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 Upper sh = FlexUpper NonUnit sh

type UnitUpper sh = Array (UpperTriangular Unit sh)

type Lower sh = FlexLower NonUnit sh

type UnitLower sh = Array (LowerTriangular Unit sh)

type Symmetric sh = Array (Symmetric sh)

type Diagonal sh = FlexDiagonal NonUnit 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

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

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

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