Math.Algebra.Polynomial.Misc

Some auxilary functions used internally

Documentation

equating :: Eq b => (a -> b) -> a -> a -> Bool

Lists

unique :: Ord a => [a] -> [a]

count :: Ord b => [b] -> Map b Integer

Synonym for histogram

histogram :: Ord b => [b] -> Map b Integer

sum' :: Num a => [a] -> a

longZipWith :: (a -> c) -> (b -> c) -> (a -> b -> c) -> [a] -> [b] -> [c]

longReplaceListElem :: a -> Int -> a -> [a] -> [a]

Maps

deleteLookup :: Ord a => a -> Map a b -> (Maybe b, Map a b)

unsafeDeleteLookup :: Ord a => a -> Map a b -> (b, Map a b)

insertMap :: Ord k => (b -> a) -> (b -> a -> a) -> k -> b -> Map k a -> Map k a

Example usage: insertMap (:[]) (:) ...

buildMap :: Ord k => (b -> a) -> (b -> a -> a) -> [(k, b)] -> Map k a

Example usage: buildMap (:[]) (:) ...

Signs

data Sign

Constructors

Plus

Minus

Instances

Eq Sign

Defined in Math.Algebra.Polynomial.Misc

Methods

(==) :: Sign -> Sign -> Bool

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

Ord Sign

Defined in Math.Algebra.Polynomial.Misc

Methods

compare :: Sign -> Sign -> Ordering

(<) :: Sign -> Sign -> Bool

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

(>) :: Sign -> Sign -> Bool

(>=) :: Sign -> Sign -> Bool

max :: Sign -> Sign -> Sign

min :: Sign -> Sign -> Sign

Read Sign

Defined in Math.Algebra.Polynomial.Misc

Methods

readsPrec :: Int -> ReadS Sign

readList :: ReadS [Sign]

readPrec :: ReadPrec Sign

readListPrec :: ReadPrec [Sign]

Show Sign

Defined in Math.Algebra.Polynomial.Misc

Methods

showsPrec :: Int -> Sign -> ShowS

show :: Sign -> String

showList :: [Sign] -> ShowS

Semigroup Sign

Defined in Math.Algebra.Polynomial.Misc

Methods

(<>) :: Sign -> Sign -> Sign

sconcat :: NonEmpty Sign -> Sign

stimes :: Integral b => b -> Sign -> Sign

Monoid Sign

Defined in Math.Algebra.Polynomial.Misc

Methods

mempty :: Sign

mappend :: Sign -> Sign -> Sign

mconcat :: [Sign] -> Sign

oppositeSign :: Sign -> Sign

mulSign :: Sign -> Sign -> Sign

productOfSigns :: [Sign] -> Sign

negateIfOdd :: (Integral a, Num b) => a -> b -> b

Negate the second argument if the first is odd

signValue :: Num a => Sign -> a

+1 or -1

signed :: Num a => Sign -> a -> a

Negate the second argument if the first is Minus

class IsSigned a where

Methods

signOf :: a -> Maybe Sign

Instances

IsSigned Int

Defined in Math.Algebra.Polynomial.Misc

Methods

signOf :: Int -> Maybe Sign

IsSigned Integer

Defined in Math.Algebra.Polynomial.Misc

Methods

signOf :: Integer -> Maybe Sign

IsSigned Rational

Defined in Math.Algebra.Polynomial.Misc

Methods

signOf :: Rational -> Maybe Sign

IsSigned (Poly c v)

Defined in Math.Algebra.Polynomial.Multivariate.Infinite

Methods

signOf :: Poly c v -> Maybe Sign

IsSigned (Poly c v)

Defined in Math.Algebra.Polynomial.Multivariate.Generic

Methods

signOf :: Poly c v -> Maybe Sign

IsSigned (Univariate c v)

Defined in Math.Algebra.Polynomial.Univariate

Methods

signOf :: Univariate c v -> Maybe Sign

IsSigned (Poly c v n)

Defined in Math.Algebra.Polynomial.Multivariate.Indexed

Methods

signOf :: Poly c v n -> Maybe Sign

IsSigned (Poly c v n)

Defined in Math.Algebra.Polynomial.Multivariate.Compact

Methods

signOf :: Poly c v n -> Maybe Sign

IsSigned (ExtAlg c v n)

Defined in Math.Algebra.Polynomial.Exterior.Indexed

Methods

signOf :: ExtAlg c v n -> Maybe Sign

signOfNum :: (Ord a, Num a) => a -> Maybe Sign

Numbers

fromRat :: Rational -> Integer

safeDiv :: Integer -> Integer -> Integer

Basic number theory

factorial :: Integral a => a -> Integer

A000142.

binomial :: Integral a => a -> a -> Integer

A007318. Note: This is zero for n<0 or k<0; see also signedBinomial below.

moebiusMu :: Num c => Int -> c

divisors :: Int -> [Int]

squareFreeDivisors :: Int -> [(Int, Sign)]

Square-free divisors together with their Mobius mu value

primes :: [Integer]

List of primes, using tree merge with wheel. Code by Will Ness.

groupIntegerFactors :: [Integer] -> [(Integer, Int)]

Groups integer factors. Example: from [2,2,2,3,3,5] we produce [(2,3),(3,2),(5,1)]

integerFactorsTrialDivision :: Integer -> [Integer]

The naive trial division algorithm.

Basic combinatorics

tuples' :: [Int] -> [[Int]]

sublists :: [a] -> [[a]]

All sublists of a list.

Integer-indexed cache

intCache :: ((Int -> a) -> Int -> a) -> Int -> a

newtype ITable a

Constructors

ITable [Array Int a]

mkITable :: [a] -> ITable a

lkpITable :: Int -> ITable a -> a

Stirling numbers

signedStirling1stArray :: Integral a => a -> Array Int Integer

Rows of (signed) Stirling numbers of the first kind. OEIS:A008275. Coefficients of the polinomial (x-1)*(x-2)*...*(x-n+1). This function uses the recursion formula.

stirling2nd :: Integral a => a -> a -> Integer

Stirling numbers of the second kind. OEIS:A008277. This function uses an explicit formula.

Argument order: stirling2nd n k