-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Numerical Linear Algebra using LAPACK
--   
--   This is a high-level interface to LAPACK. It provides solvers for
--   simultaneous linear equations, linear least-squares problems,
--   eigenvalue and singular value problems for matrices with certain kinds
--   of structures.
--   
--   Features:
--   
--   <ul>
--   <li>Based on <tt>comfort-array</tt>: Allows to precisely express
--   one-column or one-row matrices, as well as dense, square, triangular,
--   banded, symmetric, Hermitian, banded Hermitian, blocked and LU or QR
--   decomposed matrices.</li>
--   <li>Support all data types that are supported by LAPACK, i.e. Float,
--   Double, Complex Float, Complex Double</li>
--   <li>No need for c2hs, hsc, Template Haskell or C helper functions</li>
--   <li>Dependency only on BLAS and LAPACK, no GSL</li>
--   <li>Works with matrices and vectors with zero dimensions. This one is
--   tricky and still leads to surprises since different LAPACK
--   implementations consider different situations as corner cases.</li>
--   <li>No automatic (and dangerous) implicit expansion of singleton
--   vectors or matrices. Instead there are special operators for scaling
--   of vectors and matrices.</li>
--   <li>Separate formatting operator <tt>(##)</tt>: Works better for
--   tuples of matrices and vectors than <a>show</a>. <a>Show</a> is used
--   for code one-liners that can be copied back into Haskell modules.
--   Support for nice formatting in HyperHaskell.</li>
--   </ul>
--   
--   See also: <tt>hmatrix</tt>.
@package lapack
@version 0.4

module Numeric.LAPACK.Output
class Output out
text :: Output out => String -> out
above :: Output out => out -> out -> out
beside :: Output out => out -> out -> out
formatRow :: Output out => [out] -> out
formatColumn :: Output out => [out] -> out
formatAligned :: (Output out, Foldable f) => [[f out]] -> out
formatSeparateTriangle :: (Output out, Foldable f) => [[f out]] -> out
(/+/) :: Output out => out -> out -> out
(<+>) :: Output out => out -> out -> out
hyper :: Html -> Graphic
instance GHC.Show.Show Numeric.LAPACK.Output.Separator
instance GHC.Classes.Ord Numeric.LAPACK.Output.Separator
instance GHC.Classes.Eq Numeric.LAPACK.Output.Separator
instance Numeric.LAPACK.Output.Output Text.PrettyPrint.Boxes.Box
instance Numeric.LAPACK.Output.Output Numeric.LAPACK.Output.Html

module Numeric.LAPACK.Scalar
type family RealOf x
type ComplexOf x = Complex (RealOf x)
zero :: Floating a => a
one :: Floating a => a
minusOne :: Floating a => a
isZero :: Floating a => a -> Bool
selectReal :: Real a => Float -> Double -> a
selectFloating :: Floating a => Float -> Double -> Complex Float -> Complex Double -> a
data ComplexSingleton a
[Real] :: (Real a, RealOf a ~ a) => ComplexSingleton a
[Complex] :: Real a => ComplexSingleton (Complex a)
complexSingleton :: Floating a => ComplexSingleton a
complexSingletonOf :: Floating a => a -> ComplexSingleton a
complexSingletonOfFunctor :: Floating a => f a -> ComplexSingleton a
data PrecisionSingleton a
[Float] :: PrecisionSingleton Float
[Double] :: PrecisionSingleton Double
precisionSingleton :: Real a => PrecisionSingleton a
precisionOf :: Real a => a -> PrecisionSingleton a
precisionOfFunctor :: Real a => f a -> PrecisionSingleton a
equal :: Floating a => a -> a -> Bool
fromReal :: Floating a => RealOf a -> a
toComplex :: Floating a => a -> ComplexOf a
absolute :: Floating a => a -> RealOf a
absoluteSquared :: Floating a => a -> RealOf a
norm1 :: Floating a => a -> RealOf a
realPart :: Floating a => a -> RealOf a
conjugate :: Floating a => a -> a

module Numeric.LAPACK.Shape

-- | Class of shapes where indices still make sense if we permute elements.
--   We use this for all matrix factorisations involving permutations or
--   more generally orthogonal transformations, e.g. eigenvalue and
--   singular value, LU and QR decompositions. E.g. say, we have a square
--   matrix with dimension 'Shape.Enumeration Ordering'. Its vector of
--   eigenvalues has the same dimension, but it does not make sense to
--   access the eigenvalues with indices like <a>LT</a> or <a>EQ</a>. Thus
--   <a>Enumeration</a> is no instance of <a>Permutable</a> (and you should
--   not add an orphan instance).
--   
--   If you want to factor a matrix with a non-permutable shape, you should
--   convert it temporarily to a permutable one, like <a>ZeroBased</a>
--   (i.e. <a>ShapeInt</a>) or <a>IntIndexed</a>.
--   
--   The <a>Permutable</a> class has no method, so you could add any shape
--   to it. However, you should use good taste when adding an instance.
--   There is no strict criterion which shape type to add.
--   
--   We tried to use <tt>ShapeInt</tt> for eigenvalue vectors and
--   <tt>LiberalSquare</tt>s as transformation matrices of eigenvalue
--   decompositions. However, this way, the type checker cannot infer that
--   the product of the factorisation is a strict square.
--   
--   We also tried to use <a>IntIndexed</a> for eigenvalue vectors with
--   according <tt>LiberalSquare</tt>s transformations. This has also the
--   problem of inferring squares. Additionally, more such transformations
--   lead to nested <a>IntIndexed</a> wrappers and for <a>ShapeInt</a> even
--   the first wrapper is unnecessary.
class (C shape) => Permutable shape

-- | This shape type wraps any other array shape type. However, its
--   <a>Indexed</a> instance just uses zero-based <a>Int</a> indices. Thus
--   it can turn any shape type into a <a>Indexed</a> one. The main usage
--   is to make an arbitrary shape <a>Permutable</a>.
newtype IntIndexed sh
IntIndexed :: sh -> IntIndexed sh
[deconsIntIndexed] :: IntIndexed sh -> sh
shapeInt :: C sh => sh -> ZeroBased Int
instance GHC.Show.Show sh => GHC.Show.Show (Numeric.LAPACK.Shape.IntIndexed sh)
instance GHC.Classes.Eq sh => GHC.Classes.Eq (Numeric.LAPACK.Shape.IntIndexed sh)
instance Control.DeepSeq.NFData sh => Control.DeepSeq.NFData (Numeric.LAPACK.Shape.IntIndexed sh)
instance Data.Array.Comfort.Shape.C sh => Data.Array.Comfort.Shape.C (Numeric.LAPACK.Shape.IntIndexed sh)
instance Data.Array.Comfort.Shape.C sh => Data.Array.Comfort.Shape.Indexed (Numeric.LAPACK.Shape.IntIndexed sh)
instance Data.Array.Comfort.Shape.C sh => Data.Array.Comfort.Shape.InvIndexed (Numeric.LAPACK.Shape.IntIndexed sh)
instance Data.Array.Comfort.Shape.C sh => Numeric.LAPACK.Shape.Permutable (Numeric.LAPACK.Shape.IntIndexed sh)
instance Numeric.LAPACK.Shape.Permutable Data.Array.Comfort.Shape.Zero
instance Numeric.LAPACK.Shape.Permutable ()
instance GHC.Ix.Ix n => Numeric.LAPACK.Shape.Permutable (Data.Array.Comfort.Shape.Range n)
instance GHC.Real.Integral n => Numeric.LAPACK.Shape.Permutable (Data.Array.Comfort.Shape.Shifted n)
instance GHC.Real.Integral n => Numeric.LAPACK.Shape.Permutable (Data.Array.Comfort.Shape.ZeroBased n)
instance GHC.Real.Integral n => Numeric.LAPACK.Shape.Permutable (Data.Array.Comfort.Shape.OneBased n)
instance GHC.Real.Integral n => Numeric.LAPACK.Shape.Permutable (Data.Array.Comfort.Shape.Cyclic n)
instance Numeric.LAPACK.Shape.Permutable sh => Numeric.LAPACK.Shape.Permutable (Data.Array.Comfort.Shape.Deferred sh)
instance Numeric.LAPACK.Shape.Permutable sh => Numeric.LAPACK.Shape.Permutable (Data.Tagged.Tagged s sh)

module Numeric.LAPACK.Matrix.Layout
type General height width = Full Size Big Big height width
type Tall height width = Full Size Big Small height width
type Wide height width = Full Size Small Big height width
type Square size = SquareMeas Shape size size
data Full meas vert horiz height width
Full :: Order -> Extent meas vert horiz height width -> Full meas vert horiz height width
[fullOrder] :: Full meas vert horiz height width -> Order
[fullExtent] :: Full meas vert horiz height width -> Extent meas vert horiz height width
fullHeight :: (Measure meas, C vert, C horiz) => Full meas vert horiz height width -> height
fullWidth :: (Measure meas, C vert, C horiz) => Full meas vert horiz height width -> width
data Order
RowMajor :: Order
ColumnMajor :: Order
flipOrder :: Order -> Order
general :: Order -> height -> width -> General height width
square :: Order -> sh -> Square sh
liberalSquare :: (C height, C width) => Order -> height -> width -> LiberalSquare height width
wide :: (C height, C width) => Order -> height -> width -> Wide height width
tall :: (C height, C width) => Order -> height -> width -> Tall height width
data Split lower meas vert horiz height width
type SplitGeneral lower height width = Split lower Size Big Big height width
data Triangle
Triangle :: Triangle
data Reflector
Reflector :: Reflector
splitGeneral :: lower -> Order -> height -> width -> SplitGeneral lower height width
splitFromFull :: lower -> Full meas vert horiz height width -> Split lower meas vert horiz height width
data Mosaic pack mirror uplo size
Mosaic :: PackingSingleton pack -> MirrorSingleton mirror -> UpLoSingleton uplo -> Order -> size -> Mosaic pack mirror uplo size
[mosaicPack] :: Mosaic pack mirror uplo size -> PackingSingleton pack
[mosaicMirror] :: Mosaic pack mirror uplo size -> MirrorSingleton mirror
[mosaicUplo] :: Mosaic pack mirror uplo size -> UpLoSingleton uplo
[mosaicOrder] :: Mosaic pack mirror uplo size -> Order
[mosaicSize] :: Mosaic pack mirror uplo size -> size
class Packing pack
autoPacking :: Packing pack => PackingSingleton pack
data PackingSingleton pack
[Packed] :: PackingSingleton Packed
[Unpacked] :: PackingSingleton Unpacked
data Packed
data Unpacked
class Mirror mirror
autoMirror :: Mirror mirror => MirrorSingleton mirror
data MirrorSingleton mirror
[NoMirror] :: MirrorSingleton NoMirror
[SimpleMirror] :: MirrorSingleton SimpleMirror
[ConjugateMirror] :: MirrorSingleton ConjugateMirror
autoUplo :: UpLo uplo => UpLoSingleton uplo
type Triangular = TriangularP Packed
type LowerTriangular = Triangular Lower
type LowerTriangularP pack = TriangularP pack Lower
lowerTriangular :: Order -> size -> LowerTriangular size
lowerTriangularP :: PackingSingleton pack -> Order -> size -> LowerTriangularP pack size
type UpperTriangular = Triangular Upper
type UpperTriangularP pack = TriangularP pack Upper
upperTriangular :: Order -> size -> UpperTriangular size
upperTriangularP :: PackingSingleton pack -> Order -> size -> UpperTriangularP pack size
type Symmetric = SymmetricP Packed
type SymmetricP pack = Mosaic pack SimpleMirror Upper
symmetric :: Order -> size -> Symmetric size
symmetricP :: PackingSingleton pack -> Order -> size -> SymmetricP pack size
type Hermitian = HermitianP Packed
type HermitianP pack = Mosaic pack ConjugateMirror Upper
hermitian :: Order -> size -> Hermitian size
hermitianP :: PackingSingleton pack -> Order -> size -> HermitianP pack size
type Diagonal size = BandedSquare U0 U0 size
diagonal :: Order -> size -> Diagonal size
data Banded sub super meas vert horiz height width
Banded :: (UnaryProxy sub, UnaryProxy super) -> Order -> Extent meas vert horiz height width -> Banded sub super meas vert horiz height width
[bandedOffDiagonals] :: Banded sub super meas vert horiz height width -> (UnaryProxy sub, UnaryProxy super)
[bandedOrder] :: Banded sub super meas vert horiz height width -> Order
[bandedExtent] :: Banded sub super meas vert horiz height width -> Extent meas vert horiz height width
type BandedGeneral sub super = Banded sub super Size Big Big
type BandedSquare sub super size = BandedSquareMeas sub super Shape size size
type BandedLowerTriangular sub size = BandedSquare sub U0 size
type BandedUpperTriangular super size = BandedSquare U0 super size
data BandedIndex row column
InsideBox :: row -> column -> BandedIndex row column
VertOutsideBox :: Int -> column -> BandedIndex row column
HorizOutsideBox :: row -> Int -> BandedIndex row column
bandedGeneral :: (UnaryProxy sub, UnaryProxy super) -> Order -> height -> width -> BandedGeneral sub super height width
bandedSquare :: (UnaryProxy sub, UnaryProxy super) -> Order -> size -> BandedSquare sub super size
bandedFromFull :: (UnaryProxy sub, UnaryProxy super) -> Full meas vert horiz height width -> Banded sub super meas vert horiz height width
type UnaryProxy a = Proxy (Un a)
addOffDiagonals :: (Natural subA, Natural superA, Natural subB, Natural superB, (subA :+: subB) ~ subC, (superA :+: superB) ~ superC) => (UnaryProxy subA, UnaryProxy superA) -> (UnaryProxy subB, UnaryProxy superB) -> ((Nat subC, Nat superC), (UnaryProxy subC, UnaryProxy superC))
data BandedHermitian off size
BandedHermitian :: UnaryProxy off -> Order -> size -> BandedHermitian off size
[bandedHermitianOffDiagonals] :: BandedHermitian off size -> UnaryProxy off
[bandedHermitianOrder] :: BandedHermitian off size -> Order
[bandedHermitianSize] :: BandedHermitian off size -> size
bandedHermitian :: UnaryProxy off -> Order -> size -> BandedHermitian off size

