOfSprays a -> RatioOfSprays a -> RatioOfSprays a
(+) (RatioOfSprays Spray a
p Spray a
q) (RatioOfSprays Spray a
p' Spray a
q') =
Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
irreducibleFraction (Spray a
p Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray a
p' Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q) (Spray a
q Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q')
zero :: RatioOfSprays a
zero :: RatioOfSprays a
zero = Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays Spray a
forall a. (Eq a, C a) => Spray a
zeroSpray Spray a
forall a. C a => Spray a
unitSpray
negate :: RatioOfSprays a -> RatioOfSprays a
negate :: RatioOfSprays a -> RatioOfSprays a
negate (RatioOfSprays Spray a
p Spray a
q) = Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays (Spray a -> Spray a
forall a. C a => Spray a -> Spray a
negateSpray Spray a
p) Spray a
q
instance (AlgField.C a, Eq a) => AlgMod.C a (RatioOfSprays a) where
(*>) :: a -> RatioOfSprays a -> RatioOfSprays a
a
lambda *> :: a -> RatioOfSprays a -> RatioOfSprays a
*> (RatioOfSprays Spray a
p Spray a
q) = Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays (a
lambda a -> Spray a -> Spray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ Spray a
p) Spray a
q
instance (AlgField.C a, Eq a) => AlgRightMod.C a (RatioOfSprays a) where
(<*) :: RatioOfSprays a -> a -> RatioOfSprays a
RatioOfSprays a
rOS <* :: RatioOfSprays a -> a -> RatioOfSprays a
<* a
lambda = a
lambda a -> RatioOfSprays a -> RatioOfSprays a
forall a v. C a v => a -> v -> v
AlgMod.*> RatioOfSprays a
rOS
instance (AlgField.C a, Eq a) => AlgMod.C (Spray a) (RatioOfSprays a) where
(*>) :: Spray a -> RatioOfSprays a -> RatioOfSprays a
Spray a
spray *> :: Spray a -> RatioOfSprays a -> RatioOfSprays a
*> (RatioOfSprays Spray a
p Spray a
q) = Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
irreducibleFraction (Spray a
spray Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
p) Spray a
q
instance (AlgField.C a, Eq a) => AlgRightMod.C (Spray a) (RatioOfSprays a) where
(<*) :: RatioOfSprays a -> Spray a -> RatioOfSprays a
RatioOfSprays a
rOS <* :: RatioOfSprays a -> Spray a -> RatioOfSprays a
<* Spray a
spray = Spray a
spray Spray a -> RatioOfSprays a -> RatioOfSprays a
forall a v. C a v => a -> v -> v
AlgMod.*> RatioOfSprays a
rOS
instance (AlgField.C a, Eq a) => AlgRing.C (RatioOfSprays a) where
(*) :: RatioOfSprays a -> RatioOfSprays a -> RatioOfSprays a
* :: RatioOfSprays a -> RatioOfSprays a -> RatioOfSprays a
(*) (RatioOfSprays Spray a
p Spray a
q) (RatioOfSprays Spray a
p' Spray a
q') =
Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
irreducibleFraction (Spray a
p Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
p') (Spray a
q Spray a -> Spray a -> Spray a
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Spray a
q')
(^) :: RatioOfSprays a -> Integer -> RatioOfSprays a
^ :: RatioOfSprays a -> Integer -> RatioOfSprays a
(^) (RatioOfSprays Spray a
p Spray a
q) Integer
n = Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays (Spray a
p Spray a -> Integer -> Spray a
forall a. C a => a -> Integer -> a
AlgRing.^ Integer
n) (Spray a
q Spray a -> Integer -> Spray a
forall a. C a => a -> Integer -> a
AlgRing.^ Integer
n)
one :: RatioOfSprays a
one :: RatioOfSprays a
one = Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays Spray a
forall a. C a => Spray a
unitSpray Spray a
forall a. C a => Spray a
unitSpray
instance (AlgField.C a, Eq a) => AlgField.C (RatioOfSprays a) where
recip :: RatioOfSprays a -> RatioOfSprays a
recip :: RatioOfSprays a -> RatioOfSprays a
recip (RatioOfSprays Spray a
p Spray a
q) = Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays Spray a
q Spray a
p
infixl 7 %:%
(%:%) :: Spray a -> Spray a -> RatioOfSprays a
%:% :: forall a. Spray a -> Spray a -> RatioOfSprays a
(%:%) = Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays
infixl 7 %//%
(%//%) :: (Eq a, AlgField.C a) => Spray a -> Spray a -> RatioOfSprays a
%//% :: forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
(%//%) = Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
irreducibleFraction
infixl 7 ^/^
(^/^) :: (Eq a, AlgField.C a) => Spray a -> Spray a -> RatioOfSprays a
^/^ :: forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
(^/^) = Spray a -> Spray a -> RatioOfSprays a
forall a. (Eq a, C a) => Spray a -> Spray a -> RatioOfSprays a
irreducibleFraction
infixl 7 %/%
(%/%) :: (Eq a, AlgField.C a) => RatioOfSprays a -> Spray a -> RatioOfSprays a
%/% :: forall a.
(Eq a, C a) =>
RatioOfSprays a -> Spray a -> RatioOfSprays a
(%/%) RatioOfSprays a
rOS Spray a
spray = RatioOfSprays a
rOS RatioOfSprays a -> RatioOfSprays a -> RatioOfSprays a
forall a. C a => a -> a -> a
AlgRing.* Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays Spray a
forall a. C a => Spray a
unitSpray Spray a
spray
isConstantRatioOfSprays :: (Eq a, AlgField.C a) => RatioOfSprays a -> Bool
isConstantRatioOfSprays :: forall a. (Eq a, C a) => RatioOfSprays a -> Bool
isConstantRatioOfSprays = RatioOfSprays a -> Bool
forall b. HasVariables b => b -> Bool
isConstant
isPolynomialRatioOfSprays :: (Eq a, AlgRing.C a) => RatioOfSprays a -> Bool
isPolynomialRatioOfSprays :: forall a. (Eq a, C a) => RatioOfSprays a -> Bool
isPolynomialRatioOfSprays = Spray a -> Bool
forall b. HasVariables b => b -> Bool
isConstant (Spray a -> Bool)
-> (RatioOfSprays a -> Spray a) -> RatioOfSprays a -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. RatioOfSprays a -> Spray a
forall a. RatioOfSprays a -> Spray a
_denominator
zeroRatioOfSprays, zeroROS :: (AlgField.C a, Eq a) => RatioOfSprays a
zeroRatioOfSprays :: forall a. (C a, Eq a) => RatioOfSprays a
zeroRatioOfSprays = RatioOfSprays a
forall a. C a => a
AlgAdd.zero
zeroROS :: forall a. (C a, Eq a) => RatioOfSprays a
zeroROS = RatioOfSprays a
forall a. C a => a
AlgAdd.zero
unitRatioOfSprays, unitROS :: (AlgField.C a, Eq a) => RatioOfSprays a
unitRatioOfSprays :: forall a. (C a, Eq a) => RatioOfSprays a
unitRatioOfSprays = RatioOfSprays a
forall a. C a => a
AlgRing.one
unitROS :: forall a. (C a, Eq a) => RatioOfSprays a
unitROS = RatioOfSprays a
forall a. C a => a
AlgRing.one
constantRatioOfSprays :: (Eq a, AlgRing.C a) => a -> RatioOfSprays a
constantRatioOfSprays :: forall a. (Eq a, C a) => a -> RatioOfSprays a
constantRatioOfSprays a
x = Spray a -> RatioOfSprays a
forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays (a -> Spray a
forall a. (C a, Eq a) => a -> Spray a
constantSpray a
x)
evalRatioOfSprays :: (Eq a, AlgField.C a) => RatioOfSprays a -> [a] -> a
evalRatioOfSprays :: forall a. (Eq a, C a) => RatioOfSprays a -> [a] -> a
evalRatioOfSprays = RatioOfSprays a -> [a] -> a
RatioOfSprays a
-> [BaseRing (RatioOfSprays a)] -> BaseRing (RatioOfSprays a)
forall b. HasVariables b => b -> [BaseRing b] -> BaseRing b
evaluate
substituteRatioOfSprays ::
(Eq a, AlgField.C a) => [Maybe a] -> RatioOfSprays a -> RatioOfSprays a
substituteRatioOfSprays :: forall a.
