Numeric.LAPACK.Matrix.Triangular

Documentation

type Triangular lo diag up sh = TriangularP Packed lo diag up sh

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

type Upper sh = FlexUpper Arbitrary sh

type FlexUpper diag sh = Quadratic Packed diag Empty Filled sh

type UnitUpper sh = FlexUpper Unit sh

type QuasiUpper sh = Quadratic Unpacked Arbitrary (Bands U1) Filled sh

type Lower sh = FlexLower Arbitrary sh

type FlexLower diag sh = Quadratic Packed diag Filled Empty sh

type UnitLower sh = FlexLower Unit sh

size :: TriangularP pack lo diag up sh a -> sh

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

autoFromList :: (UpLo lo up, Storable a) => Order -> [a] -> Triangular lo Arbitrary up ShapeInt a

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

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

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

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

asLower :: Id (FlexLowerP pack diag sh a)

asUpper :: Id (FlexUpperP pack diag sh a)

requireUnitDiagonal :: Id (TriangularP pack lo Unit up sh a)

requireArbitraryDiagonal :: Id (TriangularP pack lo Arbitrary up sh a)

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

strictArbitraryDiagonal :: TriDiag diag => TriangularP pack lo diag up sh a -> TriangularP pack lo Arbitrary up sh a

identityOrder :: (Quadratic pack property lower upper, C sh, Floating a) => Order -> sh -> Quadratic pack property lower upper sh a

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

takeDiagonal :: (PowerStrip lo, PowerStrip up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> Vector sh a

transpose :: (PowerStrip lo, PowerStrip up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> TriangularP pack up diag lo sh a

adjoint :: (PowerStrip lo, PowerStrip up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> TriangularP pack up diag lo sh a

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

(#%%%) :: (Packing pack, TriDiag diag, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => FlexLowerP pack diag sh0 a -> (General sh1 sh0 a, FlexLowerP pack diag sh1 a) -> FlexLowerP pack diag (sh0 ::+ sh1) a infixl 2

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

(%%%#) :: (Packing pack, TriDiag diag, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => (FlexUpperP pack diag sh0 a, General sh0 sh1 a) -> FlexUpperP pack diag sh1 a -> FlexUpperP pack diag (sh0 ::+ sh1) a infixr 2

splitLower :: (Packing pack, TriDiag diag, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => FlexLowerP pack diag (sh0 ::+ sh1) a -> (FlexLowerP pack diag sh0 a, General sh1 sh0 a, FlexLowerP pack diag sh1 a)

splitUpper :: (Packing pack, TriDiag diag, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => FlexUpperP pack diag (sh0 ::+ sh1) a -> (FlexUpperP pack diag sh0 a, General sh0 sh1 a, FlexUpperP pack diag sh1 a)

takeTopLeft :: (Packing pack, UpLo lo up, TriDiag diag, C sh0, C sh1, Floating a) => TriangularP pack lo diag up (sh0 ::+ sh1) a -> TriangularP pack lo diag up sh0 a

takeTopRight :: (Packing pack, TriDiag diag, C sh0, C sh1, Floating a) => FlexUpperP pack diag (sh0 ::+ sh1) a -> General sh0 sh1 a

takeBottomLeft :: (Packing pack, TriDiag diag, C sh0, C sh1, Floating a) => FlexLowerP pack diag (sh0 ::+ sh1) a -> General sh1 sh0 a

takeBottomRight :: (Packing pack, UpLo lo up, TriDiag diag, C sh0, C sh1, Floating a) => TriangularP pack lo diag up (sh0 ::+ sh1) a -> TriangularP pack lo diag up sh1 a

pack :: (Packing pack, UpLo lo up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> Triangular lo diag up sh a

toSquare :: (PowerStrip lo, PowerStrip up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> Square sh a

takeLower :: (Property property, Strip upper) => (Measure meas, C horiz, C height, C width, Floating a) => Unpacked property Filled upper meas Small horiz height width a -> Lower height a

takeUpper :: (Property property, Strip lower) => (Measure meas, C vert, C height, C width, Floating a) => Unpacked property lower Filled meas 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 :: (PowerStrip lo, PowerStrip up, TriDiag diag, C sh, Floating a) => Order -> TriangularP pack lo diag up sh a -> TriangularP pack lo diag up sh a

adaptOrder :: (PowerStrip lo, PowerStrip up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> TriangularP pack lo diag up sh a -> TriangularP pack lo diag up sh a

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

add :: (PowerStrip lo, PowerStrip up, Eq lo, Eq up, Eq sh, C sh, Floating a) => TriangularP pack lo Arbitrary up sh a -> TriangularP pack lo Arbitrary up sh a -> TriangularP pack lo Arbitrary up sh a

sub :: (PowerStrip lo, PowerStrip up, Eq lo, Eq up, Eq sh, C sh, Floating a) => TriangularP pack lo Arbitrary up sh a -> TriangularP pack lo Arbitrary up sh a -> TriangularP pack lo Arbitrary up sh a

multiplyVector :: (Packing pack, UpLo lo up, TriDiag diag, C sh, Eq sh, Floating a) => TriangularP pack lo diag up sh a -> Vector sh a -> Vector sh a

square :: (Packing pack, UpLo lo up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> TriangularP pack lo diag up sh a

multiply :: (Packing pack, UpLo lo up, TriDiag diag, C sh, Eq sh, Floating a) => TriangularP pack lo diag up sh a -> TriangularP pack lo diag up sh a -> TriangularP pack lo diag up sh a

multiplyFull :: (Packing pack, UpLo lo up, TriDiag diag, Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => TriangularP pack lo diag up height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a

solve :: (Packing pack, UpLo lo up, TriDiag diag, Measure meas, C vert, C horiz, C sh, Eq sh, C nrhs, Floating a) => TriangularP pack lo diag up sh a -> Full meas vert horiz sh nrhs a -> Full meas vert horiz sh nrhs a

inverse :: (Packing pack, UpLo lo up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> TriangularP pack lo diag up sh a

determinant :: (Packing pack, UpLo lo up, TriDiag diag, C sh, Floating a) => TriangularP pack lo diag up sh a -> a

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

eigensystem :: (Packing pack, DiagUpLo lo up, C sh, Floating a) => TriangularP pack lo Arbitrary up sh a -> (TriangularP pack lo Arbitrary up sh a, Vector sh a, TriangularP pack lo Arbitrary 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, but not necessarily an identity matrix. This is because all eigenvectors are normalized such that normInf1 is 1. With the following scaling, the decomposition becomes perfect:

let scal = takeDiagonal $ vlAdj <> vr
a == vr <> diagonal (Vector.divide d 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.