module Numeric.LAPACK.Matrix.Shape.Omni
data Omni pack property lower upper meas vert horiz height width
[Full] :: (Property property, Strip lower, Strip upper) => Full meas vert horiz height width -> Omni Unpacked property lower upper meas vert horiz height width
[UpperTriangular] :: TriDiag diag => UpperTriangular size -> Omni Packed diag Empty Filled Shape Small Small size size
[LowerTriangular] :: TriDiag diag => LowerTriangular size -> Omni Packed diag Filled Empty Shape Small Small size size
[Symmetric] :: Symmetric size -> Omni Packed Symmetric Filled Filled Shape Small Small size size
[Hermitian] :: (C neg, C zero, C pos) => Hermitian size -> Omni Packed (Hermitian neg zero pos) Filled Filled Shape Small Small size size
[Banded] :: (Natural sub, Natural super) => Banded sub super meas vert horiz height width -> Omni Packed Arbitrary (Bands sub) (Bands super) meas vert horiz height width
[UnitBandedTriangular] :: (BandedTriangular sub super, BandedTriangular super sub) => BandedSquare sub super size -> Omni Packed Unit (Bands sub) (Bands super) Shape Small Small size size
[BandedHermitian] :: (C neg, C zero, C pos, Natural offDiag) => BandedHermitian offDiag size -> Omni Packed (Hermitian neg zero pos) (Bands offDiag) (Bands offDiag) Shape Small Small size size
data Unit
data Arbitrary
data Symmetric
data Hermitian neg zero pos
type HermitianUnknownDefiniteness = Hermitian True True True
type HermitianPositiveDefinite = Hermitian False False True
type HermitianPositiveSemidefinite = Hermitian False True True
type HermitianNegativeDefinite = Hermitian True False False
type HermitianNegativeSemidefinite = Hermitian True True False
hermitianSet :: (C neg, C zero, C pos) => Omni pack (Hermitian neg zero pos) lower upper meas vert horiz height width -> (Singleton neg, Singleton zero, Singleton pos)
class (Property diag) => TriDiag diag
switchTriDiag :: TriDiag diag => f Unit -> f Arbitrary -> f diag
data DiagSingleton diag
[Unit] :: DiagSingleton Unit
[Arbitrary] :: DiagSingleton Arbitrary
autoDiag :: TriDiag diag => DiagSingleton diag
charFromTriDiag :: TriDiag diag => DiagSingleton diag -> Char
packTag :: Packing pack => Omni pack propery lower upper meas vert horiz height width -> PackingSingleton pack
class Property property
property :: Property property => Omni pack property lower upper meas vert horiz height width -> PropertySingleton property
data PropertySingleton property
[PropArbitrary] :: PropertySingleton Arbitrary
[PropUnit] :: PropertySingleton Unit
[PropSymmetric] :: PropertySingleton Symmetric
[PropHermitian] :: (C neg, C zero, C pos) => PropertySingleton (Hermitian neg zero pos)
propertySingleton :: Property property => PropertySingleton property
class (MultipliedBands c Filled ~ Filled, MultipliedBands c Empty ~ c) => Strip c
switchStrip :: Strip c => (forall offDiag. Natural offDiag => f (Bands offDiag)) -> f Filled -> f c
strips :: (Strip lower, Strip upper) => Omni pack property lower upper meas vert horiz height width -> (StripSingleton lower, StripSingleton upper)
data StripSingleton c
[StripBands] :: Natural offDiag => HeadSingleton offDiag -> StripSingleton (Bands offDiag)
[StripFilled] :: StripSingleton Filled
stripSingleton :: Strip c => StripSingleton c

