Safe Haskell	None
Language	Haskell2010

Algebra.LinkedMatrix

Synopsis

data Matrix a
toLists :: forall a. Monoidal a => Matrix a -> [[a]]
fromLists :: DecidableZero a => [[a]] -> Matrix a
fromList :: DecidableZero a => [((Int, Int), a)] -> Matrix a
swapRows :: Int -> Int -> Matrix a -> Matrix a
identity :: Unital a => Int -> Matrix a
nonZeroRows :: Matrix r -> [Int]
nonZeroCols :: Matrix r -> [Int]
swapCols :: Int -> Int -> Matrix a -> Matrix a
switchCols :: Int -> Int -> Matrix a -> Matrix a
switchRows :: Int -> Int -> Matrix a -> Matrix a
addRow :: DecidableZero a => Vector a -> Int -> Matrix a -> Matrix a
addCol :: DecidableZero a => Vector a -> Int -> Matrix a -> Matrix a
ncols :: Matrix a -> Int
nrows :: Matrix a -> Int
getRow :: Monoidal a => Int -> Matrix a -> Vector a
getCol :: Monoidal a => Int -> Matrix a -> Vector a
triangulateModular :: Matrix (Fraction Integer) -> (Matrix (Fraction Integer), Matrix (Fraction Integer))
scaleRow :: (DecidableZero a, Multiplicative a) => a -> Int -> Matrix a -> Matrix a
combineRows :: (DecidableZero a, Multiplicative a) => a -> Int -> Int -> Matrix a -> Matrix a
combineCols :: (DecidableZero a, Multiplicative a) => a -> Int -> Int -> Matrix a -> Matrix a
transpose :: Matrix a -> Matrix a
inBound :: (Int, Int) -> Matrix a -> Bool
height :: forall a. Lens' (Matrix a) Int
width :: forall a. Lens' (Matrix a) Int
cmap :: DecidableZero a => (a1 -> a) -> Matrix a1 -> Matrix a
empty :: Matrix a
rowVector :: DecidableZero a => Vector a -> Matrix a
colVector :: DecidableZero a => Vector a -> Matrix a
rowCount :: Int -> Matrix a -> Int
colCount :: Int -> Matrix a -> Int
traverseRow :: b -> (b -> Int -> Entry a -> b) -> Int -> Matrix a -> b
traverseCol :: b -> (b -> Int -> Entry a -> b) -> Int -> Matrix a -> b
data Entry a
idx :: forall a. Lens' (Entry a) (Int, Int)
value :: forall a a. Lens (Entry a) (Entry a) a a
substMatrix :: CoeffRing r => Matrix r -> Polynomial r 1 -> Matrix r
catRow :: DecidableZero b => Matrix b -> Vector b -> Matrix b
catCol :: DecidableZero b => Matrix b -> Vector b -> Matrix b
(<||>) :: DecidableZero b => Matrix b -> Matrix b -> Matrix b
(<-->) :: DecidableZero b => Matrix b -> Matrix b -> Matrix b
toRows :: Monoidal a => Matrix a -> [Vector a]
toCols :: Monoidal a => Matrix a -> [Vector a]
zeroMat :: Int -> Int -> Matrix a
getDiag :: Monoidal a => Matrix a -> Vector a
trace :: Monoidal c => Matrix c -> c
diagProd :: (Unital c, Monoidal c) => Matrix c -> c
diag :: DecidableZero a => Vector a -> Matrix a
scaleCol :: (DecidableZero a, Multiplicative a) => a -> Int -> Matrix a -> Matrix a
clearRow :: Int -> Matrix a -> Matrix a
clearCol :: Int -> Matrix a -> Matrix a
index :: Monoidal a => Key -> Int -> Matrix a -> Maybe a
(!) :: Monoidal a => Matrix a -> (Int, Int) -> a
nonZeroEntries :: Matrix a -> Vector ((Int, Int), a)
rankLM :: (DecidableZero r, Division r, Group r) => Matrix r -> Int
splitIndependentDirs :: (DecidableZero a, Field a) => Direction -> Matrix a -> (Matrix a, [Int], [Int])
structuredGauss :: (DecidableZero a, Division a, Group a) => Matrix a -> (Matrix a, Matrix a)
multWithVector :: (Multiplicative a, Monoidal a) => Matrix a -> Vector a -> Vector a
solveWiedemann :: (Eq a, Field a, DecidableZero a, DecidableUnits a, ZeroProductSemiring a, Random a, MonadRandom m) => Matrix a -> Vector a -> m (Either (Vector a) (Vector a))
henselLift :: Integer -> Matrix Integer -> Matrix Integer -> Vector Integer -> [Vector Integer]
solveHensel :: Int -> Integer -> Matrix (Fraction Integer) -> Vector (Fraction Integer) -> Maybe (Vector (Fraction Integer))
structuredGauss' :: (DecidableZero a, Division a, Group a) => Matrix a -> (Matrix a, Matrix a, a)
intDet :: Matrix Integer -> Integer