(Eq a, C a) =>
[Maybe a] -> RatioOfSprays a -> RatioOfSprays a
substituteRatioOfSprays = [Maybe a] -> RatioOfSprays a -> RatioOfSprays a
[Maybe (BaseRing (RatioOfSprays a))]
-> RatioOfSprays a -> RatioOfSprays a
forall b. HasVariables b => [Maybe (BaseRing b)] -> b -> b
substitute
asRatioOfSprays :: AlgRing.C a => Spray a -> RatioOfSprays a
asRatioOfSprays :: forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays Spray a
spray = Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays Spray a
spray Spray a
forall a. C a => Spray a
unitSpray
fromRatioOfPolynomials ::
(Eq a, AlgRing.C a) => RatioOfPolynomials a -> RatioOfSprays a
fromRatioOfPolynomials :: forall a. (Eq a, C a) => RatioOfPolynomials a -> RatioOfSprays a
fromRatioOfPolynomials RatioOfPolynomials a
rop =
Spray a -> Spray a -> RatioOfSprays a
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays
(Polynomial a -> Spray a
forall a. (Eq a, C a) => Polynomial a -> Spray a
polynomialToSpray (Polynomial a -> Spray a) -> Polynomial a -> Spray a
forall a b. (a -> b) -> a -> b
$ RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.numerator RatioOfPolynomials a
rop)
(Polynomial a -> Spray a
forall a. (Eq a, C a) => Polynomial a -> Spray a
polynomialToSpray (Polynomial a -> Spray a) -> Polynomial a -> Spray a
forall a b. (a -> b) -> a -> b
$ RatioOfPolynomials a -> Polynomial a
forall a. T a -> a
NumberRatio.denominator RatioOfPolynomials a
rop)
fromRatioOfQPolynomials :: RatioOfQPolynomials -> RatioOfQSprays
fromRatioOfQPolynomials :: RatioOfQPolynomials -> RatioOfSprays Rational
fromRatioOfQPolynomials RatioOfQPolynomials
rop =
Spray Rational -> Spray Rational -> RatioOfSprays Rational
forall a. Spray a -> Spray a -> RatioOfSprays a
RatioOfSprays
(QPolynomial -> Spray Rational
qPolynomialToQSpray (QPolynomial -> Spray Rational) -> QPolynomial -> Spray Rational
forall a b. (a -> b) -> a -> b
$ RatioOfQPolynomials -> QPolynomial
forall a. T a -> a
NumberRatio.numerator RatioOfQPolynomials
rop)
(QPolynomial -> Spray Rational
qPolynomialToQSpray (QPolynomial -> Spray Rational) -> QPolynomial -> Spray Rational
forall a b. (a -> b) -> a -> b
$ RatioOfQPolynomials -> QPolynomial
forall a. T a -> a
NumberRatio.denominator RatioOfQPolynomials
rop)
showRatioOfSprays :: (Eq a, AlgRing.C a)
=> ((Spray a, Spray a) -> (String, String))
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfSprays :: forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays (Spray a, Spray a) -> (String, String)
spraysShower (String, String)
braces String
quotientBar (RatioOfSprays Spray a
p Spray a
q) =
String
numeratorString String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
denominatorString
where
enclose :: ShowS
enclose = (String, String) -> ShowS
bracify (String, String)
braces
(String
pString, String
qString) = (Spray a, Spray a) -> (String, String)
spraysShower (Spray a
p, Spray a
q)
numeratorString :: String
numeratorString = ShowS
enclose String
pString
denominatorString :: String
denominatorString = if Spray a
q Spray a -> Spray a -> Bool
forall a. Eq a => a -> a -> Bool
== Spray a
forall a. C a => Spray a
unitSpray
then String
""
else String
quotientBar String -> ShowS
forall a. [a] -> [a] -> [a]
++ ShowS
enclose String
qString
showTwoSpraysXYZ :: (Eq a, AlgRing.C a)
=> (a -> String)
-> (String, String)
-> [String]
-> (Spray a, Spray a)
-> (String, String)
showTwoSpraysXYZ :: forall a.
(Eq a, C a) =>
(a -> String)
-> (String, String)
-> [String]
-> (Spray a, Spray a)
-> (String, String)
showTwoSpraysXYZ a -> String
showCoef (String, String)
braces [String]
letters (Spray a
spray1, Spray a
spray2) =
(Spray a -> String) -> (Spray a, Spray a) -> (String, String)
forall a b. (a -> b) -> (a, a) -> (b, b)
both ((a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray a -> String
showCoef (String, String)
braces [Seq Int] -> [String]
showMonomials) (Spray a
spray1, Spray a
spray2)
where
n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray1) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray2)
showMonomials :: [Seq Int] -> [String]
showMonomials = (Seq Int -> String) -> [Seq Int] -> [String]
forall a b. (a -> b) -> [a] -> [b]
map (Text -> String
unpack (Text -> String) -> (Seq Int -> Text) -> Seq Int -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [String] -> Int -> Seq Int -> Text
showMonomialXYZ [String]
letters Int
n)
showTwoSpraysX1X2X3 ::
(a -> String)
-> (String, String)
-> String
-> (Spray a, Spray a)
-> (String, String)
showTwoSpraysX1X2X3 :: forall a.
(a -> String)
-> (String, String)
-> String
-> (Spray a, Spray a)
-> (String, String)
showTwoSpraysX1X2X3 a -> String
showCoef (String, String)
braces String
letter (Spray a
spray1, Spray a
spray2) =
(Spray a -> String) -> (Spray a, Spray a) -> (String, String)
forall a b. (a -> b) -> (a, a) -> (b, b)
both ((a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray a -> String
showCoef (String, String)
braces [Seq Int] -> [String]
showMonomials) (Spray a
spray1, Spray a
spray2)
where
showMonomials :: [Seq Int] -> [String]
showMonomials = String -> [Seq Int] -> [String]
showMonomialsX1X2X3 String
letter
showTwoNumSprays :: (Num a, Ord a)
=> (a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
showTwoNumSprays :: forall a.