-- | <a>PowerStrip</a> is either <a>Empty</a> or <a>Filled</a>. These are
--   the <a>Strip</a>s that are preserved in matrix powers.
--   
--   Pun intended.
class (Strip c) => PowerStrip c
switchPowerStrip :: PowerStrip c => f Empty -> f Filled -> f c
data PowerStripSingleton c
[Empty] :: PowerStripSingleton Empty
[Filled] :: PowerStripSingleton Filled
powerStripSingleton :: PowerStrip c => PowerStripSingleton c
powerStrips :: (PowerStrip lower, PowerStrip upper) => Omni pack property lower upper meas vert horiz height width -> (PowerStripSingleton lower, PowerStripSingleton upper)
class (Natural sub, Natural super) => BandedTriangular sub super
data BandedTriangularSingleton sub super
[BandedDiagonal] :: BandedTriangularSingleton Zero Zero
[BandedUpper] :: Natural offDiag => BandedTriangularSingleton Zero (Succ offDiag)
[BandedLower] :: Natural offDiag => BandedTriangularSingleton (Succ offDiag) Zero
bandedTriangularSingleton :: BandedTriangular sub super => Banded sub super meas vert horiz height width -> BandedTriangularSingleton sub super
extent :: (Measure meas, C vert, C horiz) => Omni pack property lower upper meas vert horiz height width -> Extent meas vert horiz height width
height :: (Measure meas, C vert, C horiz) => Omni pack property lower upper meas vert horiz height width -> height
width :: (Measure meas, C vert, C horiz) => Omni pack property lower upper meas vert horiz height width -> width
squareSize :: Omni pack property lower upper Shape Small Small sh sh -> sh
mapHeight :: (C heightA, C heightB, C vert, C horiz) => (heightA -> heightB) -> Omni pack property lower upper Size vert horiz heightA width -> Omni pack property lower upper Size vert horiz heightB width
mapWidth :: (C widthA, C widthB, C vert, C horiz) => (widthA -> widthB) -> Omni pack property lower upper Size vert horiz height widthA -> Omni pack property lower upper Size vert horiz height widthB
mapSquareSize :: (C shA, C shB) => (shA -> shB) -> Omni pack property lower upper Shape Small Small shA shA -> Omni pack property lower upper Shape Small Small shB shB
order :: Omni pack property lower upper meas vert horiz height width -> Order
transpose :: (Measure meas, C vert, C horiz) => Omni pack property lower upper meas vert horiz height width -> Omni pack property upper lower meas horiz vert width height
type family Plain pack property lower upper meas vert horiz height width
class FromPlain pack property lower upper meas vert horiz height width => ToPlain pack property lower upper meas vert horiz height width
toPlain :: ToPlain pack property lower upper meas vert horiz height width => Omni pack property lower upper meas vert horiz height width -> Plain pack property lower upper meas vert horiz height width
class FromPlain pack property lower upper meas vert horiz height width
fromPlain :: FromPlain pack property lower upper meas vert horiz height width => Plain pack property lower upper meas vert horiz height width -> Omni pack property lower upper meas vert horiz height width
toFull :: (Property property, Strip lower, Strip upper) => Omni Unpacked property lower upper meas vert horiz height width -> Full meas vert horiz height width
fromFull :: (Property property, Strip lower, Strip upper) => Full meas vert horiz height width -> Omni Unpacked property lower upper meas vert horiz height width
toBanded :: (Natural sub, Natural super) => Omni Packed Arbitrary (Bands sub) (Bands super) meas vert horiz height width -> Banded sub super meas vert horiz height width
toBandedHermitian :: (C neg, C zero, C pos, Natural offDiag) => Omni Packed (Hermitian neg zero pos) (Bands offDiag) (Bands offDiag) Shape Small Small size size -> BandedHermitian offDiag size
type family MultipliedBands bandsA bandsB
type family MultipliedStrip contA contB
type family MultipliedProperty propA propB
type family UnitIfTriangular lower upper
type family MergeUnit unit0 unit1
class Quadratic pack property lower upper
quadratic :: (Quadratic pack property lower upper, C sh) => Order -> sh -> Omni pack property lower upper Shape Small Small sh sh
uncheckedDiagonal :: (Packing pack, TriDiag diag) => Order -> size -> Omni pack diag Empty Empty Shape Small Small size size
data Power pack property lower upper meas vert horiz height width
[PowerIdentity] :: Packing pack => PowerQuadratic pack Unit Empty Empty sh
[PowerDiagonal] :: Packing pack => Power pack property Empty Empty meas vert horiz height width
[PowerUpperTriangular] :: (Packing pack, TriDiag diag) => PowerQuadratic pack diag Empty Filled sh
[PowerLowerTriangular] :: (Packing pack, TriDiag diag) => PowerQuadratic pack diag Filled Empty sh
[PowerSymmetric] :: Packing pack => PowerQuadratic pack Symmetric Filled Filled sh
[PowerHermitian] :: (Packing pack, C neg, C zero, C pos) => PowerQuadratic pack (Hermitian neg zero pos) Filled Filled sh
[PowerFull] :: Power Unpacked property lower upper meas vert horiz height width
powerSingleton :: (Packing pack, Property property, PowerStrip lower, PowerStrip upper, Measure meas, C vert, C horiz) => Omni pack property lower upper meas vert horiz height width -> Power pack property lower upper meas vert horiz height width
instance (Numeric.LAPACK.Matrix.Extent.Private.Measure meas, Numeric.LAPACK.Matrix.Extent.Private.C vert, Numeric.LAPACK.Matrix.Extent.Private.C horiz, GHC.Classes.Eq height, GHC.Classes.Eq width) => GHC.Classes.Eq (Numeric.LAPACK.Matrix.Shape.Omni.Omni pack property lower upper meas vert horiz height width)
instance (Numeric.LAPACK.Matrix.Extent.Private.Measure meas, Numeric.LAPACK.Matrix.Extent.Private.C vert, Numeric.LAPACK.Matrix.Extent.Private.C horiz, GHC.Show.Show height, GHC.Show.Show width) => GHC.Show.Show (Numeric.LAPACK.Matrix.Shape.Omni.Omni pack property lower upper meas vert horiz height width)
instance (offDiag GHC.Types.~ Type.Data.Num.Unary.Literal.U0) => Numeric.LAPACK.Matrix.Shape.Omni.PowerStrip (Numeric.LAPACK.Matrix.Layout.Private.Bands offDiag)
instance Numeric.LAPACK.Matrix.Shape.Omni.PowerStrip Numeric.LAPACK.Matrix.Layout.Private.Filled
instance (Numeric.LAPACK.Matrix.Shape.Omni.Strip lower, Numeric.LAPACK.Matrix.Shape.Omni.Strip upper, Numeric.LAPACK.Matrix.Extent.Strict.Measured meas vert, Numeric.LAPACK.Matrix.Extent.Strict.Measured meas horiz) => Numeric.LAPACK.Matrix.Shape.Omni.Cons Numeric.LAPACK.Matrix.Layout.Private.Unpacked Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary lower upper meas vert horiz
instance (Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag, Numeric.LAPACK.Matrix.Extent.Private.Shape GHC.Types.~ meas, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ vert, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ horiz) => Numeric.LAPACK.Matrix.Shape.Omni.Cons Numeric.LAPACK.Matrix.Layout.Private.Packed diag Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Layout.Private.Filled meas vert horiz
instance (Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag, Numeric.LAPACK.Matrix.Extent.Private.Shape GHC.Types.~ meas, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ vert, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ horiz) => Numeric.LAPACK.Matrix.Shape.Omni.Cons Numeric.LAPACK.Matrix.Layout.Private.Packed diag Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Empty meas vert horiz
instance (Numeric.LAPACK.Matrix.Extent.Private.Shape GHC.Types.~ meas, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ vert, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ horiz) => Numeric.LAPACK.Matrix.Shape.Omni.Cons Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Symmetric Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled meas vert horiz
instance (Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos, Numeric.LAPACK.Matrix.Extent.Private.Shape GHC.Types.~ meas, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ vert, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ horiz) => Numeric.LAPACK.Matrix.Shape.Omni.Cons Numeric.LAPACK.Matrix.Layout.Private.Packed (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos) Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled meas vert horiz
instance (Numeric.LAPACK.Matrix.Extent.Strict.Measured meas vert, Numeric.LAPACK.Matrix.Extent.Strict.Measured meas horiz, Type.Data.Num.Unary.Natural sub, Type.Data.Num.Unary.Natural super) => Numeric.LAPACK.Matrix.Shape.Omni.Cons Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) meas vert horiz
instance (Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular sub super, Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular super sub, Numeric.LAPACK.Matrix.Extent.Private.Shape GHC.Types.~ meas, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ vert, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ horiz) => Numeric.LAPACK.Matrix.Shape.Omni.Cons Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Unit (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) meas vert horiz
instance (Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos, Type.Data.Num.Unary.Natural sub, sub GHC.Types.~ super, Numeric.LAPACK.Matrix.Extent.Private.Shape GHC.Types.~ meas, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ vert, Numeric.LAPACK.Matrix.Extent.Private.Small GHC.Types.~ horiz) => Numeric.LAPACK.Matrix.Shape.Omni.Cons Numeric.LAPACK.Matrix.Layout.Private.Packed (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos) (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) meas vert horiz
instance (Numeric.LAPACK.Matrix.Shape.Omni.Strip lower, Numeric.LAPACK.Matrix.Shape.Omni.Strip upper) => Numeric.LAPACK.Matrix.Shape.Omni.Quadratic Numeric.LAPACK.Matrix.Layout.Private.Unpacked Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary lower upper
instance Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag => Numeric.LAPACK.Matrix.Shape.Omni.Quadratic Numeric.LAPACK.Matrix.Layout.Private.Packed diag Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Layout.Private.Filled
instance Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag => Numeric.LAPACK.Matrix.Shape.Omni.Quadratic Numeric.LAPACK.Matrix.Layout.Private.Packed diag Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Empty
instance Numeric.LAPACK.Matrix.Shape.Omni.Quadratic Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Symmetric Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled
instance (Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos) => Numeric.LAPACK.Matrix.Shape.Omni.Quadratic Numeric.LAPACK.Matrix.Layout.Private.Packed (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos) Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled
instance (Type.Data.Num.Unary.Natural sub, Type.Data.Num.Unary.Natural super) => Numeric.LAPACK.Matrix.Shape.Omni.Quadratic Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super)
instance (Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular sub super, Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular super sub) => Numeric.LAPACK.Matrix.Shape.Omni.Quadratic Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Unit (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super)
instance (Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos, sub GHC.Types.~ super, Type.Data.Num.Unary.Natural super) => Numeric.LAPACK.Matrix.Shape.Omni.Quadratic Numeric.LAPACK.Matrix.Layout.Private.Packed (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos) (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super)
instance Numeric.LAPACK.Matrix.Shape.Omni.ToPlain Numeric.LAPACK.Matrix.Layout.Private.Unpacked Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled meas vert horiz height width
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.ToPlain pack diag Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.ToPlain pack diag Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.ToPlain pack Numeric.LAPACK.Matrix.Shape.Omni.Symmetric Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.ToPlain pack (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos) Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Type.Data.Num.Unary.Natural sub, Type.Data.Num.Unary.Natural super) => Numeric.LAPACK.Matrix.Shape.Omni.ToPlain Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) meas vert horiz height width
instance (Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular sub super, Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular super sub, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.ToPlain Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Unit (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Type.Data.Num.Unary.Natural sub, sub GHC.Types.~ super, height GHC.Types.~ width, Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos) => Numeric.LAPACK.Matrix.Shape.Omni.ToPlain Numeric.LAPACK.Matrix.Layout.Private.Packed (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos) (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance Numeric.LAPACK.Matrix.Shape.Omni.FromPlain Numeric.LAPACK.Matrix.Layout.Private.Unpacked Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled meas vert horiz height width
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.FromPlain pack diag Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.FromPlain pack diag Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.FromPlain pack Numeric.LAPACK.Matrix.Shape.Omni.Symmetric Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.FromPlain pack (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos) Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Type.Data.Num.Unary.Natural sub, Type.Data.Num.Unary.Natural super) => Numeric.LAPACK.Matrix.Shape.Omni.FromPlain Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) meas vert horiz height width
instance (Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular sub super, Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular super sub, height GHC.Types.~ width) => Numeric.LAPACK.Matrix.Shape.Omni.FromPlain Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Unit (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Type.Data.Num.Unary.Natural sub, sub GHC.Types.~ super, height GHC.Types.~ width, Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos) => Numeric.LAPACK.Matrix.Shape.Omni.FromPlain Numeric.LAPACK.Matrix.Layout.Private.Packed (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos) (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super) Numeric.LAPACK.Matrix.Extent.Private.Shape Numeric.LAPACK.Matrix.Extent.Private.Small Numeric.LAPACK.Matrix.Extent.Private.Small height width
instance (Numeric.LAPACK.Matrix.Extent.Private.Measure meas, Numeric.LAPACK.Matrix.Extent.Private.C vert, Numeric.LAPACK.Matrix.Extent.Private.C horiz, Control.DeepSeq.NFData height, Control.DeepSeq.NFData width) => Control.DeepSeq.NFData (Numeric.LAPACK.Matrix.Shape.Omni.Omni pack property lower upper meas vert horiz height width)
instance (Numeric.LAPACK.Matrix.Extent.Private.Measure meas, Numeric.LAPACK.Matrix.Extent.Private.C vert, Numeric.LAPACK.Matrix.Extent.Private.C horiz, Data.Array.Comfort.Shape.C height, Data.Array.Comfort.Shape.C width) => Data.Array.Comfort.Shape.C (Numeric.LAPACK.Matrix.Shape.Omni.Omni pack property lower upper meas vert horiz height width)
instance Type.Data.Num.Unary.Natural offDiag => Numeric.LAPACK.Matrix.Shape.Omni.Strip (Numeric.LAPACK.Matrix.Layout.Private.Bands offDiag)
instance Numeric.LAPACK.Matrix.Shape.Omni.Strip Numeric.LAPACK.Matrix.Layout.Private.Filled
instance Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular Type.Data.Num.Unary.Zero Type.Data.Num.Unary.Zero
instance Type.Data.Num.Unary.Natural super => Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular Type.Data.Num.Unary.Zero (Type.Data.Num.Unary.Succ super)
instance Type.Data.Num.Unary.Natural sub => Numeric.LAPACK.Matrix.Shape.Omni.BandedTriangular (Type.Data.Num.Unary.Succ sub) Type.Data.Num.Unary.Zero
instance Numeric.LAPACK.Matrix.Shape.Omni.TriDiag Numeric.LAPACK.Matrix.Shape.Omni.Unit
instance Numeric.LAPACK.Matrix.Shape.Omni.TriDiag Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance Numeric.LAPACK.Matrix.Shape.Omni.Property Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance Numeric.LAPACK.Matrix.Shape.Omni.Property Numeric.LAPACK.Matrix.Shape.Omni.Unit
instance Numeric.LAPACK.Matrix.Shape.Omni.Property Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance (Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos) => Numeric.LAPACK.Matrix.Shape.Omni.Property (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos)
instance Numeric.LAPACK.Matrix.Shape.Omni.Definiteness Type.Data.Bool.True Type.Data.Bool.True Type.Data.Bool.True
instance Numeric.LAPACK.Matrix.Shape.Omni.Definiteness Type.Data.Bool.True Type.Data.Bool.True Type.Data.Bool.False
instance Numeric.LAPACK.Matrix.Shape.Omni.Definiteness Type.Data.Bool.False Type.Data.Bool.True Type.Data.Bool.True
instance Numeric.LAPACK.Matrix.Shape.Omni.Definiteness Type.Data.Bool.True Type.Data.Bool.False Type.Data.Bool.False
instance Numeric.LAPACK.Matrix.Shape.Omni.Definiteness Type.Data.Bool.False Type.Data.Bool.True Type.Data.Bool.False
instance Numeric.LAPACK.Matrix.Shape.Omni.Definiteness Type.Data.Bool.False Type.Data.Bool.False Type.Data.Bool.True
instance GHC.Classes.Eq (Numeric.LAPACK.Matrix.Shape.Omni.DiagSingleton diag)
instance GHC.Show.Show (Numeric.LAPACK.Matrix.Shape.Omni.DiagSingleton diag)
instance Control.DeepSeq.NFData (Numeric.LAPACK.Matrix.Shape.Omni.DiagSingleton diag)

module Numeric.LAPACK.Matrix.Shape
type UpperQuasitriangular size = Quadratic Unpacked Arbitrary (Bands U1) Filled size

-- | For singular values
--   
--   However, diagonal matrices produced by singular value decomposition
--   may be non-square and Hermitian must be square.
type PositiveDiagonal size = Quadratic Packed HermitianPositiveDefinite Empty Empty size

-- | For Hermitian eigenvalues
type RealDiagonal size = BandedHermitian U0 size
type BandedHermitian offDiag size = Quadratic Packed HermitianUnknownDefiniteness (Bands offDiag) (Bands offDiag) size
type BandedUnitUpper super size = BandedUnitTriangular U0 super size
type BandedUnitLower sub size = BandedUnitTriangular sub U0 size
type BandedUnitTriangular sub super size = Quadratic Packed Unit (Bands sub) (Bands super) size
type BandedUpper super size = BandedTriangular U0 super size
type BandedLower sub size = BandedTriangular sub U0 size
type BandedTriangular sub super size = Quadratic Packed Arbitrary (Bands sub) (Bands super) size
type BandedGeneral sub super = Omni Packed Arbitrary (Bands sub) (Bands super) Size Big Big
type Banded sub super meas vert horiz = Omni Packed Arbitrary (Bands sub) (Bands super) meas vert horiz
type Symmetric size = Quadratic Packed Symmetric Filled Filled size
type UpperTriangular size = Quadratic Packed Arbitrary Empty Filled size
type LowerTriangular size = Quadratic Packed Arbitrary Filled Empty size
newtype GenTriangularLoUp diag size a lo up
GenTriangularLoUp :: Triangular lo diag up size -> GenTriangularLoUp diag size a lo up
[runGenTriangularLoUp] :: GenTriangularLoUp diag size a lo up -> Triangular lo diag up size
newtype GenTriangularDiag lo up size a diag
GenTriangularDiag :: Triangular lo diag up size -> GenTriangularDiag lo up size a diag
[runGenTriangularDiag] :: GenTriangularDiag lo up size a diag -> Triangular lo diag up size
type Triangular lo diag up size = Quadratic Packed diag lo up size
data UpLoSingleton lo up
[Lower] :: UpLoSingleton Filled Empty
[Upper] :: UpLoSingleton Empty Filled
class (PowerStrip lo, PowerStrip up) => DiagUpLoC lo up
switchDiagUpLo :: DiagUpLoC lo up => f Empty Empty -> f Empty Filled -> f Filled Empty -> f lo up
type DiagUpLo lo up = (DiagUpLoC lo up, DiagUpLoC up lo)
class (DiagUpLoC lo up) => UpLoC lo up
switchUpLo :: UpLoC lo up => f Empty Filled -> f Filled Empty -> f lo up
type UpLo lo up = (UpLoC lo up, UpLoC up lo)
type Identity size = Quadratic Packed Unit Empty Empty size
type Diagonal size = Quadratic Packed Arbitrary Empty Empty size
type Hermitian size = Quadratic Packed HermitianUnknownDefiniteness Filled Filled size
type QuadraticMeas pack property lower upper meas height width = Omni pack property lower upper meas Small Small height width
type Quadratic pack property lower upper size = QuadraticMeas pack property lower upper Shape size size
type Square sh = Full Shape Small Small sh sh
type LiberalSquare = Full Size Small Small
type Wide = Full Size Small Big
type Tall = Full Size Big Small
type General = Full Size Big Big
type Full = Omni Unpacked Arbitrary Filled Filled
general :: Order -> height -> width -> General height width
tall :: (C height, C width) => Order -> height -> width -> Tall height width
wide :: (C height, C width) => Order -> height -> width -> Wide height width
liberalSquare :: (C height, C width) => Order -> height -> width -> LiberalSquare height width
square :: C sh => Order -> sh -> Square sh
hermitian :: Order -> sh -> Hermitian sh
diagonal :: Order -> size -> Diagonal size
identity :: Order -> size -> Identity size
autoUplo :: UpLo lo up => UpLoSingleton lo up
triangular :: (DiagUpLo lo up, TriDiag diag) => Order -> size -> Triangular lo diag up size
unitTriangular :: DiagUpLo lo up => Order -> size -> Triangular lo Unit up size
arbitraryTriangular :: DiagUpLo lo up => Order -> size -> Triangular lo Arbitrary up size
lowerTriangular :: Order -> size -> LowerTriangular size
upperTriangular :: Order -> size -> UpperTriangular size
symmetric :: Order -> size -> Symmetric size
bandedOffDiagonals :: Omni Packed property (Bands sub) (Bands super) meas vert horiz height width -> (UnaryProxy sub, UnaryProxy super)
bandedGeneral :: (Natural sub, Natural super, C height, C width) => (UnaryProxy sub, UnaryProxy super) -> Order -> height -> width -> BandedGeneral sub super height width
bandedHermitian :: Natural offDiag => UnaryProxy offDiag -> Order -> size -> BandedHermitian offDiag size
data Order
RowMajor :: Order
ColumnMajor :: Order
flipOrder :: Order -> Order
height :: (Measure meas, C vert, C horiz) => Omni pack property lower upper meas vert horiz height width -> height
width :: (Measure meas, C vert, C horiz) => Omni pack property lower upper meas vert horiz height width -> width
extent :: (Measure meas, C vert, C horiz) => Omni pack property lower upper meas vert horiz height width -> Extent meas vert horiz height width
squareSize :: Omni pack property lower upper Shape Small Small sh sh -> sh
order :: Omni pack property lower upper meas vert horiz height width -> Order
class (Property diag) => TriDiag diag
data DiagSingleton diag
[Unit] :: DiagSingleton Unit
[Arbitrary] :: DiagSingleton Arbitrary
class Property property