Documentation

data Matrix a Source #

Instances

Instances details

Matrix Matrix Source #
Instance details Defined in Algebra.Matrix Associated Types type Elem Matrix a Source # Methods cmap :: (Elem Matrix a, Elem Matrix b) => (a -> b) -> Matrix a -> Matrix b Source # empty :: Elem Matrix b => Matrix b Source # fromLists :: Elem Matrix a => [[a]] -> Matrix a Source # fromCols :: Elem Matrix a => [Vector a] -> Matrix a Source # fromRows :: Elem Matrix a => [Vector a] -> Matrix a Source # toCols :: Elem Matrix a => Matrix a -> [Vector a] Source # toRows :: Elem Matrix a => Matrix a -> [Vector a] Source # ncols :: Matrix a -> Int Source # nrows :: Matrix a -> Int Source # identity :: Elem Matrix a => Int -> Matrix a Source # diag :: Elem Matrix a => Vector a -> Matrix a Source # getDiag :: Elem Matrix a => Matrix a -> Vector a Source # trace :: Elem Matrix a => Matrix a -> a Source # diagProd :: Elem Matrix a => Matrix a -> a Source # zero :: Elem Matrix a => Int -> Int -> Matrix a Source # colVector :: Elem Matrix a => Vector a -> Matrix a Source # rowVector :: Elem Matrix a => Vector a -> Matrix a Source # getCol :: Elem Matrix a => Int -> Matrix a -> Vector a Source # getRow :: Elem Matrix a => Int -> Matrix a -> Vector a Source # switchRows :: Elem Matrix a => Int -> Int -> Matrix a -> Matrix a Source # scaleRow :: Elem Matrix a => a -> Int -> Matrix a -> Matrix a Source # combineRows :: Elem Matrix a => Int -> a -> Int -> Matrix a -> Matrix a Source # trans :: Elem Matrix a => Matrix a -> Matrix a Source # buildMatrix :: Elem Matrix a => Int -> Int -> ((Int, Int) -> a) -> Matrix a Source # index :: Elem Matrix a => Int -> Int -> Matrix a -> Maybe a Source # (!) :: Elem Matrix a => Matrix a -> (Int, Int) -> a Source # (<\|\|>) :: Elem Matrix a => Matrix a -> Matrix a -> Matrix a Source # (<-->) :: Elem Matrix a => Matrix a -> Matrix a -> Matrix a Source # nonZeroRows :: (DecidableZero a, Elem Matrix a) => Matrix a -> [Int] Source # nonZeroCols :: (DecidableZero a, Elem Matrix a) => Matrix a -> [Int] Source #
(DecidableZero r, RightModule Integer r) => RightModule Integer (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (*.) :: Matrix r -> Integer -> Matrix r #
(DecidableZero r, RightModule Natural r) => RightModule Natural (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (*.) :: Matrix r -> Natural -> Matrix r #
(DecidableZero r, LeftModule Integer r) => LeftModule Integer (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (.*) :: Integer -> Matrix r -> Matrix r #
(DecidableZero r, LeftModule Natural r) => LeftModule Natural (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (.*) :: Natural -> Matrix r -> Matrix r #
Eq a => Eq (Matrix a) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (==) :: Matrix a -> Matrix a -> Bool # (/=) :: Matrix a -> Matrix a -> Bool #
Read a => Read (Matrix a) Source #
Instance details Defined in Algebra.LinkedMatrix Methods readsPrec :: Int -> ReadS (Matrix a) # readList :: ReadS [Matrix a] # readPrec :: ReadPrec (Matrix a) # readListPrec :: ReadPrec [Matrix a] #
Show a => Show (Matrix a) Source #
Instance details Defined in Algebra.LinkedMatrix Methods showsPrec :: Int -> Matrix a -> ShowS # show :: Matrix a -> String # showList :: [Matrix a] -> ShowS #
(DecidableZero r, Semiring r) => Semiring (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix
(DecidableZero r, Multiplicative r) => Multiplicative (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (*) :: Matrix r -> Matrix r -> Matrix r # pow1p :: Matrix r -> Natural -> Matrix r # productWith1 :: Foldable1 f => (a -> Matrix r) -> f a -> Matrix r #
DecidableZero r => Monoidal (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods zero :: Matrix r # sinnum :: Natural -> Matrix r -> Matrix r # sumWith :: Foldable f => (a -> Matrix r) -> f a -> Matrix r #
(DecidableZero r, Group r) => Group (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (-) :: Matrix r -> Matrix r -> Matrix r # negate :: Matrix r -> Matrix r # subtract :: Matrix r -> Matrix r -> Matrix r # times :: Integral n => n -> Matrix r -> Matrix r #
DecidableZero r => Additive (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (+) :: Matrix r -> Matrix r -> Matrix r # sinnum1p :: Natural -> Matrix r -> Matrix r # sumWith1 :: Foldable1 f => (a -> Matrix r) -> f a -> Matrix r #
(DecidableZero r, Abelian r) => Abelian (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix
(DecidableZero r, Semiring r, Multiplicative r) => RightModule (Scalar r) (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (*.) :: Matrix r -> Scalar r -> Matrix r #
(DecidableZero r, Semiring r, Multiplicative r) => LeftModule (Scalar r) (Matrix r) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (.*) :: Scalar r -> Matrix r -> Matrix r #
type Elem Matrix a Source #
Instance details Defined in Algebra.Matrix type Elem Matrix a = (CoeffRing a, Unital a, Monoidal a, Multiplicative a, Additive a)