(Num a, Ord a) =>
(a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
showTwoNumSprays a -> String
showPositiveCoef [Seq Int] -> [String]
showMonomials =
(Spray a -> String) -> (Spray a, Spray a) -> (String, String)
forall a b. (a -> b) -> (a, a) -> (b, b)
both (([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
forall a.
(Num a, Ord a) =>
([Seq Int] -> [String]) -> (a -> String) -> Spray a -> String
showNumSpray [Seq Int] -> [String]
showMonomials a -> String
showPositiveCoef)
showTwoQSprays ::
([Seq Int] -> [String])
-> (QSpray, QSpray)
-> (String, String)
showTwoQSprays :: ([Seq Int] -> [String])
-> (Spray Rational, Spray Rational) -> (String, String)
showTwoQSprays = (Rational -> String)
-> ([Seq Int] -> [String])
-> (Spray Rational, Spray Rational)
-> (String, String)
forall a.
(Num a, Ord a) =>
(a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
showTwoNumSprays Rational -> String
showRatio
showTwoNumSpraysXYZ :: (AlgRing.C a, Num a, Ord a)
=> (a -> String)
-> [String]
-> (Spray a, Spray a)
-> (String, String)
showTwoNumSpraysXYZ :: forall a.
(C a, Num a, Ord a) =>
(a -> String) -> [String] -> (Spray a, Spray a) -> (String, String)
showTwoNumSpraysXYZ a -> String
showPositiveCoef [String]
letters (Spray a
spray1, Spray a
spray2) =
(a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
forall a.
(Num a, Ord a) =>
(a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
showTwoNumSprays a -> String
showPositiveCoef [Seq Int] -> [String]
showMonomials (Spray a
spray1, Spray a
spray2)
where
n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
max (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray1) (Spray a -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray a
spray2)
showMonomials :: [Seq Int] -> [String]
showMonomials = (Seq Int -> String) -> [Seq Int] -> [String]
forall a b. (a -> b) -> [a] -> [b]
map (Text -> String
unpack (Text -> String) -> (Seq Int -> Text) -> Seq Int -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [String] -> Int -> Seq Int -> Text
showMonomialXYZ [String]
letters Int
n)
showTwoQSpraysXYZ ::
[String]
-> (QSpray, QSpray)
-> (String, String)
showTwoQSpraysXYZ :: [String] -> (Spray Rational, Spray Rational) -> (String, String)
showTwoQSpraysXYZ = (Rational -> String)
-> [String] -> (Spray Rational, Spray Rational) -> (String, String)
forall a.
(C a, Num a, Ord a) =>
(a -> String) -> [String] -> (Spray a, Spray a) -> (String, String)
showTwoNumSpraysXYZ Rational -> String
showRatio
showTwoNumSpraysX1X2X3 :: (Num a, Ord a)
=> (a -> String)
-> String
-> (Spray a, Spray a)
-> (String, String)
showTwoNumSpraysX1X2X3 :: forall a.
(Num a, Ord a) =>
(a -> String) -> String -> (Spray a, Spray a) -> (String, String)
showTwoNumSpraysX1X2X3 a -> String
showPositiveCoef String
letter (Spray a
spray1, Spray a
spray2) =
(a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
forall a.
(Num a, Ord a) =>
(a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
showTwoNumSprays a -> String
showPositiveCoef [Seq Int] -> [String]
showMonomials (Spray a
spray1, Spray a
spray2)
where
showMonomials :: [Seq Int] -> [String]
showMonomials = String -> [Seq Int] -> [String]
showMonomialsX1X2X3 String
letter
showTwoQSpraysX1X2X3 ::
String
-> (QSpray, QSpray)
-> (String, String)
showTwoQSpraysX1X2X3 :: String -> (Spray Rational, Spray Rational) -> (String, String)
showTwoQSpraysX1X2X3 = (Rational -> String)
-> String -> (Spray Rational, Spray Rational) -> (String, String)
forall a.
(Num a, Ord a) =>
(a -> String) -> String -> (Spray a, Spray a) -> (String, String)
showTwoNumSpraysX1X2X3 Rational -> String
showRatio
showRatioOfNumSprays :: (Num a, Ord a, AlgRing.C a)
=> (a -> String)
-> ([Seq Int] -> [String])
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfNumSprays :: forall a.
(Num a, Ord a, C a) =>
(a -> String)
-> ([Seq Int] -> [String])
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfNumSprays a -> String
showPositiveCoef [Seq Int] -> [String]
showMonomials =
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays ((a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
forall a.
(Num a, Ord a) =>
(a -> String)
-> ([Seq Int] -> [String])
-> (Spray a, Spray a)
-> (String, String)
showTwoNumSprays a -> String
showPositiveCoef [Seq Int] -> [String]
showMonomials)
showRatioOfQSprays ::
([Seq Int] -> [String])
-> (String, String)
-> String
-> RatioOfQSprays
-> String
showRatioOfQSprays :: ([Seq Int] -> [String])
-> (String, String) -> String -> RatioOfSprays Rational -> String
showRatioOfQSprays [Seq Int] -> [String]
showMonomials =
((Spray Rational, Spray Rational) -> (String, String))
-> (String, String) -> String -> RatioOfSprays Rational -> String
forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays (([Seq Int] -> [String])
-> (Spray Rational, Spray Rational) -> (String, String)
showTwoQSprays [Seq Int] -> [String]
showMonomials)
showRatioOfNumSpraysXYZ :: (Num a, Ord a, AlgRing.C a)
=> (a -> String)
-> [String]
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfNumSpraysXYZ :: forall a.
(Num a, Ord a, C a) =>
(a -> String)
-> [String]
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfNumSpraysXYZ a -> String
showPositiveCoef [String]
letters =
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays ((a -> String) -> [String] -> (Spray a, Spray a) -> (String, String)
forall a.
(C a, Num a, Ord a) =>
(a -> String) -> [String] -> (Spray a, Spray a) -> (String, String)
showTwoNumSpraysXYZ a -> String
showPositiveCoef [String]
letters)
showRatioOfNumSpraysX1X2X3 :: (Num a, Ord a, AlgRing.C a)
=> (a -> String)
-> String
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfNumSpraysX1X2X3 :: forall a.
(Num a, Ord a, C a) =>
(a -> String)
-> String
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfNumSpraysX1X2X3 a -> String
showPositiveCoef String
letter =
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays ((a -> String) -> String -> (Spray a, Spray a) -> (String, String)
forall a.
(Num a, Ord a) =>
(a -> String) -> String -> (Spray a, Spray a) -> (String, String)
showTwoNumSpraysX1X2X3 a -> String
showPositiveCoef String
letter)
showRatioOfQSpraysXYZ ::
[String]
-> (String, String)
-> String
-> RatioOfQSprays
-> String
showRatioOfQSpraysXYZ :: [String]
-> (String, String) -> String -> RatioOfSprays Rational -> String
showRatioOfQSpraysXYZ [String]
letters = ((Spray Rational, Spray Rational) -> (String, String))
-> (String, String) -> String -> RatioOfSprays Rational -> String
forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays ([String] -> (Spray Rational, Spray Rational) -> (String, String)
showTwoQSpraysXYZ [String]
letters)
showRatioOfQSpraysX1X2X3 ::
String
-> (String, String)
-> String
-> RatioOfQSprays
-> String
showRatioOfQSpraysX1X2X3 :: String
-> (String, String) -> String -> RatioOfSprays Rational -> String
showRatioOfQSpraysX1X2X3 String
letter = ((Spray Rational, Spray Rational) -> (String, String))
-> (String, String) -> String -> RatioOfSprays Rational -> String
forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays (String -> (Spray Rational, Spray Rational) -> (String, String)
showTwoQSpraysX1X2X3 String
letter)
showRatioOfSpraysXYZ :: forall a. (Eq a, AlgField.C a)
=> [String]
-> (a -> String)
-> (String, String)
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfSpraysXYZ :: forall a.