-- | <a>PowerStrip</a> is either <a>Empty</a> or <a>Filled</a>. These are
--   the <a>Strip</a>s that are preserved in matrix powers.
--   
--   Pun intended.
class (Strip c) => PowerStrip c
data PowerStripSingleton c
[Empty] :: PowerStripSingleton Empty
[Filled] :: PowerStripSingleton Filled
powerStripSingleton :: PowerStrip c => PowerStripSingleton c
class (MultipliedBands c Filled ~ Filled, MultipliedBands c Empty ~ c) => Strip c
data StripSingleton c
[StripBands] :: Natural offDiag => HeadSingleton offDiag -> StripSingleton (Bands offDiag)
[StripFilled] :: StripSingleton Filled
stripSingleton :: Strip c => StripSingleton c
data Arbitrary
data Unit
data Filled
type Empty = Bands U0
data Bands offDiag
addOffDiagonals :: (Natural subA, Natural superA, Natural subB, Natural superB, (subA :+: subB) ~ subC, (superA :+: superB) ~ superC) => (UnaryProxy subA, UnaryProxy superA) -> (UnaryProxy subB, UnaryProxy superB) -> ((Nat subC, Nat superC), (UnaryProxy subC, UnaryProxy superC))
class Packing pack
data Packed
data Unpacked
instance Numeric.LAPACK.Matrix.Shape.UpLoC Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Layout.Private.Filled
instance Numeric.LAPACK.Matrix.Shape.UpLoC Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Empty
instance Numeric.LAPACK.Matrix.Shape.DiagUpLoC Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Layout.Private.Empty
instance Numeric.LAPACK.Matrix.Shape.DiagUpLoC Numeric.LAPACK.Matrix.Layout.Private.Empty Numeric.LAPACK.Matrix.Layout.Private.Filled
instance Numeric.LAPACK.Matrix.Shape.DiagUpLoC Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Empty

module Numeric.LAPACK.Matrix.Extent
class C tag
switchTag :: C tag => f Small -> f Big -> f tag
data Small
data Big
class Measure meas
switchMeasure :: Measure meas => f Shape -> f Size -> f meas
data Shape
data Size

-- | Admissible tag combinations are:
--   
--   <pre>
--   meas  vert  horiz
--   Shape Small Small - Square
--   Size  Small Small - LiberalSquare
--   Size  Big   Small - Tall
--   Size  Small Big   - Wide
--   Size  Big   Big   - General
--   </pre>
--   
--   We can enforce this set with the constraints
--   
--   <pre>
--   (Extent.Measured meas vert, Extent.Measured meas horiz)
--   </pre>
--   
--   However, in some cases it leads to constraints like <tt>Measured meas
--   Small</tt> or <tt>Measured meas Big</tt>. The former one is morally
--   equivalent to <tt>Measure meas</tt> and the latter one is morally
--   equivalent to <tt>meas ~ Size</tt>. However, in order to convince the
--   compiler you would have to go through <a>switchMeasured</a>.
--   
--   In order to circumvent this trouble we use internal functions with
--   weaker constraints:
--   
--   <pre>
--   (Extent.Measure meas, Extent.C vert, Extent.C horiz)
--   </pre>
--   
--   This is typesafe whenever the input is based on one of the five
--   admissible extent types. We only need the strict constraints when
--   constructing matrices of arbitrary extent type, i.e. this almost only
--   concerns <a>fromSquare</a>.
class (Measure meas, C tag) => Measured meas tag
switchMeasured :: Measured meas tag => f Shape Small -> f Size Small -> f Size Big -> f meas tag
data Extent meas vert horiz height width
data Map measA vertA horizA measB vertB horizB height width
height :: (Measure meas, C vert, C horiz) => Extent meas vert horiz height width -> height
width :: (Measure meas, C vert, C horiz) => Extent meas vert horiz height width -> width
squareSize :: Square shape -> shape
dimensions :: (Measure meas, C vert, C horiz) => Extent meas vert horiz height width -> (height, width)
transpose :: (Measure meas, C vert, C horiz) => Extent meas vert horiz height width -> Extent meas horiz vert width height
fuse :: (Measure meas, C vert, C horiz, Eq fuse) => Extent meas vert horiz height fuse -> Extent meas vert horiz fuse width -> Maybe (Extent meas vert horiz height width)
type family MultiplyMeasure a b
square :: sh -> Square sh
toGeneral :: (Measure meas, C vert, C horiz) => Map meas vert horiz Size Big Big height width
fromSquare :: (Measured meas vert, Measured meas horiz) => Map Shape Small Small meas vert horiz size size
fromSquareLiberal :: (Measured meas vert, Measured meas horiz) => Map Shape Small Small meas vert horiz height width
fromLiberalSquare :: (C vert, C horiz) => Map Size Small Small Size vert horiz height width
generalizeTall :: (Measure meas, C vert, C horiz) => Map meas vert Small Size vert horiz height width
generalizeWide :: (Measure meas, C vert, C horiz) => Map meas Small horiz Size vert horiz height width
weakenTall :: (Measured meas horiz, C vert) => Map meas vert Small meas vert horiz height width
weakenWide :: (Measured meas vert, C horiz) => Map meas Small horiz meas vert horiz height width
data AppendMode vertA vertB vertC height widthA widthB
appendSame :: C vert => AppendMode vert vert vert height widthA widthB
appendLeft :: C vert => AppendMode vert Big vert height widthA widthB
appendRight :: C vert => AppendMode Big vert vert height widthA widthB
type family Append a b
appendAny :: (C vertA, C vertB) => AppendMode vertA vertB (Append vertA vertB) height widthA widthB

module Numeric.LAPACK.Vector
type Vector = Array
type family RealOf x
type ComplexOf x = Complex (RealOf x)
toList :: (C sh, Storable a) => Vector sh a -> [a]
fromList :: (C sh, Storable a) => sh -> [a] -> Vector sh a
autoFromList :: Storable a => [a] -> Vector (ZeroBased Int) a
append :: (C shx, C shy, Storable a) => Array shx a -> Array shy a -> Array (shx ::+ shy) a
infixr 5 `append`

-- | Precedence and associativity (right) of (List.++). This also matches
--   <a>(::+)</a>.
(+++) :: (C shx, C shy, Storable a) => Vector shx a -> Vector shy a -> Vector (shx ::+ shy) a
infixr 5 +++

-- | <pre>
--   \(QC.NonNegative n) (Array16 x)  -&gt;  x == Array.mapShape (Shape.ZeroBased . Shape.size) (Array.append (Array.take n x) (Array.drop n x))
--   </pre>
take :: (Integral n, Storable a) => n -> Array (ZeroBased n) a -> Array (ZeroBased n) a

-- | <pre>
--   \(QC.NonNegative n) (Array16 x)  -&gt;  x == Array.mapShape (Shape.ZeroBased . Shape.size) (Array.append (Array.take n x) (Array.drop n x))
--   </pre>
drop :: (Integral n, Storable a) => n -> Array (ZeroBased n) a -> Array (ZeroBased n) a

-- | <pre>
--   \(Array16 x) (Array16 y) -&gt; let xy = Array.append x y in x == Array.takeLeft xy  &amp;&amp;  y == Array.takeRight xy
--   </pre>
takeLeft :: (C sh0, C sh1, Storable a) => Array (sh0 ::+ sh1) a -> Array sh0 a
takeRight :: (C sh0, C sh1, Storable a) => Array (sh0 ::+ sh1) a -> Array sh1 a
swap :: (Indexed sh, Storable a) => Index sh -> Index sh -> Vector sh a -> Vector sh a

-- | <pre>
--   \x  -&gt;  Array.singleton x ! () == (x::Word16)
--   </pre>
singleton :: Storable a => a -> Array () a

-- | <pre>
--   constant () = singleton
--   </pre>
--   
--   However, singleton does not need <a>Floating</a> constraint.
constant :: (C sh, Floating a) => sh -> a -> Vector sh a
zero :: (C sh, Floating a) => sh -> Vector sh a
one :: (C sh, Floating a) => sh -> Vector sh a
unit :: (Indexed sh, Floating a) => sh -> Index sh -> Vector sh a

-- | <pre>
--   dot x y = Matrix.toScalar (singleRow x &lt;#&gt; singleColumn y)
--   </pre>
dot :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> a

-- | <pre>
--   inner x y = dot (conjugate x) y
--   </pre>
inner :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> a

-- | <pre>
--   dot x y = Matrix.toScalar (singleRow x &lt;#&gt; singleColumn y)
--   </pre>
(-*|) :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> a
infixl 7 -*|
sum :: (C sh, Floating a) => Vector sh a -> a

-- | Sum of the absolute values of real numbers or components of complex
--   numbers. For real numbers it is equivalent to <a>norm1</a>.
absSum :: (C sh, Floating a) => Vector sh a -> RealOf a
norm1 :: (C sh, Floating a) => Vector sh a -> RealOf a

-- | Euclidean norm of a vector or Frobenius norm of a matrix.
norm2 :: (C sh, Floating a) => Vector sh a -> RealOf a
norm2Squared :: (C sh, Floating a) => Vector sh a -> RealOf a
normInf :: (C sh, Floating a) => Vector sh a -> RealOf a

-- | Computes (almost) the infinity norm of the vector. For complex numbers
--   every element is replaced by the sum of the absolute component values
--   first.
normInf1 :: (C sh, Floating a) => Vector sh a -> RealOf a

-- | Returns the index and value of the element with the maximal absolute
--   value. Caution: It actually returns the value of the element, not its
--   absolute value!
argAbsMaximum :: (InvIndexed sh, Floating a) => Vector sh a -> (Index sh, a)

-- | Returns the index and value of the element with the maximal absolute
--   value. The function does not strictly compare the absolute value of a
--   complex number but the sum of the absolute complex components.
--   Caution: It actually returns the value of the element, not its
--   absolute value!
argAbs1Maximum :: (InvIndexed sh, Floating a) => Vector sh a -> (Index sh, a)
product :: (C sh, Floating a) => Vector sh a -> a
scale :: (C sh, Floating a) => a -> Vector sh a -> Vector sh a
scaleReal :: (C sh, Floating a) => RealOf a -> Vector sh a -> Vector sh a
(.*|) :: (C sh, Floating a) => a -> Vector sh a -> Vector sh a
infixl 7 .*|
add :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> Vector sh a
infixl 6 `add`
sub :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> Vector sh a
infixl 6 `sub`
(|+|) :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> Vector sh a
infixl 6 |+|
(|-|) :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> Vector sh a
infixl 6 |-|
negate :: (C sh, Floating a) => Vector sh a -> Vector sh a
raise :: (C sh, Floating a) => a -> Array sh a -> Array sh a
mac :: (C sh, Eq sh, Floating a) => a -> Vector sh a -> Vector sh a -> Vector sh a
mul :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> Vector sh a