toLists :: forall a. Monoidal a => Matrix a -> [[a]] Source #

fromLists :: DecidableZero a => [[a]] -> Matrix a Source #

fromList :: DecidableZero a => [((Int, Int), a)] -> Matrix a Source #

swapRows :: Int -> Int -> Matrix a -> Matrix a Source #

identity :: Unital a => Int -> Matrix a Source #

nonZeroRows :: Matrix r -> [Int] Source #

nonZeroCols :: Matrix r -> [Int] Source #

swapCols :: Int -> Int -> Matrix a -> Matrix a Source #

switchCols :: Int -> Int -> Matrix a -> Matrix a Source #

switchRows :: Int -> Int -> Matrix a -> Matrix a Source #

addRow :: DecidableZero a => Vector a -> Int -> Matrix a -> Matrix a Source #

addCol :: DecidableZero a => Vector a -> Int -> Matrix a -> Matrix a Source #

ncols :: Matrix a -> Int Source #

nrows :: Matrix a -> Int Source #

getRow :: Monoidal a => Int -> Matrix a -> Vector a Source #

getCol :: Monoidal a => Int -> Matrix a -> Vector a Source #

triangulateModular :: Matrix (Fraction Integer) -> (Matrix (Fraction Integer), Matrix (Fraction Integer)) Source #

scaleRow :: (DecidableZero a, Multiplicative a) => a -> Int -> Matrix a -> Matrix a Source #

combineRows :: (DecidableZero a, Multiplicative a) => a -> Int -> Int -> Matrix a -> Matrix a Source #

combineCols :: (DecidableZero a, Multiplicative a) => a -> Int -> Int -> Matrix a -> Matrix a Source #