(Eq a, C a) =>
[String]
-> (a -> String)
-> (String, String)
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfSpraysXYZ [String]
letters a -> String
showCoef (String, String)
coeffBraces =
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays ((a -> String)
-> (String, String)
-> [String]
-> (Spray a, Spray a)
-> (String, String)
forall a.
(Eq a, C a) =>
(a -> String)
-> (String, String)
-> [String]
-> (Spray a, Spray a)
-> (String, String)
showTwoSpraysXYZ a -> String
showCoef (String, String)
coeffBraces [String]
letters)
showRatioOfSpraysXYZ' :: (Eq a, AlgField.C a)
=> [String]
-> (a -> String)
-> RatioOfSprays a
-> String
showRatioOfSpraysXYZ' :: forall a.
(Eq a, C a) =>
[String] -> (a -> String) -> RatioOfSprays a -> String
showRatioOfSpraysXYZ' [String]
letters a -> String
showCoef =
[String]
-> (a -> String)
-> (String, String)
-> (String, String)
-> String
-> RatioOfSprays a
-> String
forall a.
(Eq a, C a) =>
[String]
-> (a -> String)
-> (String, String)
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfSpraysXYZ [String]
letters a -> String
showCoef (String
"(", String
")") (String
"[ ", String
" ]") String
" %//% "
showRatioOfSpraysX1X2X3 :: forall a. (Eq a, AlgField.C a)
=> String
-> (a -> String)
-> (String, String)
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfSpraysX1X2X3 :: forall a.
(Eq a, C a) =>
String
-> (a -> String)
-> (String, String)
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfSpraysX1X2X3 String
letter a -> String
showCoef (String, String)
coeffBraces =
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
forall a.
(Eq a, C a) =>
((Spray a, Spray a) -> (String, String))
-> (String, String) -> String -> RatioOfSprays a -> String
showRatioOfSprays ((a -> String)
-> (String, String)
-> String
-> (Spray a, Spray a)
-> (String, String)
forall a.
(a -> String)
-> (String, String)
-> String
-> (Spray a, Spray a)
-> (String, String)
showTwoSpraysX1X2X3 a -> String
showCoef (String, String)
coeffBraces String
letter)
showRatioOfSpraysX1X2X3' :: (Eq a, AlgField.C a)
=> String
-> (a -> String)
-> RatioOfSprays a
-> String
showRatioOfSpraysX1X2X3' :: forall a.
(Eq a, C a) =>
String -> (a -> String) -> RatioOfSprays a -> String
showRatioOfSpraysX1X2X3' String
letter a -> String
showCoef =
String
-> (a -> String)
-> (String, String)
-> (String, String)
-> String
-> RatioOfSprays a
-> String
forall a.
(Eq a, C a) =>
String
-> (a -> String)
-> (String, String)
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfSpraysX1X2X3 String
letter a -> String
showCoef (String
"(", String
")") (String
"[ ", String
" ]") String
" %//% "
prettyRatioOfQSpraysXYZ ::
[String]
-> RatioOfQSprays
-> String
prettyRatioOfQSpraysXYZ :: [String] -> RatioOfSprays Rational -> String
prettyRatioOfQSpraysXYZ [String]
letters =
[String]
-> (String, String) -> String -> RatioOfSprays Rational -> String
showRatioOfQSpraysXYZ [String]
letters (String
"[ ", String
" ]") String
" %//% "
prettyRatioOfQSprays :: RatioOfQSprays -> String
prettyRatioOfQSprays :: RatioOfSprays Rational -> String
prettyRatioOfQSprays = [String] -> RatioOfSprays Rational -> String
prettyRatioOfQSpraysXYZ [String
"x", String
"y", String
"z"]
prettyRatioOfQSprays' :: RatioOfQSprays -> String
prettyRatioOfQSprays' :: RatioOfSprays Rational -> String
prettyRatioOfQSprays' = [String] -> RatioOfSprays Rational -> String
prettyRatioOfQSpraysXYZ [String
"X", String
"Y", String
"Z"]
prettyRatioOfQSpraysX1X2X3 ::
String
-> RatioOfQSprays
-> String
prettyRatioOfQSpraysX1X2X3 :: String -> RatioOfSprays Rational -> String
prettyRatioOfQSpraysX1X2X3 String
letter =
String
-> (String, String) -> String -> RatioOfSprays Rational -> String
showRatioOfQSpraysX1X2X3 String
letter (String
"[ ", String
" ]") String
" %//% "
prettyRatioOfNumSpraysXYZ :: (Num a, Ord a, AlgRing.C a, Show a)
=> [String]
-> RatioOfSprays a
-> String
prettyRatioOfNumSpraysXYZ :: forall a.
(Num a, Ord a, C a, Show a) =>
[String] -> RatioOfSprays a -> String
prettyRatioOfNumSpraysXYZ [String]
letters =
(a -> String)
-> [String]
-> (String, String)
-> String
-> RatioOfSprays a
-> String
forall a.
(Num a, Ord a, C a) =>
(a -> String)
-> [String]
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfNumSpraysXYZ a -> String
forall a. Show a => a -> String
show [String]
letters (String
"[ ", String
" ]") String
" %//% "
prettyRatioOfNumSprays ::
(Num a, Ord a, AlgRing.C a, Show a) => RatioOfSprays a -> String
prettyRatioOfNumSprays :: forall a. (Num a, Ord a, C a, Show a) => RatioOfSprays a -> String
prettyRatioOfNumSprays = [String] -> RatioOfSprays a -> String
forall a.
(Num a, Ord a, C a, Show a) =>
[String] -> RatioOfSprays a -> String
prettyRatioOfNumSpraysXYZ [String
"x", String
"y", String
"z"]
prettyRatioOfNumSprays' ::
(Num a, Ord a, AlgRing.C a, Show a) => RatioOfSprays a -> String
prettyRatioOfNumSprays' :: forall a. (Num a, Ord a, C a, Show a) => RatioOfSprays a -> String
prettyRatioOfNumSprays' = [String] -> RatioOfSprays a -> String
forall a.
(Num a, Ord a, C a, Show a) =>
[String] -> RatioOfSprays a -> String
prettyRatioOfNumSpraysXYZ [String
"X", String
"Y", String
"Z"]
prettyRatioOfNumSpraysX1X2X3 :: (Num a, Ord a, AlgRing.C a, Show a)
=> String
-> RatioOfSprays a
-> String
prettyRatioOfNumSpraysX1X2X3 :: forall a.
(Num a, Ord a, C a, Show a) =>
String -> RatioOfSprays a -> String
prettyRatioOfNumSpraysX1X2X3 String
letter =
(a -> String)
-> String
-> (String, String)
-> String
-> RatioOfSprays a
-> String
forall a.
