Numeric.LAPACK.Matrix.Triangular

Documentation

type Triangular lo diag up sh = Array (Triangular lo diag up sh)

type UpLo lo up = (UpLoC lo up, UpLoC up lo)

type Diagonal sh = FlexDiagonal NonUnit sh

type Upper sh = FlexUpper NonUnit sh

type FlexUpper diag sh = Array (UpperTriangular diag sh)

type UnitUpper sh = FlexUpper Unit sh

type Lower sh = FlexLower NonUnit sh

type FlexLower diag sh = Array (LowerTriangular diag sh)

type UnitLower sh = FlexLower Unit sh

type Symmetric sh = Array (Symmetric sh)

type FlexSymmetric diag sh = Array (FlexSymmetric diag sh)

fromList :: (Content lo, Content up, C sh, Storable a) => Order -> sh -> [a] -> Triangular lo NonUnit up sh a

autoFromList :: (Content lo, Content up, Storable a) => Order -> [a] -> Triangular lo NonUnit up ZeroInt a

lowerFromList :: (C sh, Storable a) => Order -> sh -> [a] -> Lower sh a

autoLowerFromList :: Storable a => Order -> [a] -> Lower ZeroInt a

upperFromList :: (C sh, Storable a) => Order -> sh -> [a] -> Upper sh a

autoUpperFromList :: Storable a => Order -> [a] -> Upper ZeroInt a

symmetricFromList :: (C sh, Storable a) => Order -> sh -> [a] -> Symmetric sh a

autoSymmetricFromList :: Storable a => Order -> [a] -> Symmetric ZeroInt a

diagonalFromList :: (C sh, Storable a) => Order -> sh -> [a] -> Diagonal sh a

autoDiagonalFromList :: Storable a => Order -> [a] -> Diagonal ZeroInt a

relaxUnitDiagonal :: TriDiag diag => Triangular lo Unit up sh a -> Triangular lo diag up sh a

strictNonUnitDiagonal :: TriDiag diag => Triangular lo diag up sh a -> Triangular lo NonUnit up sh a

asDiagonal :: FlexDiagonal diag sh a -> FlexDiagonal diag sh a

asLower :: FlexLower diag sh a -> FlexLower diag sh a

asUpper :: FlexUpper diag sh a -> FlexUpper diag sh a

asSymmetric :: FlexSymmetric diag sh a -> FlexSymmetric diag sh a

forceUnitDiagonal :: Triangular lo Unit up sh a -> Triangular lo Unit up sh a

forceNonUnitDiagonal :: Triangular lo NonUnit up sh a -> Triangular lo NonUnit up sh a

identity :: (Content lo, Content up, C sh, Floating a) => Order -> sh -> Triangular lo Unit up sh a

diagonal :: (Content lo, Content up, C sh, Floating a) => Order -> Vector sh a -> Triangular lo NonUnit up sh a

takeDiagonal :: (Content lo, Content up, C sh, Floating a) => Triangular lo diag up sh a -> Vector sh a

transpose :: (Content lo, Content up, TriDiag diag) => Triangular lo diag up sh a -> Triangular up diag lo sh a

adjoint :: (Content lo, Content up, TriDiag diag, C sh, Floating a) => Triangular lo diag up sh a -> Triangular up diag lo sh a

stackLower :: (TriDiag diag, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => FlexLower diag sh0 a -> General sh1 sh0 a -> FlexLower diag sh1 a -> FlexLower diag (sh0 :+: sh1) a

stackUpper :: (TriDiag diag, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => FlexUpper diag sh0 a -> General sh0 sh1 a -> FlexUpper diag sh1 a -> FlexUpper diag (sh0 :+: sh1) a

For upper triangular matrices it holds:

toSquare (stack a b c)

