Numeric.LAPACK.Matrix.Hermitian

Documentation

type Hermitian sh = ArrayMatrix (Hermitian sh)

data Transposition

Constructors

NonTransposed

Transposed

Instances

Bounded Transposition

Defined in Numeric.LAPACK.Matrix.Modifier

Methods

minBound :: Transposition

maxBound :: Transposition

Enum Transposition

Defined in Numeric.LAPACK.Matrix.Modifier

Methods

succ :: Transposition -> Transposition

pred :: Transposition -> Transposition

toEnum :: Int -> Transposition

fromEnum :: Transposition -> Int

enumFrom :: Transposition -> [Transposition]

enumFromThen :: Transposition -> Transposition -> [Transposition]

enumFromTo :: Transposition -> Transposition -> [Transposition]

enumFromThenTo :: Transposition -> Transposition -> Transposition -> [Transposition]

Eq Transposition

Defined in Numeric.LAPACK.Matrix.Modifier

Methods

(==) :: Transposition -> Transposition -> Bool

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

Show Transposition

Defined in Numeric.LAPACK.Matrix.Modifier

Methods

showsPrec :: Int -> Transposition -> ShowS

show :: Transposition -> String

showList :: [Transposition] -> ShowS

Semigroup Transposition

Defined in Numeric.LAPACK.Matrix.Modifier

Methods

(<>) :: Transposition -> Transposition -> Transposition

sconcat :: NonEmpty Transposition -> Transposition

stimes :: Integral b => b -> Transposition -> Transposition

Monoid Transposition

Defined in Numeric.LAPACK.Matrix.Modifier

Methods

mempty :: Transposition

mappend :: Transposition -> Transposition -> Transposition

mconcat :: [Transposition] -> Transposition

size :: Hermitian sh a -> sh

fromList :: (C sh, Storable a) => Order -> sh -> [a] -> Hermitian sh a

autoFromList :: Storable a => Order -> [a] -> Hermitian ShapeInt a

identity :: (C sh, Floating a) => Order -> sh -> Hermitian sh a

diagonal :: (C sh, Floating a) => Order -> Vector sh (RealOf a) -> Hermitian sh a

takeDiagonal :: (C sh, Floating a) => Hermitian sh a -> Vector sh (RealOf a)

forceOrder :: (C sh, Floating a) => Order -> Hermitian sh a -> Hermitian sh a

stack :: (C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => Hermitian sh0 a -> General sh0 sh1 a -> Hermitian sh1 a -> Hermitian (sh0 :+: sh1) a

toSquare (stack a b c)

=

toSquare a ||| b
===
adjoint b ||| toSquare c

It holds order (stack a b c) = order b. The function is most efficient when the order of all blocks match.

(*%%%#) :: (C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => (Hermitian sh0 a, General sh0 sh1 a) -> Hermitian sh1 a -> Hermitian (sh0 :+: sh1) a infixr 2

split :: (C sh0, C sh1, Floating a) => Hermitian (sh0 :+: sh1) a -> (Hermitian sh0 a, General sh0 sh1 a, Hermitian sh1 a)

takeTopLeft :: (C sh0, C sh1, Floating a) => Hermitian (sh0 :+: sh1) a -> Hermitian sh0 a

takeTopRight :: (C sh0, C sh1, Floating a) => Hermitian (sh0 :+: sh1) a -> General sh0 sh1 a

takeBottomRight :: (C sh0, C sh1, Floating a) => Hermitian (sh0 :+: sh1) a -> Hermitian sh1 a

multiplyVector :: (C sh, Eq sh, Floating a) => Transposition -> Hermitian sh a -> Vector sh a -> Vector sh a

square :: (C sh, Eq sh, Floating a) => Hermitian sh a -> Hermitian sh a

multiplyFull :: (C vert, C horiz, C height, Eq height, C width, Floating a) => Transposition -> Hermitian height a -> Full vert horiz height width a -> Full vert horiz height width a

outer :: (C sh, Floating a) => Order -> Vector sh a -> Hermitian sh a

sumRank1 :: (C sh, Eq sh, Floating a) => Order -> sh -> [(RealOf a, Vector sh a)] -> Hermitian sh a

sumRank1NonEmpty :: (C sh, Eq sh, Floating a) => Order -> T [] (RealOf a, Vector sh a) -> Hermitian sh a

sumRank2 :: (C sh, Eq sh, Floating a) => Order -> sh -> [(a, (Vector sh a, Vector sh a))] -> Hermitian sh a

sumRank2NonEmpty :: (C sh, Eq sh, Floating a) => Order -> T [] (a, (Vector sh a, Vector sh a)) -> Hermitian sh a

toSquare :: (C sh, Floating a) => Hermitian sh a -> Square sh a

fromSymmetric :: (C sh, Real a) => Symmetric sh a -> Hermitian sh a

gramian :: (C height, C width, Floating a) => General height width a -> Hermitian width a

gramian A = A^H * A

gramianAdjoint :: (C height, C width, Floating a) => General height width a -> Hermitian height a

gramianAdjoint A = A * A^H = gramian (A^H)

congruenceDiagonal :: (C height, Eq height, C width, Floating a) => Vector height (RealOf a) -> General height width a -> Hermitian width a

congruenceDiagonal D A = A^H * D * A

congruenceDiagonalAdjoint :: (C height, C width, Eq width, Floating a) => General height width a -> Vector width (RealOf a) -> Hermitian height a

congruenceDiagonalAdjoint A D = A * D * A^H

congruence :: (C height, Eq height, C width, Floating a) => Hermitian height a -> General height width a -> Hermitian width a

congruence B A = A^H * B * A

congruenceAdjoint :: (C height, C width, Eq width, Floating a) => General height width a -> Hermitian width a -> Hermitian height a

congruenceAdjoint B A = A * B * A^H

anticommutator :: (C vert, C horiz, C height, Eq height, C width, Eq width, Floating a) => Full vert horiz height width a -> Full vert horiz height width a -> Hermitian width a

anticommutator A B = A^H * B + B^H * A

Not exactly a matrix anticommutator, thus I like to call it Hermitian anticommutator.

anticommutatorAdjoint :: (C vert, C horiz, C height, Eq height, C width, Eq width, Floating a) => Full vert horiz height width a -> Full vert horiz height width a -> Hermitian height a

anticommutatorAdjoint A B = A * B^H + B * A^H = anticommutator (adjoint A) (adjoint B)

addAdjoint :: (C sh, Floating a) => Square sh a -> Hermitian sh a

addAdjoint A = A^H + A

solve :: (C vert, C horiz, C sh, Eq sh, C nrhs, Floating a) => Hermitian sh a -> Full vert horiz sh nrhs a -> Full vert horiz sh nrhs a

inverse :: (C sh, Floating a) => Hermitian sh a -> Hermitian sh a

determinant :: (C sh, Floating a) => Hermitian sh a -> RealOf a

eigenvalues :: (C sh, Floating a) => Hermitian sh a -> Vector sh (RealOf a)

eigensystem :: (C sh, Floating a) => Hermitian sh a -> (Square sh a, Vector sh (RealOf a))

For symmetric eigenvalue problems, eigensystem and schur coincide.