Safe Haskell	None
Language	Haskell98

Numeric.LAPACK.Matrix.Triangular

Synopsis

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
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
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
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)

Documentation

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

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

type Diagonal sh = FlexDiagonal NonUnit sh Source #

type Upper sh = FlexUpper NonUnit sh Source #

type FlexUpper diag sh = Array (UpperTriangular diag sh) Source #

type UnitUpper sh = FlexUpper Unit sh Source #

type Lower sh = FlexLower NonUnit sh Source #

type FlexLower diag sh = Array (LowerTriangular diag sh) Source #

type UnitLower sh = FlexLower Unit sh Source #

type Symmetric sh = Array (Symmetric sh) Source #

type FlexSymmetric diag sh = Array (FlexSymmetric diag sh) Source #

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Source #

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 Source #

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 Source #

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

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

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

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

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

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

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

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

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 Source #

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 Source #

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 Source #

type family PowerDiag lo up diag Source #

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 Source #

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

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 Source #

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 Source #

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 Source #

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 Source #

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

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 Source #

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

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

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

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) Source #

(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.