=

toSquare a ||| b
===
constant 0 ||| toSquare c

For the other triangular flavors it holds accordingly

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

stackSymmetric :: (TriDiag diag, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => FlexSymmetric diag sh0 a -> General sh0 sh1 a -> FlexSymmetric diag sh1 a -> FlexSymmetric diag (sh0 :+: sh1) a

toSquare :: (Content lo, Content up, C sh, Floating a) => Triangular lo diag up sh a -> Square sh a

takeLower :: (C horiz, C height, C width, Floating a) => Full Small horiz height width a -> Lower height a

takeUpper :: (C vert, C height, C width, Floating a) => Full vert Small height width a -> Upper width a

fromLowerRowMajor :: (C sh, Floating a) => Array (Triangular Lower sh) a -> Lower sh a

toLowerRowMajor :: (C sh, Floating a) => Lower sh a -> Array (Triangular Lower sh) a

fromUpperRowMajor :: (C sh, Floating a) => Array (Triangular Upper sh) a -> Upper sh a

toUpperRowMajor :: (C sh, Floating a) => Upper sh a -> Array (Triangular Upper sh) a

forceOrder :: (Content lo, Content up, TriDiag diag, C sh, Floating a) => Order -> Triangular lo diag up sh a -> Triangular lo diag up sh a

adaptOrder :: (Content lo, Content up, TriDiag diag, C sh, Floating a) => Triangular lo diag up sh a -> Triangular lo diag up sh a -> Triangular lo diag up sh a

adaptOrder x y contains the data of y with the layout of x.

add :: (Content lo, Content up, Eq lo, Eq up, Eq sh, C sh, Floating a) => Triangular lo NonUnit up sh a -> Triangular lo NonUnit up sh a -> Triangular lo NonUnit up sh a

sub :: (Content lo, Content up, Eq lo, Eq up, Eq sh, C sh, Floating a) => Triangular lo NonUnit up sh a -> Triangular lo NonUnit up sh a -> Triangular lo NonUnit up sh a

type family PowerDiag lo up diag

multiplyVector :: (Content lo, Content up, TriDiag diag, C sh, Eq sh, Floating a) => Triangular lo diag up sh a -> Vector sh a -> Vector sh a

square :: (DiagUpLo lo up, TriDiag diag, C sh, Eq sh, Floating a) => Triangular lo diag up sh a -> Triangular lo diag up sh a

squareGeneric :: (Content lo, Content up, TriDiag diag, C sh, Eq sh, Floating a) => Triangular lo diag up sh a -> Triangular lo (PowerDiag lo up diag) up sh a

Include symmetric matrices. However, symmetric matrices do not preserve unit diagonals.

multiply :: (DiagUpLo lo up, TriDiag diag, C sh, Eq sh, Floating a) => Triangular lo diag up sh a -> Triangular lo diag up sh a -> Triangular lo diag up sh a

multiplyFull :: (Content lo, Content up, TriDiag diag, C vert, C horiz, C height, Eq height, C width, Floating a) => Triangular lo diag up height a -> Full vert horiz height width a -> Full vert horiz height width a

solve :: (Content lo, Content up, TriDiag diag, C vert, C horiz, C sh, Eq sh, C nrhs, Floating a) => Triangular lo diag up sh a -> Full vert horiz sh nrhs a -> Full vert horiz sh nrhs a

inverse :: (DiagUpLo lo up, TriDiag diag, C sh, Floating a) => Triangular lo diag up sh a -> Triangular lo diag up sh a

inverseGeneric :: (Content lo, Content up, TriDiag diag, C sh, Floating a) => Triangular lo diag up sh a -> Triangular lo (PowerDiag lo up diag) up sh a

determinant :: (Content lo, Content up, TriDiag diag, C sh, Floating a) => Triangular lo diag up sh a -> a

size :: Triangular lo diag up sh a -> sh

eigenvalues :: (DiagUpLo lo up, C sh, Floating a) => Triangular lo diag up sh a -> Vector sh a

eigensystem :: (DiagUpLo lo up, C sh, Floating a) => Triangular lo NonUnit up sh a -> (Triangular lo NonUnit up sh a, Vector sh a, Triangular lo NonUnit up sh a)

(vr,d,vlAdj) = eigensystem a

Counterintuitively, vr contains the right eigenvectors as columns and vlAdj contains the left conjugated eigenvectors as rows. The idea is to provide a decomposition of a. If a is diagonalizable, then vr and vlAdj are almost inverse to each other. More precisely, vlAdj <#> vr is a diagonal matrix. This is because the eigenvectors are not normalized. With the following scaling, the decomposition becomes perfect:

let scal = Array.map recip $ takeDiagonal $ vlAdj <#> vr
a == vr <#> diagonal d <#> diagonal scal <#> vlAdj

If a is non-diagonalizable then some columns of vr and corresponding rows of vlAdj are left zero and the above property does not hold.