(Num a, Ord a, C a) =>
(a -> String)
-> String
-> (String, String)
-> String
-> RatioOfSprays a
-> String
showRatioOfNumSpraysX1X2X3 a -> String
forall a. Show a => a -> String
show String
letter (String
"[ ", String
" ]") String
" %//% "
type SimpleParametricSpray a = Spray (Spray a)
type SimpleParametricQSpray = SimpleParametricSpray Rational
type ParametricSpray a = Spray (RatioOfSprays a)
type ParametricQSpray = ParametricSpray Rational
instance (Eq a, AlgRing.C a) => AlgMod.C a (SimpleParametricSpray a) where
(*>) :: a -> SimpleParametricSpray a -> SimpleParametricSpray a
a
lambda *> :: a -> SimpleParametricSpray a -> SimpleParametricSpray a
*> SimpleParametricSpray a
pspray = (HashMap Powers a -> HashMap Powers a)
-> SimpleParametricSpray a -> SimpleParametricSpray a
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map (a
lambda a -> HashMap Powers a -> HashMap Powers a
forall a v. C a v => a -> v -> v
AlgMod.*>) SimpleParametricSpray a
pspray
instance (Eq a, AlgRing.C a) => AlgRightMod.C a (SimpleParametricSpray a) where
(<*) :: SimpleParametricSpray a -> a -> SimpleParametricSpray a
SimpleParametricSpray a
pspray <* :: SimpleParametricSpray a -> a -> SimpleParametricSpray a
<* a
lambda = (HashMap Powers a -> HashMap Powers a)
-> SimpleParametricSpray a -> SimpleParametricSpray a
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map (HashMap Powers a -> a -> HashMap Powers a
forall a b. C a b => b -> a -> b
AlgRightMod.<* a
lambda) SimpleParametricSpray a
pspray
instance (Eq a, AlgField.C a) => AlgMod.C a (ParametricSpray a) where
(*>) :: a -> ParametricSpray a -> ParametricSpray a
a
lambda *> :: a -> ParametricSpray a -> ParametricSpray a
*> ParametricSpray a
pspray = (RatioOfSprays a -> RatioOfSprays a)
-> ParametricSpray a -> ParametricSpray a
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map (a
lambda a -> RatioOfSprays a -> RatioOfSprays a
forall a v. C a v => a -> v -> v
AlgMod.*>) ParametricSpray a
pspray
instance (Eq a, AlgField.C a) => AlgRightMod.C a (ParametricSpray a) where
(<*) :: ParametricSpray a -> a -> ParametricSpray a
ParametricSpray a
pspray <* :: ParametricSpray a -> a -> ParametricSpray a
<* a
lambda = (RatioOfSprays a -> RatioOfSprays a)
-> ParametricSpray a -> ParametricSpray a
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map (RatioOfSprays a -> a -> RatioOfSprays a
forall a b. C a b => b -> a -> b
AlgRightMod.<* a
lambda) ParametricSpray a
pspray
instance (Eq a, AlgField.C a) => AlgMod.C (Spray a) (ParametricSpray a) where
(*>) :: Spray a -> ParametricSpray a -> ParametricSpray a
Spray a
spray *> :: Spray a -> ParametricSpray a -> ParametricSpray a
*> ParametricSpray a
pspray = Spray a -> RatioOfSprays a
forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays Spray a
spray RatioOfSprays a -> ParametricSpray a -> ParametricSpray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ ParametricSpray a
pspray
instance (Eq a, AlgField.C a) => AlgRightMod.C (Spray a) (ParametricSpray a) where
(<*) :: ParametricSpray a -> Spray a -> ParametricSpray a
ParametricSpray a
pspray <* :: ParametricSpray a -> Spray a -> ParametricSpray a
<* Spray a
spray = Spray a -> RatioOfSprays a
forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays Spray a
spray RatioOfSprays a -> ParametricSpray a -> ParametricSpray a
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ ParametricSpray a
pspray
numberOfParameters :: HasVariables b => Spray b -> Int
numberOfParameters :: forall b. HasVariables b => Spray b -> Int
numberOfParameters Spray b
pspray =
if Spray b -> Bool
forall a. Spray a -> Bool
isZeroSpray Spray b
pspray
then Int
0
else
[Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum ((b -> Int) -> [b] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map b -> Int
forall b. HasVariables b => b -> Int
numberOfVariables (Spray b -> [b]
forall k v. HashMap k v -> [v]
HM.elems Spray b
pspray))
changeParameters :: HasVariables b => Spray b -> [VariablesType b] -> Spray b
changeParameters :: forall b. HasVariables b => Spray b -> [VariablesType b] -> Spray b
changeParameters Spray b
pspray [VariablesType b]
newParameters =
if [VariablesType b] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [VariablesType b]
newParameters Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Spray b -> Int
forall b. HasVariables b => Spray b -> Int
numberOfParameters Spray b
pspray
then
String -> Spray b
forall a. HasCallStack => String -> a
error String
"changeParameters: not enough new parameters provided."
else
(b -> b) -> Spray b -> Spray b
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map (b -> [VariablesType b] -> b
forall b. HasVariables b => b -> [VariablesType b] -> b
`changeVariables` [VariablesType b]
newParameters) Spray b
pspray
substituteParameters ::
(HasVariables b, Eq (BaseRing b), AlgAdd.C (BaseRing b))
=> Spray b
-> [BaseRing b]
-> Spray (BaseRing b)
substituteParameters :: forall b.
(HasVariables b, Eq (BaseRing b), C (BaseRing b)) =>
Spray b -> [BaseRing b] -> Spray (BaseRing b)
substituteParameters Spray b
pspray [BaseRing b]
values =
if [BaseRing b] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [BaseRing b]
values Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Spray b -> Int
forall b. HasVariables b => Spray b -> Int
numberOfParameters Spray b
pspray
then
String -> HashMap Powers (BaseRing b)
forall a. HasCallStack => String -> a
error String
"substituteParameters: not enough values provided."
else
HashMap Powers (BaseRing b) -> HashMap Powers (BaseRing b)
forall a. (C a, Eq a) => Spray a -> Spray a
removeZeroTerms (HashMap Powers (BaseRing b) -> HashMap Powers (BaseRing b))
-> HashMap Powers (BaseRing b) -> HashMap Powers (BaseRing b)
forall a b. (a -> b) -> a -> b
$ (b -> BaseRing b) -> Spray b -> HashMap Powers (BaseRing b)
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map ([BaseRing b] -> b -> BaseRing b
forall b. HasVariables b => [BaseRing b] -> b -> BaseRing b
evaluateAt [BaseRing b]
values) Spray b
pspray
evalMonomial' ::
(AlgMod.C (BaseRing b) b) => [BaseRing b] -> Monomial b -> b
evalMonomial' :: forall b. C (BaseRing b) b => [BaseRing b] -> Monomial b -> b
evalMonomial' [BaseRing b]
xs (Powers
powers, b
coeff) =
[BaseRing b] -> BaseRing b
forall a. C a => [a] -> a
AlgRing.product ((BaseRing b -> Integer -> BaseRing b)
-> [BaseRing b] -> [Integer] -> [BaseRing b]
forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
zipWith BaseRing b -> Integer -> BaseRing b
forall a. C a => a -> Integer -> a
(AlgRing.^) [BaseRing b]
xs [Integer]
pows) BaseRing b -> b -> b
forall a v. C a v => a -> v -> v
AlgMod.*> b
coeff
where
pows :: [Integer]
pows = Seq Integer -> [Integer]
forall a. Seq a -> [a]
forall (t :: * -> *) a. Foldable t => t a -> [a]
DF.toList (Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Seq Int -> Seq Integer
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Powers -> Seq Int
exponents Powers
powers)
evalParametricSpray ::
(Eq b, AlgMod.C (BaseRing b) b, AlgRing.C b)
=> Spray b -> [BaseRing b] -> b
evalParametricSpray :: forall b.