-- | <pre>
--   \xs ys -&gt; mulConj xs ys == mul (conjugate xs) ys
--   </pre>
mulConj :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> Vector sh a
divide :: (C sh, Eq sh, Floating a) => Vector sh a -> Vector sh a -> Vector sh a
recip :: (C sh, Floating a) => Vector sh a -> Vector sh a

-- | For restrictions see <a>limits</a>.
minimum :: (C sh, Real a) => Vector sh a -> a

-- | For restrictions see <a>limits</a>.
argMinimum :: (InvIndexed sh, Index sh ~ ix, Real a) => Vector sh a -> (ix, a)

-- | For restrictions see <a>limits</a>.
maximum :: (C sh, Real a) => Vector sh a -> a

-- | For restrictions see <a>limits</a>.
argMaximum :: (InvIndexed sh, Index sh ~ ix, Real a) => Vector sh a -> (ix, a)

-- | It should hold <tt>limits x = Array.limits x</tt>. The function is
--   based on fast BLAS functions. It should be faster than
--   <tt>Array.minimum</tt> and <tt>Array.maximum</tt> although it is
--   certainly not as fast as possible. It is less precise if minimum and
--   maximum differ considerably in magnitude and there are several minimum
--   or maximum candidates of similar value. E.g. you cannot rely on the
--   property that <tt>raise (- minimum x) x</tt> has only non-negative
--   elements.
limits :: (C sh, Real a) => Vector sh a -> (a, a)
argLimits :: (InvIndexed sh, Index sh ~ ix, Real a) => Vector sh a -> ((ix, a), (ix, a))
foldl :: (C sh, Storable a) => (b -> a -> b) -> b -> Array sh a -> b
foldl1 :: (C sh, Storable a) => (a -> a -> a) -> Array sh a -> a
foldMap :: (C sh, Storable a, Ord a, Semigroup m) => (a -> m) -> Array sh a -> m
conjugate :: (C sh, Floating a) => Vector sh a -> Vector sh a
fromReal :: (C sh, Floating a) => Vector sh (RealOf a) -> Vector sh a
toComplex :: (C sh, Floating a) => Vector sh a -> Vector sh (ComplexOf a)
realPart :: (C sh, Floating a) => Vector sh a -> Vector sh (RealOf a)
imaginaryPart :: (C sh, Real a) => Vector sh (Complex a) -> Vector sh a
zipComplex :: (C sh, Eq sh, Real a) => Vector sh a -> Vector sh a -> Vector sh (Complex a)
unzipComplex :: (C sh, Real a) => Vector sh (Complex a) -> (Vector sh a, Vector sh a)
random :: (C sh, Floating a) => RandomDistribution -> sh -> Word64 -> Vector sh a
data RandomDistribution
UniformBox01 :: RandomDistribution
UniformBoxPM1 :: RandomDistribution
Normal :: RandomDistribution
UniformDisc :: RandomDistribution
UniformCircle :: RandomDistribution
instance GHC.Enum.Enum Numeric.LAPACK.Vector.RandomDistribution
instance GHC.Show.Show Numeric.LAPACK.Vector.RandomDistribution
instance GHC.Classes.Ord Numeric.LAPACK.Vector.RandomDistribution
instance GHC.Classes.Eq Numeric.LAPACK.Vector.RandomDistribution

module Numeric.LAPACK.Matrix.Array
data family Matrix typ extraLower extraUpper lower upper meas vert horiz height width a
type ArrayMatrix pack property = Matrix (Array pack property) () ()
data Array pack property
type OmniArray pack prop lower upper meas vert horiz height width a = Array (Omni pack prop lower upper meas vert horiz height width) a
type PlainArray pack prop lower upper meas vert horiz height width = Array (Plain pack prop lower upper meas vert horiz height width)
type Full meas vert horiz height width = UnpackedMatrix Arbitrary Filled Filled meas vert horiz height width
type General height width = Full Size Big Big height width
type Tall height width = Full Size Big Small height width
type Wide height width = Full Size Small Big height width
type LiberalSquare height width = SquareMeas Size height width
type Square sh = SquareMeas Shape sh sh
type SquareMeas meas height width = Full meas Small Small height width
type Quadratic pack property lower upper sh = QuadraticMeas pack property lower upper Shape sh sh
type FullQuadratic pack property sh = Quadratic pack property Filled Filled sh
type QuadraticMeas pack property lower upper meas height width = ArrayMatrix pack property lower upper meas Small Small height width
plainShape :: ToPlain pack property lower upper meas vert horiz height width => ArrayMatrix pack property lower upper meas vert horiz height width a -> Plain pack property lower upper meas vert horiz height width
shape :: ArrayMatrix pack property lower upper meas vert horiz height width a -> Omni pack property lower upper meas vert horiz height width
extent :: (Measure meas, C vert, C horiz) => ArrayMatrix pack property lower upper meas vert horiz height width a -> Extent meas vert horiz height width
subBandsSingleton :: Natural sub => ArrayMatrix pack property (Bands sub) upper meas vert horiz height width a -> HeadSingleton sub
superBandsSingleton :: Natural super => ArrayMatrix pack property lower (Bands super) meas vert horiz height width a -> HeadSingleton super
packTag :: Packing pack => ArrayMatrix pack diag lower upper meas vert horiz height width a -> PackingSingleton pack
diagTag :: TriDiag diag => ArrayMatrix pack diag lower upper meas vert horiz height width a -> DiagSingleton diag
asPacked :: Id (ArrayMatrix Packed property lower upper meas vert horiz height width a)
asUnpacked :: Id (ArrayMatrix Unpacked property lower upper meas vert horiz height width a)
requirePacking :: PackingSingleton pack -> Id (ArrayMatrix pack property lower upper meas vert horiz height width a)
reshape :: (Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => (C heightA, C widthA) => (C heightB, C widthB) => Omni packB propB lowerB upperB measB vertB horizB heightB widthB -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB a
mapShape :: (Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => (C heightA, C widthA) => (C heightB, C widthB) => (Omni packA propA lowerA upperA measA vertA horizA heightA widthA -> Omni packB propB lowerB upperB measB vertB horizB heightB widthB) -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB a
unwrap :: ArrayMatrix pack property lower upper meas vert horiz height width a -> OmniArray pack property lower upper meas vert horiz height width a
toVector :: ToPlain pack property lower upper meas vert horiz height width => ArrayMatrix pack property lower upper meas vert horiz height width a -> PlainArray pack property lower upper meas vert horiz height width a
fromVector :: FromPlain pack prop lower upper meas vert horiz height width => Plain pack prop lower upper meas vert horiz height width ~ shape => (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => Array shape a -> ArrayMatrix pack prop lower upper meas vert horiz height width a

-- | <a>lift0</a> is a synonym for <a>fromVector</a> but lacks the
--   admissibility check. You may thus fool the type tags. This applies to
--   the other lift functions, too.
lift0 :: FromPlain pack prop lower upper meas vert horiz height width => PlainArray pack prop lower upper meas vert horiz height width a -> ArrayMatrix pack prop lower upper meas vert horiz height width a
lift1 :: ToPlain packA propA lowerA upperA measA vertA horizA heightA widthA => FromPlain packB propB lowerB upperB measB vertB horizB heightB widthB => (PlainArray packA propA lowerA upperA measA vertA horizA heightA widthA a -> PlainArray packB propB lowerB upperB measB vertB horizB heightB widthB b) -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB b
lift2 :: ToPlain packA propA lowerA upperA measA vertA horizA heightA widthA => ToPlain packB propB lowerB upperB measB vertB horizB heightB widthB => FromPlain packC propC lowerC upperC measC vertC horizC heightC widthC => (PlainArray packA propA lowerA upperA measA vertA horizA heightA widthA a -> PlainArray packB propB lowerB upperB measB vertB horizB heightB widthB b -> PlainArray packC propC lowerC upperC measC vertC horizC heightC widthC c) -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB b -> ArrayMatrix packC propC lowerC upperC measC vertC horizC heightC widthC c
lift3 :: ToPlain packA propA lowerA upperA measA vertA horizA heightA widthA => ToPlain packB propB lowerB upperB measB vertB horizB heightB widthB => ToPlain packC propC lowerC upperC measC vertC horizC heightC widthC => FromPlain packD propD lowerD upperD measD vertD horizD heightD widthD => (PlainArray packA propA lowerA upperA measA vertA horizA heightA widthA a -> PlainArray packB propB lowerB upperB measB vertB horizB heightB widthB b -> PlainArray packC propC lowerC upperC measC vertC horizC heightC widthC c -> PlainArray packD propD lowerD upperD measD vertD horizD heightD widthD d) -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB b -> ArrayMatrix packC propC lowerC upperC measC vertC horizC heightC widthC c -> ArrayMatrix packD propD lowerD upperD measD vertD horizD heightD widthD d
lift4 :: ToPlain packA propA lowerA upperA measA vertA horizA heightA widthA => ToPlain packB propB lowerB upperB measB vertB horizB heightB widthB => ToPlain packC propC lowerC upperC measC vertC horizC heightC widthC => ToPlain packD propD lowerD upperD measD vertD horizD heightD widthD => FromPlain packE propE lowerE upperE measE vertE horizE heightE widthE => (PlainArray packA propA lowerA upperA measA vertA horizA heightA widthA a -> PlainArray packB propB lowerB upperB measB vertB horizB heightB widthB b -> PlainArray packC propC lowerC upperC measC vertC horizC heightC widthC c -> PlainArray packD propD lowerD upperD measD vertD horizD heightD widthD d -> PlainArray packE propE lowerE upperE measE vertE horizE heightE widthE e) -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB b -> ArrayMatrix packC propC lowerC upperC measC vertC horizC heightC widthC c -> ArrayMatrix packD propD lowerD upperD measD vertD horizD heightD widthD d -> ArrayMatrix packE propE lowerE upperE measE vertE horizE heightE widthE e
unlift1 :: FromPlain packA propA lowerA upperA measA vertA horizA heightA widthA => ToPlain packB propB lowerB upperB measB vertB horizB heightB widthB => (ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB b) -> PlainArray packA propA lowerA upperA measA vertA horizA heightA widthA a -> PlainArray packB propB lowerB upperB measB vertB horizB heightB widthB b
unlift2 :: (FromPlain packA propA lowerA upperA measA vertA horizA heightA widthA, FromPlain packB propB lowerB upperB measB vertB horizB heightB widthB) => ToPlain packC propC lowerC upperC measC vertC horizC heightC widthC => (ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB b -> ArrayMatrix packC propC lowerC upperC measC vertC horizC heightC widthC c) -> PlainArray packA propA lowerA upperA measA vertA horizA heightA widthA a -> PlainArray packB propB lowerB upperB measB vertB horizB heightB widthB b -> PlainArray packC propC lowerC upperC measC vertC horizC heightC widthC c
unliftRow :: Order -> (General () height0 a -> General () height1 b) -> Vector height0 a -> Vector height1 b
unliftColumn :: Order -> (General height0 () a -> General height1 () b) -> Vector height0 a -> Vector height1 b
unpackedToVector :: (Property property, Strip lower, Strip upper) => UnpackedMatrix property lower upper meas vert horiz height width a -> FullArray meas vert horiz height width a
liftUnpacked0 :: (Property propertyA, Strip lowerA, Strip upperA) => FullArray measA vertA horizA heightA widthA a -> UnpackedMatrix propertyA lowerA upperA measA vertA horizA heightA widthA a
liftUnpacked1 :: (Property propertyA, Strip lowerA, Strip upperA) => (Property propertyB, Strip lowerB, Strip upperB) => (FullArray measA vertA horizA heightA widthA a -> FullArray measB vertB horizB heightB widthB b) -> UnpackedMatrix propertyA lowerA upperA measA vertA horizA heightA widthA a -> UnpackedMatrix propertyB lowerB upperB measB vertB horizB heightB widthB b
liftUnpacked2 :: (Property propertyA, Strip lowerA, Strip upperA) => (Property propertyB, Strip lowerB, Strip upperB) => (Property propertyC, Strip lowerC, Strip upperC) => (FullArray measA vertA horizA heightA widthA a -> FullArray measB vertB horizB heightB widthB b -> FullArray measC vertC horizC heightC widthC c) -> UnpackedMatrix propertyA lowerA upperA measA vertA horizA heightA widthA a -> UnpackedMatrix propertyB lowerB upperB measB vertB horizB heightB widthB b -> UnpackedMatrix propertyC lowerC upperC measC vertC horizC heightC widthC c
liftUnpacked3 :: (Property propertyA, Strip lowerA, Strip upperA) => (Property propertyB, Strip lowerB, Strip upperB) => (Property propertyC, Strip lowerC, Strip upperC) => (Property propertyD, Strip lowerD, Strip upperD) => (FullArray measA vertA horizA heightA widthA a -> FullArray measB vertB horizB heightB widthB b -> FullArray measC vertC horizC heightC widthC c -> FullArray measD vertD horizD heightD widthD d) -> UnpackedMatrix propertyA lowerA upperA measA vertA horizA heightA widthA a -> UnpackedMatrix propertyB lowerB upperB measB vertB horizB heightB widthB b -> UnpackedMatrix propertyC lowerC upperC measC vertC horizC heightC widthC c -> UnpackedMatrix propertyD lowerD upperD measD vertD horizD heightD widthD d
liftOmni1 :: (OmniArray packA propA lowerA upperA measA vertA horizA heightA widthA a -> OmniArray packB propB lowerB upperB measB vertB horizB heightB widthB b) -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB b
liftOmni2 :: (OmniArray packA propA lowerA upperA measA vertA horizA heightA widthA a -> OmniArray packB propB lowerB upperB measB vertB horizB heightB widthB b -> OmniArray packC propC lowerC upperC measC vertC horizC heightC widthC c) -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB b -> ArrayMatrix packC propC lowerC upperC measC vertC horizC heightC widthC c
class (Property property) => Homogeneous property
class (Homogeneous property) => Scale property
zero :: Homogeneous property => (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => Omni pack property lower upper meas vert horiz height width -> ArrayMatrix pack property lower upper meas vert horiz height width a
negate :: Homogeneous property => (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
scaleReal :: Homogeneous property => (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => RealOf a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
scale :: (Measure meas, C vert, C horiz) => (Scale property, C height, C width, Floating a) => a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
scaleRealReal :: Homogeneous property => (Measure meas, C vert, C horiz) => (C height, C width, Real a) => a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
(.*#) :: (Measure meas, C vert, C horiz) => (Scale property, C height, C width, Floating a) => a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 7 .*#
order :: ArrayMatrix pack property lower upper meas vert horiz height width a -> Order
forceOrder :: (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => Order -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a