transpose :: Matrix a -> Matrix a Source #

inBound :: (Int, Int) -> Matrix a -> Bool Source #

height :: forall a. Lens' (Matrix a) Int Source #

width :: forall a. Lens' (Matrix a) Int Source #

cmap :: DecidableZero a => (a1 -> a) -> Matrix a1 -> Matrix a Source #

empty :: Matrix a Source #

rowVector :: DecidableZero a => Vector a -> Matrix a Source #

colVector :: DecidableZero a => Vector a -> Matrix a Source #

rowCount :: Int -> Matrix a -> Int Source #

colCount :: Int -> Matrix a -> Int Source #

traverseRow :: b -> (b -> Int -> Entry a -> b) -> Int -> Matrix a -> b Source #

traverseCol :: b -> (b -> Int -> Entry a -> b) -> Int -> Matrix a -> b Source #

data Entry a Source #

Instances

Instances details

Eq a => Eq (Entry a) Source #
Instance details Defined in Algebra.LinkedMatrix Methods (==) :: Entry a -> Entry a -> Bool # (/=) :: Entry a -> Entry a -> Bool #
Ord a => Ord (Entry a) Source #
Instance details Defined in Algebra.LinkedMatrix Methods compare :: Entry a -> Entry a -> Ordering # (<) :: Entry a -> Entry a -> Bool # (<=) :: Entry a -> Entry a -> Bool # (>) :: Entry a -> Entry a -> Bool # (>=) :: Entry a -> Entry a -> Bool # max :: Entry a -> Entry a -> Entry a # min :: Entry a -> Entry a -> Entry a #
Read a => Read (Entry a) Source #
Instance details Defined in Algebra.LinkedMatrix Methods readsPrec :: Int -> ReadS (Entry a) # readList :: ReadS [Entry a] # readPrec :: ReadPrec (Entry a) # readListPrec :: ReadPrec [Entry a] #
Show a => Show (Entry a) Source #
Instance details Defined in Algebra.LinkedMatrix Methods showsPrec :: Int -> Entry a -> ShowS # show :: Entry a -> String # showList :: [Entry a] -> ShowS #

idx :: forall a. Lens' (Entry a) (Int, Int) Source #

value :: forall a a. Lens (Entry a) (Entry a) a a Source #

substMatrix :: CoeffRing r => Matrix r -> Polynomial r 1 -> Matrix r Source #

catRow :: DecidableZero b => Matrix b -> Vector b -> Matrix b Source #

catCol :: DecidableZero b => Matrix b -> Vector b -> Matrix b Source #

(<||>) :: DecidableZero b => Matrix b -> Matrix b -> Matrix b Source #

(<-->) :: DecidableZero b => Matrix b -> Matrix b -> Matrix b Source #

toRows :: Monoidal a => Matrix a -> [Vector a] Source #

toCols :: Monoidal a => Matrix a -> [Vector a] Source #

zeroMat :: Int -> Int -> Matrix a Source #

getDiag :: Monoidal a => Matrix a -> Vector a Source #

trace :: Monoidal c => Matrix c -> c Source #

diagProd :: (Unital c, Monoidal c) => Matrix c -> c Source #

diag :: DecidableZero a => Vector a -> Matrix a Source #

scaleCol :: (DecidableZero a, Multiplicative a) => a -> Int -> Matrix a -> Matrix a Source #

clearRow :: Int -> Matrix a -> Matrix a Source #

clearCol :: Int -> Matrix a -> Matrix a Source #

index :: Monoidal a => Key -> Int -> Matrix a -> Maybe a Source #

(!) :: Monoidal a => Matrix a -> (Int, Int) -> a Source #

nonZeroEntries :: Matrix a -> Vector ((Int, Int), a) Source #

rankLM :: (DecidableZero r, Division r, Group r) => Matrix r -> Int Source #

splitIndependentDirs Source #

Arguments

:: (DecidableZero a, Field a)
=> Direction
-> Matrix a
-> (Matrix a, [Int], [Int])	`(m', bs, as)` with `m` is full-rank submatrix, `bs` are independent and `as` are dependent.

structuredGauss :: (DecidableZero a, Division a, Group a) => Matrix a -> (Matrix a, Matrix a) Source #

multWithVector :: (Multiplicative a, Monoidal a) => Matrix a -> Vector a -> Vector a Source #

solveWiedemann :: (Eq a, Field a, DecidableZero a, DecidableUnits a, ZeroProductSemiring a, Random a, MonadRandom m) => Matrix a -> Vector a -> m (Either (Vector a) (Vector a)) Source #

Solving linear equation using linearly recurrent sequence (Wiedemann algorithm).

henselLift Source #

Arguments

:: Integer	prime number `p`
-> Matrix Integer	original matrix `M`
-> Matrix Integer	inverse matrix of `M` mod `p`
-> Vector Integer	coefficient vector `v`
-> [Vector Integer]	vector `x` with `Mx = b mod p`

solveHensel :: Int -> Integer -> Matrix (Fraction Integer) -> Vector (Fraction Integer) -> Maybe (Vector (Fraction Integer)) Source #

structuredGauss' :: (DecidableZero a, Division a, Group a) => Matrix a -> (Matrix a, Matrix a, a) Source #

intDet :: Matrix Integer -> Integer Source #