(Eq b, C (BaseRing b) b, C b) =>
Spray b -> [BaseRing b] -> b
evalParametricSpray Spray b
spray [BaseRing b]
xs = if [BaseRing b] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [BaseRing b]
xs Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>= Spray b -> Int
forall b. HasVariables b => b -> Int
numberOfVariables Spray b
spray
then [b] -> b
forall a. C a => [a] -> a
AlgAdd.sum ([b] -> b) -> [b] -> b
forall a b. (a -> b) -> a -> b
$ (Monomial b -> b) -> [Monomial b] -> [b]
forall a b. (a -> b) -> [a] -> [b]
map ([BaseRing b] -> Monomial b -> b
forall b. C (BaseRing b) b => [BaseRing b] -> Monomial b -> b
evalMonomial' [BaseRing b]
xs) (Spray b -> [Monomial b]
forall k v. HashMap k v -> [(k, v)]
HM.toList Spray b
spray)
else String -> b
forall a. HasCallStack => String -> a
error String
"evalParametricSpray: not enough values provided."
canCoerceToSimpleParametricSpray ::
(Eq a, AlgRing.C a) => ParametricSpray a -> Bool
canCoerceToSimpleParametricSpray :: forall a. (Eq a, C a) => ParametricSpray a -> Bool
canCoerceToSimpleParametricSpray ParametricSpray a
spray =
(RatioOfSprays a -> Bool) -> [RatioOfSprays a] -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
all RatioOfSprays a -> Bool
forall a. (Eq a, C a) => RatioOfSprays a -> Bool
isPolynomialRatioOfSprays (ParametricSpray a -> [RatioOfSprays a]
forall k v. HashMap k v -> [v]
HM.elems ParametricSpray a
spray)
asSimpleParametricSprayUnsafe :: ParametricSpray a -> SimpleParametricSpray a
asSimpleParametricSprayUnsafe :: forall a. ParametricSpray a -> SimpleParametricSpray a
asSimpleParametricSprayUnsafe = (RatioOfSprays a -> Spray a)
-> HashMap Powers (RatioOfSprays a) -> HashMap Powers (Spray a)
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map RatioOfSprays a -> Spray a
forall a. RatioOfSprays a -> Spray a
_numerator
asSimpleParametricSpray ::
(Eq a, AlgRing.C a) => ParametricSpray a -> SimpleParametricSpray a
asSimpleParametricSpray :: forall a.
(Eq a, C a) =>
ParametricSpray a -> SimpleParametricSpray a
asSimpleParametricSpray ParametricSpray a
spray =
if ParametricSpray a -> Bool
forall a. (Eq a, C a) => ParametricSpray a -> Bool
canCoerceToSimpleParametricSpray ParametricSpray a
spray
then ParametricSpray a -> SimpleParametricSpray a
forall a. ParametricSpray a -> SimpleParametricSpray a
asSimpleParametricSprayUnsafe ParametricSpray a
spray
else String -> SimpleParametricSpray a
forall a. HasCallStack => String -> a
error (String -> SimpleParametricSpray a)
-> String -> SimpleParametricSpray a
forall a b. (a -> b) -> a -> b
$
String
"asSimpleParametricSpray: this parametric spray is not coercable" String -> ShowS
forall a. [a] -> [a] -> [a]
++
String
" to a simple parametric spray."
fromOneParameterSpray ::
(Eq a, AlgRing.C a) => OneParameterSpray a -> ParametricSpray a
fromOneParameterSpray :: forall a. (Eq a, C a) => OneParameterSpray a -> ParametricSpray a
fromOneParameterSpray = (RatioOfPolynomials a -> RatioOfSprays a)
-> HashMap Powers (RatioOfPolynomials a)
-> HashMap Powers (RatioOfSprays a)
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map RatioOfPolynomials a -> RatioOfSprays a
forall a. (Eq a, C a) => RatioOfPolynomials a -> RatioOfSprays a
fromRatioOfPolynomials
fromOneParameterQSpray :: OneParameterQSpray -> ParametricQSpray
fromOneParameterQSpray :: OneParameterQSpray -> Spray (RatioOfSprays Rational)
fromOneParameterQSpray = (RatioOfQPolynomials -> RatioOfSprays Rational)
-> OneParameterQSpray -> Spray (RatioOfSprays Rational)
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map RatioOfQPolynomials -> RatioOfSprays Rational
fromRatioOfQPolynomials
parametricSprayToOneParameterSpray ::
forall a. (AlgRing.C a) => ParametricSpray a -> OneParameterSpray a
parametricSprayToOneParameterSpray :: forall a. C a => ParametricSpray a -> OneParameterSpray a
parametricSprayToOneParameterSpray = (RatioOfSprays a -> RatioOfPolynomials a)
-> HashMap Powers (RatioOfSprays a)
-> HashMap Powers (RatioOfPolynomials a)
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map RatioOfSprays a -> RatioOfPolynomials a
toRatioOfPolynomials
where
toRatioOfPolynomials :: RatioOfSprays a -> RatioOfPolynomials a
toRatioOfPolynomials :: RatioOfSprays a -> RatioOfPolynomials a
toRatioOfPolynomials (RatioOfSprays Spray a
p Spray a
q) =
Spray a -> Polynomial a
toPolynomial Spray a
p Polynomial a -> Polynomial a -> RatioOfPolynomials a
forall a. a -> a -> T a
:% Spray a -> Polynomial a
toPolynomial Spray a
q
where
toPolynomial :: Spray a -> Polynomial a
toPolynomial :: Spray a -> Polynomial a
toPolynomial Spray a
spray = [a] -> Polynomial a
forall a. [a] -> Polynomial a
polyFromCoeffs [a]
coeffs
where
coeffs :: [a]
coeffs = (Int -> a) -> [Int] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> [Int] -> Spray a -> a
forall a. C a => [Int] -> Spray a -> a
getCoefficient [Int
i] Spray a
spray) [Int
0 .. Int
deg]
deg :: Int
deg = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum (Int
0 Int -> [Int] -> [Int]
forall a. a -> [a] -> [a]
: (Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) [Seq Int]
expnts)
powers :: [Powers]
powers = Spray a -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray a
spray
expnts :: [Seq Int]
expnts = (Seq Int -> Bool) -> [Seq Int] -> [Seq Int]
forall a. (a -> Bool) -> [a] -> [a]
filter (Bool -> Bool
not (Bool -> Bool) -> (Seq Int -> Bool) -> Seq Int -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Seq Int -> Bool