-- | <tt>adaptOrder x y</tt> contains the data of <tt>y</tt> with the
--   layout of <tt>x</tt>.
adaptOrder :: (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
class (Property property) => Additive property
add :: (Measure meas, C vert, C horiz) => (Additive property, C height, Eq height, C width, Eq width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 6 `add`
(#+#) :: (Measure meas, C vert, C horiz) => (Additive property, C height, Eq height, C width, Eq width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 6 #+#
class (Additive property) => Subtractive property
sub :: (Measure meas, C vert, C horiz) => (Subtractive property, C height, Eq height, C width, Eq width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 6 `sub`
(#-#) :: (Measure meas, C vert, C horiz) => (Subtractive property, C height, Eq height, C width, Eq width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 6 #-#
class MapExtent pack property lower upper
mapExtent :: (MapExtent pack property lower upper, Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => Map measA vertA horizA measB vertB horizB height width -> ArrayMatrix pack property lower upper measA vertA horizA height width a -> ArrayMatrix pack property lower upper measB vertB horizB height width a
instance (Numeric.LAPACK.Matrix.Extent.Private.Measure meas, Numeric.LAPACK.Matrix.Extent.Private.C vert, Numeric.LAPACK.Matrix.Extent.Private.C horiz, Data.Array.Comfort.Shape.C height, Data.Array.Comfort.Shape.C width, Foreign.Storable.Storable a, GHC.Show.Show height, GHC.Show.Show width, GHC.Show.Show a) => GHC.Show.Show (Numeric.LAPACK.Matrix.Type.Matrix (Numeric.LAPACK.Matrix.Array.Array pack prop) xl xu lower upper meas vert horiz height width a)
instance (Numeric.LAPACK.Matrix.Array.MapExtent pack property lower upper, xl GHC.Types.~ (), xu GHC.Types.~ ()) => Numeric.LAPACK.Matrix.Type.MapExtent (Numeric.LAPACK.Matrix.Array.Array pack property) xl xu lower upper
instance Numeric.LAPACK.Matrix.Array.MapExtent Numeric.LAPACK.Matrix.Layout.Private.Unpacked Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary Numeric.LAPACK.Matrix.Layout.Private.Filled Numeric.LAPACK.Matrix.Layout.Private.Filled
instance (Type.Data.Num.Unary.Natural sub, Type.Data.Num.Unary.Natural super) => Numeric.LAPACK.Matrix.Array.MapExtent Numeric.LAPACK.Matrix.Layout.Private.Packed Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary (Numeric.LAPACK.Matrix.Layout.Private.Bands sub) (Numeric.LAPACK.Matrix.Layout.Private.Bands super)
instance Numeric.LAPACK.Matrix.Array.Subtractive Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance Numeric.LAPACK.Matrix.Array.Subtractive Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance (Type.Data.Bool.C neg, Type.Data.Bool.C zero, neg GHC.Types.~ pos) => Numeric.LAPACK.Matrix.Array.Subtractive (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos)
instance Numeric.LAPACK.Matrix.Array.Additive Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance Numeric.LAPACK.Matrix.Array.Additive Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance (Type.Data.Bool.C neg, Type.Data.Bool.C zero, Type.Data.Bool.C pos) => Numeric.LAPACK.Matrix.Array.Additive (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos)
instance Numeric.LAPACK.Matrix.Array.Scale Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance Numeric.LAPACK.Matrix.Array.Scale Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance Numeric.LAPACK.Matrix.Array.Homogeneous Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance Numeric.LAPACK.Matrix.Array.Homogeneous Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance (zero GHC.Types.~ Type.Data.Bool.True, neg GHC.Types.~ pos, Type.Data.Bool.C pos) => Numeric.LAPACK.Matrix.Array.Homogeneous (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos)
instance Numeric.LAPACK.Matrix.Type.NFData (Numeric.LAPACK.Matrix.Array.Array pack property)
instance Numeric.LAPACK.Matrix.Type.Box (Numeric.LAPACK.Matrix.Array.Array pack property)
instance Numeric.LAPACK.Matrix.Type.Transpose (Numeric.LAPACK.Matrix.Array.Array pack property)
instance Numeric.LAPACK.Matrix.Type.FormatMatrix (Numeric.LAPACK.Matrix.Array.Array pack property)
instance (Numeric.LAPACK.Matrix.Layout.Private.Packing pack, Numeric.LAPACK.Matrix.Shape.Omni.TriDiag diag, xl GHC.Types.~ (), xu GHC.Types.~ ()) => Numeric.LAPACK.Matrix.Type.MultiplySame (Numeric.LAPACK.Matrix.Array.Array pack diag) xl xu
instance Numeric.LAPACK.Matrix.Type.ToQuadratic (Numeric.LAPACK.Matrix.Array.Array pack property)

module Numeric.LAPACK.Permutation
data Permutation sh
newtype Shape sh
Shape :: sh -> Shape sh
newtype Element sh
Element :: CInt -> Element sh
size :: Permutation sh -> sh
identity :: C sh => sh -> Permutation sh
data Inversion
NonInverted :: Inversion
Inverted :: Inversion

-- | <pre>
--   QC.forAll QC.arbitraryBoundedEnum $ \inv -&gt; QC.forAll (QC.arbitrary &gt;&gt;= genPivots) $ \xs -&gt; xs == Perm.toPivots inv (Perm.fromPivots inv xs)
--   </pre>
fromPivots :: C sh => Inversion -> Vector (Shape sh) (Element sh) -> Permutation sh
toPivots :: C sh => Inversion -> Permutation sh -> Vector (Shape sh) (Element sh)
toMatrix :: (C sh, Floating a) => Permutation sh -> Square sh a
data Sign
Positive :: Sign
Negative :: Sign

-- | <pre>
--   QC.forAll genPerm2 $ \(p0,p1) -&gt; determinant (multiply p0 p1) == determinant p0 &lt;&gt; determinant p1
--   </pre>
determinant :: C sh => Permutation sh -> Sign

-- | <pre>
--   numberFromSign s == (-1)^fromEnum s
--   </pre>
numberFromSign :: Floating a => Sign -> a

-- | <pre>
--   QC.forAll genPerm2 $ \(p0,p1) -&gt; transpose (multiply p0 p1) == multiply (transpose p1) (transpose p0)
--   </pre>
transpose :: C sh => Permutation sh -> Permutation sh
inversionFromTransposition :: Transposition -> Inversion
multiply :: (C sh, Eq sh) => Permutation sh -> Permutation sh -> Permutation sh
apply :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Inversion -> Permutation height -> Full meas vert horiz height width a -> Full meas vert horiz height width a

module Numeric.LAPACK.Matrix.Permutation
type Permutation sh = FlexPermutation Filled Filled sh
size :: FlexPermutation lower upper sh a -> sh
identity :: C sh => sh -> FlexPermutation lower upper sh a
data Inversion
NonInverted :: Inversion
Inverted :: Inversion
inversionFromTransposition :: Transposition -> Inversion
fromPermutation :: C sh => Permutation sh -> Permutation sh a
toPermutation :: C sh => FlexPermutation lower upper sh a -> Permutation sh
toSquare :: (Strip lower, Strip upper, C sh, Floating a) => FlexPermutation lower upper sh a -> Quadratic Unpacked Arbitrary lower upper sh a
determinant :: (C sh, Floating a) => FlexPermutation lower upper sh a -> a
transpose :: C sh => FlexPermutation lower upper sh a -> FlexPermutation upper lower sh a
multiplyVector :: (C size, Eq size, Floating a) => Inversion -> FlexPermutation lower upper size a -> Vector size a -> Vector size a
multiplyFull :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Inversion -> FlexPermutation lower upper height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a

module Numeric.LAPACK.Matrix.Full
type Full meas vert horiz height width = UnpackedMatrix Arbitrary Filled Filled meas vert horiz height width
type Unpacked property lower upper meas vert horiz height width = ArrayMatrix Unpacked property lower upper meas vert horiz height width
identity :: (Measure meas, C vert, C horiz) => (C sh, Floating a) => sh -> Full meas vert horiz sh sh a
diagonal :: (Measure meas, C vert, C horiz) => (C sh, Floating a) => Vector sh a -> Full meas vert horiz sh sh a
mapExtent :: (Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => Map measA vertA horizA measB vertB horizB height width -> Full measA vertA horizA height width a -> Full measB vertB horizB height width a

-- | The number of rows must be maintained by the height mapping function.
mapHeight :: (C vert, C horiz, C heightA, C heightB, C width) => (heightA -> heightB) -> Full Size vert horiz heightA width a -> Full Size vert horiz heightB width a

-- | The number of columns must be maintained by the width mapping
--   function.
mapWidth :: (C vert, C horiz, C widthA, C widthB, C height) => (widthA -> widthB) -> Full Size vert horiz height widthA a -> Full Size vert horiz height widthB a
transpose :: (Measure meas, C vert, C horiz) => Full meas vert horiz height width a -> Full meas horiz vert width height a