forall a. Seq a -> Bool
S.null) ((Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
powers)
parametricQSprayToOneParameterQSpray :: ParametricQSpray -> OneParameterQSpray
parametricQSprayToOneParameterQSpray :: Spray (RatioOfSprays Rational) -> OneParameterQSpray
parametricQSprayToOneParameterQSpray = (RatioOfSprays Rational -> RatioOfQPolynomials)
-> Spray (RatioOfSprays Rational) -> OneParameterQSpray
forall v1 v2 k. (v1 -> v2) -> HashMap k v1 -> HashMap k v2
HM.map RatioOfSprays Rational -> RatioOfQPolynomials
toRatioOfQPolynomials
where
toRatioOfQPolynomials :: RatioOfQSprays -> RatioOfQPolynomials
toRatioOfQPolynomials :: RatioOfSprays Rational -> RatioOfQPolynomials
toRatioOfQPolynomials (RatioOfSprays Spray Rational
p Spray Rational
q) =
Spray Rational -> QPolynomial
toQPolynomial Spray Rational
p QPolynomial -> QPolynomial -> RatioOfQPolynomials
forall a. a -> a -> T a
:% Spray Rational -> QPolynomial
toQPolynomial Spray Rational
q
where
toQPolynomial :: QSpray -> QPolynomial
toQPolynomial :: Spray Rational -> QPolynomial
toQPolynomial Spray Rational
spray = [Rational'] -> QPolynomial
forall a. [a] -> Polynomial a
polyFromCoeffs [Rational']
coeffs'
where
coeffs' :: [Rational']
coeffs' = (Int -> Rational') -> [Int] -> [Rational']
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> Rational -> Rational'
f ([Int] -> Spray Rational -> Rational
forall a. C a => [Int] -> Spray a -> a
getCoefficient [Int
i] Spray Rational
spray)) [Int
0 .. Int
deg]
f :: Rational -> Rational'
f :: Rational -> Rational'
f Rational
r = Rational -> Integer
forall a. Ratio a -> a
DR.numerator Rational
r Integer -> Integer -> Rational'
forall a. a -> a -> T a
:% Rational -> Integer
forall a. Ratio a -> a
DR.denominator Rational
r
deg :: Int
deg = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum (Int
0 Int -> [Int] -> [Int]
forall a. a -> [a] -> [a]
: (Seq Int -> Int) -> [Seq Int] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (Seq Int -> Int -> Int
forall a. Seq a -> Int -> a
`index` Int
0) [Seq Int]
expnts)
powers :: [Powers]
powers = Spray Rational -> [Powers]
forall k v. HashMap k v -> [k]
HM.keys Spray Rational
spray
expnts :: [Seq Int]
expnts = (Seq Int -> Bool) -> [Seq Int] -> [Seq Int]
forall a. (a -> Bool) -> [a] -> [a]
filter (Bool -> Bool
not (Bool -> Bool) -> (Seq Int -> Bool) -> Seq Int -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Seq Int -> Bool
forall a. Seq a -> Bool
S.null) ((Powers -> Seq Int) -> [Powers] -> [Seq Int]
forall a b. (a -> b) -> [a] -> [b]
map Powers -> Seq Int
exponents [Powers]
powers)
gegenbauerPolynomial :: Int -> SimpleParametricQSpray
gegenbauerPolynomial :: Int -> SimpleParametricQSpray
gegenbauerPolynomial Int
n
| Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = SimpleParametricQSpray
forall a. C a => Spray a
unitSpray
| Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 = (Int
2Int -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Int -> a -> a
.^Spray Rational
a) Spray Rational -> SimpleParametricQSpray -> SimpleParametricQSpray
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ SimpleParametricQSpray
x
| Bool
otherwise =
(Int
2Int -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Int -> a -> a
.^(Spray Rational
n'' Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray Rational
a) Spray Rational -> Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> a -> Spray a
/^ Rational
n') Spray Rational -> SimpleParametricQSpray -> SimpleParametricQSpray
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ (SimpleParametricQSpray
x SimpleParametricQSpray
-> SimpleParametricQSpray -> SimpleParametricQSpray
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Int -> SimpleParametricQSpray
gegenbauerPolynomial (Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
1))
SimpleParametricQSpray
-> SimpleParametricQSpray -> SimpleParametricQSpray
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ ((Spray Rational
n'' Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Int
2Int -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Int -> a -> a
.^Spray Rational
a Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray Rational
forall a. C a => Spray a
unitSpray) Spray Rational -> Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> a -> Spray a
/^ Rational
n') Spray Rational -> SimpleParametricQSpray -> SimpleParametricQSpray
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ Int -> SimpleParametricQSpray
gegenbauerPolynomial (Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
2)
where
x :: SimpleParametricQSpray
x = Int -> SimpleParametricQSpray
forall a. C a => Int -> Spray a
lone Int
1 :: SimpleParametricQSpray
a :: Spray Rational
a = Int -> Spray Rational
forall a. C a => Int -> Spray a
lone Int
1 :: QSpray
n' :: Rational
n' = Int -> Rational
forall a. Real a => a -> Rational
toRational Int
n
n'' :: Spray Rational
n'' = Rational -> Spray Rational
forall a. (C a, Eq a) => a -> Spray a
constantSpray (Rational
n' Rational -> Rational -> Rational
forall a. Num a => a -> a -> a
- Rational
1)
jacobiPolynomial :: Int -> ParametricQSpray
jacobiPolynomial :: Int -> Spray (RatioOfSprays Rational)
jacobiPolynomial Int
n
| Int
n Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
0 = String -> Spray (RatioOfSprays Rational)
forall a. HasCallStack => String -> a
error String
"jacobiPolynomial: `n` must be positive."
| Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = Spray (RatioOfSprays Rational)
forall a. C a => Spray a
unitSpray
| Int
n Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 =
Spray Rational -> Spray (RatioOfSprays Rational)
asParametricQSpray (Spray Rational
alpha0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Rational -> Spray Rational
cst Rational
1) Spray (RatioOfSprays Rational)
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^
(Spray Rational -> RatioOfSprays Rational
forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays ((Spray Rational
alpha0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray Rational
beta0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Rational -> Spray Rational
cst Rational
2) Spray Rational -> Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> a -> Spray a
/^ Rational
2) RatioOfSprays Rational
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^
(Spray (RatioOfSprays Rational)
x Spray (RatioOfSprays Rational)
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray (RatioOfSprays Rational)
forall a. C a => Spray a
unitSpray))
| Bool
otherwise =
Spray (RatioOfSprays Rational)
lambda1 Spray (RatioOfSprays Rational)
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Int -> Spray (RatioOfSprays Rational)
jacobiPolynomial (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
1) Spray (RatioOfSprays Rational)
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray (RatioOfSprays Rational)
lambda2 Spray (RatioOfSprays Rational)
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^ Int -> Spray (RatioOfSprays Rational)
jacobiPolynomial (Int
nInt -> Int -> Int
forall a. Num a => a -> a -> a
-Int
2)
where
cst :: Rational -> QSpray
cst :: Rational -> Spray Rational
cst = Rational -> Spray Rational
forall a. (C a, Eq a) => a -> Spray a
constantSpray
alpha0 :: Spray Rational
alpha0 = Int -> Spray Rational
qlone Int
1
beta0 :: Spray Rational
beta0 = Int -> Spray Rational
qlone Int
2
x :: Spray (RatioOfSprays Rational)
x = Int -> Spray (RatioOfSprays Rational)
forall a. C a => Int -> Spray a
lone Int
1 :: ParametricQSpray
n0 :: Spray Rational
n0 = Rational -> Spray Rational
cst (Int -> Rational
forall a. Real a => a -> Rational
toRational Int
n)
a0 :: Spray Rational
a0 = Spray Rational
n0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray Rational
alpha0
b0 :: Spray Rational
b0 = Spray Rational
n0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray Rational
beta0
c0 :: Spray Rational
c0 = Spray Rational
a0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^ Spray Rational
b0
asParametricQSpray :: QSpray -> ParametricQSpray
asParametricQSpray :: Spray Rational -> Spray (RatioOfSprays Rational)
asParametricQSpray = RatioOfSprays Rational -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => a -> Spray a
constantSpray (RatioOfSprays Rational -> Spray (RatioOfSprays Rational))
-> (Spray Rational -> RatioOfSprays Rational)
-> Spray Rational
-> Spray (RatioOfSprays Rational)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Spray Rational -> RatioOfSprays Rational
forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays
lambda0 :: RatioOfSprays Rational
lambda0 = Spray Rational -> RatioOfSprays Rational
forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays (Spray Rational -> RatioOfSprays Rational)
-> Spray Rational -> RatioOfSprays Rational
forall a b. (a -> b) -> a -> b
$ Int
2Int -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Int -> a -> a
.^(Spray Rational
n0Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^(Spray Rational
c0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray Rational
n0)Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^(Spray Rational
c0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Rational -> Spray Rational
cst Rational
2))
lambda1 :: Spray (RatioOfSprays Rational)
lambda1 = (Spray Rational -> RatioOfSprays Rational
forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays (Spray Rational
c0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Rational -> Spray Rational
cst Rational
1) RatioOfSprays Rational
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a v. C a v => a -> v -> v
AlgMod.*>
((Spray Rational -> RatioOfSprays Rational
forall a. C a => Spray a -> RatioOfSprays a
asRatioOfSprays (Spray Rational
c0Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^(Spray Rational
c0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Rational -> Spray Rational
cst Rational
2)) RatioOfSprays Rational
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => a -> Spray a -> Spray a
*^ Spray (RatioOfSprays Rational)
x ) Spray (RatioOfSprays Rational)
-> Spray (RatioOfSprays Rational) -> Spray (RatioOfSprays Rational)
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^+^
Spray Rational -> Spray (RatioOfSprays Rational)
asParametricQSpray ((Spray Rational
a0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Spray Rational
b0)Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^(Spray Rational
c0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Int
2Int -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Int -> a -> a
.^Spray Rational
n0)))) Spray (RatioOfSprays Rational)
-> RatioOfSprays Rational -> Spray (RatioOfSprays Rational)
forall k a. (C k, C k a) => a -> k -> a
/> RatioOfSprays Rational
lambda0
lambda2 :: Spray (RatioOfSprays Rational)
lambda2 =
Spray Rational -> Spray (RatioOfSprays Rational)
asParametricQSpray (Int
2Int -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Int -> a -> a
.^((Spray Rational
a0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Rational -> Spray Rational
cst Rational
1)Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^(Spray Rational
b0 Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^-^ Rational -> Spray Rational
cst Rational
1)Spray Rational -> Spray Rational -> Spray Rational
forall a. (C a, Eq a) => Spray a -> Spray a -> Spray a
^*^Spray Rational
c0)) Spray (RatioOfSprays Rational)
-> RatioOfSprays Rational -> Spray (RatioOfSprays Rational)
forall k a. (C k, C k a) => a -> k -> a
/> RatioOfSprays Rational
lambda0
prettyParametricQSprayABCXYZ ::
[String]
-> [String]
-> ParametricQSpray
-> String
prettyParametricQSprayABCXYZ :: [String] -> [String] -> Spray (RatioOfSprays Rational) -> String
prettyParametricQSprayABCXYZ [String]
abc [String]
xyz Spray (RatioOfSprays Rational)
spray =
(RatioOfSprays Rational -> String)
-> (String, String)
-> ([Seq Int] -> [String])
-> Spray (RatioOfSprays Rational)
-> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray RatioOfSprays Rational -> String
rOSShower (String
"{ ", String
" }") ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String]
xyz) Spray (RatioOfSprays Rational)
spray
where
rOSShower :: RatioOfSprays Rational -> String
rOSShower = if Spray (RatioOfSprays Rational) -> Int
forall b. HasVariables b => Spray b -> Int
numberOfParameters Spray (RatioOfSprays Rational)
spray Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= [String] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [String]
abc
then [String] -> RatioOfSprays Rational -> String
prettyRatioOfQSpraysXYZ [String]
abc
else String -> RatioOfSprays Rational -> String
prettyRatioOfQSpraysX1X2X3 ([String]
abc [String] -> Int -> String
forall a. HasCallStack => [a] -> Int -> a
!! Int
0)
prettyParametricQSpray :: ParametricQSpray -> String
prettyParametricQSpray :: Spray (RatioOfSprays Rational) -> String
prettyParametricQSpray =
(RatioOfSprays Rational -> String)
-> (String, String)
-> ([Seq Int] -> [String])
-> Spray (RatioOfSprays Rational)
-> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray (String -> RatioOfSprays Rational -> String
prettyRatioOfQSpraysX1X2X3 String
"a") (String
"{ ", String
" }") ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String
"X", String
"Y", String
"Z"])
prettySimpleParametricQSprayABCXYZ ::
[String]
-> [String]
-> SimpleParametricQSpray
-> String
prettySimpleParametricQSprayABCXYZ :: [String] -> [String] -> SimpleParametricQSpray -> String
prettySimpleParametricQSprayABCXYZ [String]
abc [String]
xyz SimpleParametricQSpray
spray =
(Spray Rational -> String)
-> (String, String)
-> ([Seq Int] -> [String])
-> SimpleParametricQSpray
-> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray Spray Rational -> String
sprayShower (String
"{ ", String
" }") ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String]
xyz) SimpleParametricQSpray
spray
where
sprayShower :: Spray Rational -> String
sprayShower = if SimpleParametricQSpray -> Int
forall b. HasVariables b => Spray b -> Int
numberOfParameters SimpleParametricQSpray
spray Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= [String] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [String]
abc
then [String] -> Spray Rational -> String
prettyQSprayXYZ [String]
abc
else String -> Spray Rational -> String
prettyQSprayX1X2X3 ([String]
abc [String] -> Int -> String
forall a. HasCallStack => [a] -> Int -> a
!! Int
0)
prettySimpleParametricQSpray :: SimpleParametricQSpray -> String
prettySimpleParametricQSpray :: SimpleParametricQSpray -> String
prettySimpleParametricQSpray =
(Spray Rational -> String)
-> (String, String)
-> ([Seq Int] -> [String])
-> SimpleParametricQSpray
-> String
forall a.
(a -> String)
-> (String, String) -> ([Seq Int] -> [String]) -> Spray a -> String
showSpray (String -> Spray Rational -> String
prettyQSprayX1X2X3 String
"a") (String
"{ ", String
" }") ([String] -> [Seq Int] -> [String]
showMonomialsXYZ [String
"X", String
"Y", String
"Z"])