-- | conjugate transpose
--   
--   Problem: <tt>adjoint a &lt;&gt; a</tt> is always square, but how to
--   convince the type checker to choose the Square type?
--   
--   Anser: Use <tt>Hermitian.toSquare $ Hermitian.gramian a</tt> instead.
adjoint :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> Full meas horiz vert width height a
multiplyVector :: (Measure meas, C vert, C horiz, C height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Vector width a -> Vector height a
multiply :: (Measure meas, C vert, C horiz, C height, C fuse, Eq fuse, C width, Floating a) => Full meas vert horiz height fuse a -> Full meas vert horiz fuse width a -> Full meas vert horiz height width a

module Numeric.LAPACK.Matrix.BandedHermitian
type BandedHermitian offDiag sh = Hermitian offDiag sh
data Transposition
NonTransposed :: Transposition
Transposed :: Transposition
class (C neg, C pos) => Semidefinite neg pos
assureFullRank :: (Semidefinite neg pos, C zero) => AnyHermitianP pack neg zero pos bands sh a -> AnyHermitianP pack neg False pos bands sh a
assureAnyRank :: (Semidefinite neg pos, C zero) => AnyHermitianP pack neg True pos bands sh a -> AnyHermitianP pack neg zero pos bands sh a
relaxSemidefinite :: (C neg, C zero, C pos) => AnyHermitianP pack neg False pos bands sh a -> AnyHermitianP pack neg zero pos bands sh a
relaxIndefinite :: (C neg, C zero, C pos) => AnyHermitianP pack neg zero pos bands sh a -> Quadratic pack HermitianUnknownDefiniteness bands bands sh a
assurePositiveDefiniteness :: (C neg, C zero, C pos) => AnyHermitianP pack neg zero pos bands sh a -> Quadratic pack HermitianPositiveDefinite bands bands sh a
relaxDefiniteness :: (C neg, C zero, C pos) => Quadratic pack HermitianPositiveDefinite bands bands sh a -> AnyHermitianP pack neg zero pos bands sh a
size :: BandedHermitian offDiag sh a -> sh
fromList :: (Natural offDiag, C size, Floating a) => UnaryProxy offDiag -> Order -> size -> [a] -> BandedHermitian offDiag size a
identity :: (C sh, Floating a) => sh -> HermitianPosDef U0 sh a
diagonal :: (C sh, Floating a) => Vector sh (RealOf a) -> Diagonal sh a
takeDiagonal :: (C neg, C zero, C pos) => (Natural offDiag, C size, Floating a) => FlexHermitian neg zero pos offDiag size a -> Vector size (RealOf a)
toHermitian :: (C neg, C zero, C pos) => (Natural offDiag, C size, Floating a) => FlexHermitian neg zero pos offDiag size a -> FlexHermitian neg zero pos size a
toBanded :: (C neg, C zero, C pos, Natural offDiag, C size, Floating a) => FlexHermitian neg zero pos offDiag size a -> Square offDiag offDiag size a
forceOrder :: (C neg, C zero, C pos, Natural offDiag, C size, Floating a) => Order -> FlexHermitian neg zero pos offDiag size a -> FlexHermitian neg zero pos offDiag size a
takeTopLeft :: (C neg, C zero, C pos, Natural offDiag, C sh0, C sh1, Floating a) => FlexHermitian neg zero pos offDiag (sh0 ::+ sh1) a -> FlexHermitian neg zero pos offDiag sh0 a
takeBottomRight :: (C neg, C zero, C pos, Natural offDiag, C sh0, C sh1, Floating a) => FlexHermitian neg zero pos offDiag (sh0 ::+ sh1) a -> FlexHermitian neg zero pos offDiag sh1 a
negate :: (C neg, C zero, C pos, Natural offDiag, C sh, Floating a) => FlexHermitian neg zero pos offDiag sh a -> FlexHermitian pos zero neg offDiag sh a
multiplyVector :: (C neg, C zero, C pos, Natural offDiag, C size, Eq size, Floating a) => Transposition -> FlexHermitian neg zero pos offDiag size a -> Vector size a -> Vector size a
multiplyFull :: (C neg, C zero, C pos, Natural offDiag, Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Transposition -> FlexHermitian neg zero pos offDiag height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
gramian :: (C size, Eq size, Floating a, Natural sub, Natural super) => Square sub super size a -> HermitianPosSemidef (sub :+: super) size a

-- | The list represents ragged rows of a sparse matrix.
sumRank1 :: (Natural k, Indexed sh, Floating a) => Order -> sh -> [(RealOf a, (Index sh, StaticVector (Succ k) a))] -> HermitianPosSemidef k sh a
eigenvalues :: (C neg, C zero, C pos, Natural offDiag) => (Permutable sh, Floating a) => FlexHermitian neg zero pos offDiag sh a -> Vector sh (RealOf a)

-- | For symmetric eigenvalue problems, <tt>eigensystem</tt> and
--   <tt>schur</tt> coincide.
eigensystem :: (C neg, C zero, C pos, Natural offDiag) => (Permutable sh, Floating a) => FlexHermitian neg zero pos offDiag sh a -> (Square sh a, Vector sh (RealOf a))

module Numeric.LAPACK.Matrix.Triangular
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

-- | <tt>adaptOrder x y</tt> contains the data of <tt>y</tt> with the
--   layout of <tt>x</tt>.
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
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

-- | <pre>
--   (vr,d,vlAdj) = eigensystem a
--   </pre>
--   
--   Counterintuitively, <tt>vr</tt> contains the right eigenvectors as
--   columns and <tt>vlAdj</tt> contains the left conjugated eigenvectors
--   as rows. The idea is to provide a decomposition of <tt>a</tt>. If
--   <tt>a</tt> is diagonalizable, then <tt>vr</tt> and <tt>vlAdj</tt> are
--   almost inverse to each other. More precisely, <tt>vlAdj &lt;&gt;
--   vr</tt> is a diagonal matrix, but not necessarily an identity matrix.
--   This is because all eigenvectors are normalized such that
--   <a>normInf1</a> is 1. With the following scaling, the decomposition
--   becomes perfect:
--   
--   <pre>
--   let scal = takeDiagonal $ vlAdj &lt;&gt; vr
--   a == vr &lt;&gt; diagonal (Vector.divide d scal) &lt;&gt; vlAdj
--   </pre>
--   
--   If <tt>a</tt> is non-diagonalizable then some columns of <tt>vr</tt>
--   and corresponding rows of <tt>vlAdj</tt> are left zero and the above
--   property does not hold.
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)

module Numeric.LAPACK.Matrix.Symmetric
type Symmetric sh = SymmetricP Packed sh
takeUpper :: Symmetric sh a -> Upper sh a
fromUpper :: Upper sh a -> Symmetric sh a
pack :: (Packing pack, C sh, Floating a) => SymmetricP pack sh a -> Symmetric sh a
assureSymmetry :: (Packing pack, C sh, Floating a) => Square sh a -> SymmetricP pack sh a
size :: SymmetricP pack sh a -> sh
fromList :: (C sh, Storable a) => Order -> sh -> [a] -> Symmetric sh a
autoFromList :: Storable a => Order -> [a] -> Symmetric ShapeInt a
identity :: (C sh, Floating a) => Order -> sh -> Symmetric sh a
diagonal :: (C sh, Floating a) => Order -> Vector sh a -> Symmetric sh a
takeDiagonal :: (C sh, Floating a) => Symmetric sh a -> Vector sh a
forceOrder :: (Packing pack, C sh, Floating a) => Order -> SymmetricP pack sh a -> SymmetricP pack sh a
transpose :: SymmetricP pack sh a -> SymmetricP pack sh a
adjoint :: (Packing pack, C sh, Floating a) => SymmetricP pack sh a -> SymmetricP pack sh a
stack :: (Packing pack, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => SymmetricP pack sh0 a -> General sh0 sh1 a -> SymmetricP pack sh1 a -> SymmetricP pack (sh0 ::+ sh1) a
(#%%%#) :: (Packing pack, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => (SymmetricP pack sh0 a, General sh0 sh1 a) -> SymmetricP pack sh1 a -> SymmetricP pack (sh0 ::+ sh1) a
infixr 2 #%%%#
split :: (Packing pack, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => SymmetricP pack (sh0 ::+ sh1) a -> (SymmetricP pack sh0 a, General sh0 sh1 a, SymmetricP pack sh1 a)
takeTopLeft :: (Packing pack, C sh0, C sh1, Floating a) => SymmetricP pack (sh0 ::+ sh1) a -> SymmetricP pack sh0 a
takeTopRight :: (Packing pack, C sh0, C sh1, Floating a) => SymmetricP pack (sh0 ::+ sh1) a -> General sh0 sh1 a
takeBottomRight :: (Packing pack, C sh0, C sh1, Floating a) => SymmetricP pack (sh0 ::+ sh1) a -> SymmetricP pack sh1 a
toSquare :: (Packing pack, C sh, Floating a) => SymmetricP pack sh a -> Square sh a
fromHermitian :: (Packing pack, C neg, C zero, C pos, C sh, Real a) => FlexHermitianP pack neg zero pos sh a -> SymmetricP pack sh a
multiplyVector :: Packing pack => (C sh, Eq sh, Floating a) => SymmetricP pack sh a -> Vector sh a -> Vector sh a
multiplyFull :: Packing pack => (Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => SymmetricP pack height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
square :: (Packing pack, C sh, Floating a) => SymmetricP pack sh a -> SymmetricP pack sh a
tensorProduct :: (Packing pack, C sh, Floating a) => Order -> Vector sh a -> SymmetricP pack sh a
sumRank1 :: (Packing pack, C sh, Eq sh, Floating a) => Order -> sh -> [(a, Vector sh a)] -> SymmetricP pack sh a
sumRank1NonEmpty :: (Packing pack, C sh, Eq sh, Floating a) => Order -> T [] (a, Vector sh a) -> SymmetricP pack sh a

-- | gramian A = A^T * A
gramian :: Packing pack => (C height, C width, Floating a) => General height width a -> SymmetricP pack width a

-- | gramianTransposed A = A * A^T = gramian (A^T)
gramianTransposed :: Packing pack => (C height, C width, Floating a) => General height width a -> SymmetricP pack height a

-- | congruenceDiagonal D A = A^T * D * A
congruenceDiagonal :: Packing pack => (C height, Eq height, C width, Floating a) => Vector height a -> General height width a -> SymmetricP pack width a

-- | congruenceDiagonalTransposed A D = A * D * A^T
congruenceDiagonalTransposed :: Packing pack => (C height, C width, Eq width, Floating a) => General height width a -> Vector width a -> SymmetricP pack height a

-- | congruence B A = A^T * B * A
congruence :: Packing pack => (C height, Eq height, C width, Floating a) => SymmetricP pack height a -> General height width a -> SymmetricP pack width a

-- | congruenceTransposed B A = A * B * A^T
congruenceTransposed :: Packing pack => (C height, C width, Eq width, Floating a) => General height width a -> SymmetricP pack width a -> SymmetricP pack height a

-- | anticommutator A B = A^T * B + B^T * A
--   
--   Not exactly a matrix anticommutator, thus I like to call it Symmetric
--   anticommutator.
anticommutator :: (Packing pack, Measure meas, C vert, C horiz, C height, Eq height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Full meas vert horiz height width a -> SymmetricP pack width a

-- | anticommutatorTransposed A B = A * B^T + B * A^T = anticommutator
--   (transpose A) (transpose B)
anticommutatorTransposed :: (Packing pack, Measure meas, C vert, C horiz, C height, Eq height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Full meas vert horiz height width a -> SymmetricP pack height a

-- | addTransposed A = A^T + A
addTransposed :: (Packing pack, C sh, Floating a) => Square sh a -> SymmetricP pack sh a
solve :: Packing pack => (Measure meas, C vert, C horiz, C sh, Eq sh, C nrhs, Floating a) => SymmetricP pack sh a -> Full meas vert horiz sh nrhs a -> Full meas vert horiz sh nrhs a
inverse :: (Packing pack, C sh, Floating a) => SymmetricP pack sh a -> SymmetricP pack sh a
determinant :: (Packing pack, C sh, Floating a) => SymmetricP pack sh a -> a

module Numeric.LAPACK.Matrix.Quadratic
asDiagonal :: Id (FlexDiagonal diag sh a)
asSymmetric :: Id (Symmetric sh a)
size :: Quadratic pack property lower upper sh a -> sh

-- | The number of rows and columns must be maintained by the shape mapping
--   function.
mapSize :: (C shA, C shB) => (shA -> shB) -> Quadratic pack property lower upper shA a -> Quadratic pack property lower upper shB a
identity :: (Quadratic pack property lower upper, C sh, Floating a) => Order -> sh -> Quadratic pack property lower upper sh a
diagonal :: (Diagonal property, Quadratic pack property lower upper, C sh, Floating a) => Order -> Vector sh a -> Quadratic pack property lower upper sh a
class (Property property) => Diagonal property
takeDiagonal :: (C sh, Floating a) => Quadratic pack property lower upper sh a -> Vector sh a
takeTopLeft :: (C sh0, C sh1, Floating a) => Quadratic pack property lower upper (sh0 ::+ sh1) a -> Quadratic pack property lower upper sh0 a
takeTopRight :: (C sh0, C sh1, Floating a) => Quadratic pack property lower Filled (sh0 ::+ sh1) a -> General sh0 sh1 a
takeBottomLeft :: (C sh0, C sh1, Floating a) => Quadratic pack property Filled upper (sh0 ::+ sh1) a -> General sh1 sh0 a
takeBottomRight :: (C sh0, C sh1, Floating a) => Quadratic pack property lower upper (sh0 ::+ sh1) a -> Quadratic pack property lower upper sh1 a
instance Numeric.LAPACK.Matrix.Quadratic.Diagonal Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance Numeric.LAPACK.Matrix.Quadratic.Diagonal Numeric.LAPACK.Matrix.Shape.Omni.Symmetric

module Numeric.LAPACK.Matrix.Square
type Square sh = SquareMeas Shape sh sh
size :: Square sh a -> sh
mapSize :: (sh0 -> sh1) -> Square sh0 a -> Square sh1 a
toFull :: (Measured meas vert, Measured meas horiz) => Square sh a -> Full meas vert horiz sh sh a
toGeneral :: Square sh a -> General sh sh a
fromFull :: (Measure meas, C vert, C horiz, Eq sh) => Full meas vert horiz sh sh a -> Square sh a
liberalFromFull :: (Measure meas, C vert, C horiz, C height, C width) => Full meas vert horiz height width a -> LiberalSquare height width a
fromScalar :: Storable a => a -> Square () a
toScalar :: Storable a => Square () a -> a
fromList :: (C sh, Storable a) => sh -> [a] -> Square sh a
autoFromList :: Storable a => [a] -> Square ShapeInt a
transpose :: Square sh a -> Square sh a

-- | conjugate transpose
adjoint :: (C sh, Floating a) => Square sh a -> Square sh a
identity :: (C sh, Floating a) => sh -> Square sh a
identityFrom :: (C sh, Floating a) => Square sh a -> Square sh a
identityFromWidth :: (C height, C width, Floating a) => General height width a -> Square width a
identityFromHeight :: (C height, C width, Floating a) => General height width a -> Square height a
diagonal :: (C sh, Floating a) => Vector sh a -> Square sh a
takeDiagonal :: (C sh, Floating a) => Square sh a -> Vector sh a
trace :: (C sh, Floating a) => Square sh a -> a
stack :: (Measure meas, C vert, C horiz, C sizeA, Eq sizeA, C sizeB, Eq sizeB, Floating a) => Square sizeA a -> Full meas vert horiz sizeA sizeB a -> Full meas horiz vert sizeB sizeA a -> Square sizeB a -> Square (sizeA ::+ sizeB) a
(|=|) :: (Measure meas, C vert, C horiz, C sizeA, Eq sizeA, C sizeB, Eq sizeB, Floating a) => (Square sizeA a, Full meas vert horiz sizeA sizeB a) -> (Full meas horiz vert sizeB sizeA a, Square sizeB a) -> Square (sizeA ::+ sizeB) a
infix 3 |=|
takeTopLeft :: (C sizeA, C sizeB, Floating a) => Square (sizeA ::+ sizeB) a -> Square sizeA a
takeBottomRight :: (C sizeA, C sizeB, Floating a) => Square (sizeA ::+ sizeB) a -> Square sizeB a
multiply :: (C sh, Eq sh, Floating a) => Square sh a -> Square sh a -> Square sh a
square :: (C sh, Floating a) => Square sh a -> Square sh a
power :: (C sh, Floating a) => Integer -> Square sh a -> Square sh a

-- | congruence B A = A^H * B * A
--   
--   The meaning and order of matrix factors of these functions is
--   consistent:
--   
--   <ul>
--   <li><a>congruence</a></li>
--   <li><a>gramian</a></li>
--   <li><a>anticommutator</a></li>
--   <li><a>congruence</a></li>
--   <li><a>congruenceDiagonal</a></li>
--   </ul>
congruence :: (C height, Eq height, C width, Floating a) => Square height a -> General height width a -> Square width a

-- | congruenceAdjoint A B = A * B * A^H
congruenceAdjoint :: (C height, C width, Eq width, Floating a) => General height width a -> Square width a -> Square height a
solve :: (Measure meas, C vert, C horiz, C sh, Eq sh, C nrhs, Floating a) => Square sh a -> Full meas vert horiz sh nrhs a -> Full meas vert horiz sh nrhs a
inverse :: (Measure meas, C height, C width, Floating a) => SquareMeas meas height width a -> SquareMeas meas width height a
determinant :: (C sh, Floating a) => Square sh a -> a
eigenvalues :: (Permutable sh, Floating a) => Square sh a -> Vector sh (ComplexOf a)

-- | If <tt>(q,r) = schur a</tt>, then <tt>a = q &lt;&gt; r &lt;&gt;
--   adjoint q</tt>, where <tt>q</tt> is unitary (orthogonal) and
--   <tt>r</tt> is a right-upper triangular matrix for complex <tt>a</tt>
--   and a 1x1-or-2x2-block upper triangular matrix for real <tt>a</tt>.
--   With <tt>takeDiagonal r</tt> you get all eigenvalues of <tt>a</tt> if
--   <tt>a</tt> is complex and the real parts of the eigenvalues if
--   <tt>a</tt> is real. Complex conjugated eigenvalues of a real matrix
--   <tt>a</tt> are encoded as 2x2 blocks along the diagonal.
--   
--   The meaning and order of matrix factors of these functions is
--   consistent:
--   
--   <ul>
--   <li><a>schur</a></li>
--   <li><a>schurComplex</a></li>
--   <li><a>eigensystem</a></li>
--   <li><a>eigensystem</a></li>
--   <li><a>congruenceAdjoint</a></li>
--   <li><a>gramianAdjoint</a></li>
--   <li><a>anticommutatorAdjoint</a></li>
--   <li><a>congruenceAdjoint</a></li>
--   <li><a>congruenceDiagonalAdjoint</a></li>
--   </ul>
schur :: (Permutable sh, Floating a) => Square sh a -> (Square sh a, QuasiUpper sh a)
schurComplex :: (Permutable sh, Real a, Complex a ~ ac) => Square sh ac -> (Square sh ac, Upper sh ac)

-- | <pre>
--   (vr,d,vlAdj) = eigensystem a
--   </pre>
--   
--   Counterintuitively, <tt>vr</tt> contains the right eigenvectors as
--   columns and <tt>vlAdj</tt> contains the left conjugated eigenvectors
--   as rows. The idea is to provide a decomposition of <tt>a</tt>. If
--   <tt>a</tt> is diagonalizable, then <tt>vr</tt> and <tt>vlAdj</tt> are
--   almost inverse to each other. More precisely, <tt>vlAdj &lt;&gt;
--   vr</tt> is a diagonal matrix, but not necessarily an identity matrix.
--   This is because all eigenvectors are normalized to Euclidean norm 1.
--   With the following scaling, the decomposition becomes perfect:
--   
--   <pre>
--   let scal = takeDiagonal $ vlAdj &lt;&gt; vr
--   a == vr #*\ Vector.divide d scal ##*# vlAdj
--   </pre>
--   
--   If <tt>a</tt> is non-diagonalizable then some columns of <tt>vr</tt>
--   and corresponding rows of <tt>vlAdj</tt> are left zero and the above
--   property does not hold.
--   
--   The meaning and order of result matrices of these functions is
--   consistent:
--   
--   <ul>
--   <li><a>eigensystem</a></li>
--   <li><a>eigensystem</a></li>
--   <li><a>decompose</a></li>
--   <li><a>decomposeTall</a></li>
--   <li><a>decomposeWide</a></li>
--   </ul>
eigensystem :: (Permutable sh, Floating a, ComplexOf a ~ ac) => Square sh a -> (Square sh ac, Vector sh ac, Square sh ac)
type ComplexOf x = Complex (RealOf x)

module Numeric.LAPACK.Matrix.HermitianPositiveDefinite
class (C neg, C pos) => Semidefinite neg pos
assureFullRank :: (Semidefinite neg pos, C zero) => AnyHermitianP pack neg zero pos bands sh a -> AnyHermitianP pack neg False pos bands sh a
assureAnyRank :: (Semidefinite neg pos, C zero) => AnyHermitianP pack neg True pos bands sh a -> AnyHermitianP pack neg zero pos bands sh a
relaxSemidefinite :: (C neg, C zero, C pos) => AnyHermitianP pack neg False pos bands sh a -> AnyHermitianP pack neg zero pos bands sh a
relaxIndefinite :: (C neg, C zero, C pos) => AnyHermitianP pack neg zero pos bands sh a -> Quadratic pack HermitianUnknownDefiniteness bands bands sh a
assurePositiveDefiniteness :: (C neg, C zero, C pos) => AnyHermitianP pack neg zero pos bands sh a -> Quadratic pack HermitianPositiveDefinite bands bands sh a
relaxDefiniteness :: (C neg, C zero, C pos) => Quadratic pack HermitianPositiveDefinite bands bands sh a -> AnyHermitianP pack neg zero pos bands sh a
solve :: (Measure meas, C vert, C horiz, Packing pack, C sh, Eq sh, C nrhs, Floating a) => HermitianPosDefP pack sh a -> Full meas vert horiz sh nrhs a -> Full meas vert horiz sh nrhs a

-- | <pre>
--   solve a b == solveDecomposed (decompose a) b
--   solve (gramian u) b == solveDecomposed u b
--   </pre>
solveDecomposed :: (Measure meas, C vert, C horiz, Packing pack, C sh, Eq sh, C nrhs, Floating a) => UpperP pack sh a -> Full meas vert horiz sh nrhs a -> Full meas vert horiz sh nrhs a
inverse :: (Packing pack, C sh, Floating a) => HermitianPosDefP pack sh a -> HermitianPosDefP pack sh a

-- | Cholesky decomposition
decompose :: (Packing pack, C sh, Floating a) => HermitianPosDefP pack sh a -> UpperP pack sh a
determinant :: (Packing pack, C sh, Floating a) => HermitianPosDefP pack sh a -> RealOf a

module Numeric.LAPACK.Matrix.Hermitian

-- | The definiteness tags mean:
--   
--   <ul>
--   <li><tt>neg == False</tt>: There is no <tt>x</tt> with <tt>x^T * A * x
--   &lt; 0</tt>.</li>
--   <li><tt>zero == False</tt>: There is no <tt>x</tt> with <tt>x^T * A *
--   x = 0</tt>.</li>
--   <li><tt>pos == False</tt>: There is no <tt>x</tt> with <tt>x^T * A * x
--   &gt; 0</tt>.</li>
--   </ul>
--   
--   If a tag is <tt>True</tt> then this imposes no further restriction on
--   the matrix.
type FlexHermitian neg zero pos sh = FlexHermitianP Packed neg zero pos sh
type Hermitian sh = HermitianP Packed sh
type HermitianPosDef sh = HermitianPosDefP Packed sh
type HermitianPosSemidef sh = HermitianPosSemidefP Packed sh
data Transposition
NonTransposed :: Transposition
Transposed :: Transposition
class (C neg, C pos) => Semidefinite neg pos
assureFullRank :: (Semidefinite neg pos, C zero) => AnyHermitianP pack neg zero pos bands sh a -> AnyHermitianP pack neg False pos bands sh a
assureAnyRank :: (Semidefinite neg pos, C zero) => AnyHermitianP pack neg True pos bands sh a -> AnyHermitianP pack neg zero pos bands sh a
relaxSemidefinite :: (C neg, C zero, C pos) => AnyHermitianP pack neg False pos bands sh a -> AnyHermitianP pack neg zero pos bands sh a
relaxIndefinite :: (C neg, C zero, C pos) => AnyHermitianP pack neg zero pos bands sh a -> Quadratic pack HermitianUnknownDefiniteness bands bands sh a
assurePositiveDefiniteness :: (C neg, C zero, C pos) => AnyHermitianP pack neg zero pos bands sh a -> Quadratic pack HermitianPositiveDefinite bands bands sh a
relaxDefiniteness :: (C neg, C zero, C pos) => Quadratic pack HermitianPositiveDefinite bands bands sh a -> AnyHermitianP pack neg zero pos bands sh a
asUnknownDefiniteness :: Id (Quadratic pack HermitianUnknownDefiniteness bands bands sh a)
pack :: (Packing pack, C neg, C zero, C pos, C sh, Floating a) => FlexHermitianP pack neg zero pos sh a -> FlexHermitian neg zero pos sh a
size :: FlexHermitianP pack neg zero pos sh a -> sh
fromList :: (C sh, Floating a) => Order -> sh -> [a] -> Hermitian sh a
autoFromList :: Floating a => Order -> [a] -> Hermitian ShapeInt a
identity :: (C sh, Floating a) => Order -> sh -> HermitianPosDef sh a
diagonal :: (C sh, Floating a) => Order -> Vector sh (RealOf a) -> Hermitian sh a
takeDiagonal :: (C neg, C zero, C pos, C sh, Floating a) => FlexHermitian neg zero pos sh a -> Vector sh (RealOf a)
forceOrder :: (Packing pack, C neg, C zero, C pos, C sh, Floating a) => Order -> FlexHermitianP pack neg zero pos sh a -> FlexHermitianP pack neg zero pos sh a

-- | <pre>
--   toSquare (stack a b c)
--   
--   =
--   
--   toSquare a ||| b
--   ===
--   adjoint b ||| toSquare c
--   </pre>
--   
--   It holds <tt>order (stack a b c) = order b</tt>. The function is most
--   efficient when the order of all blocks match.
stack :: (Packing pack, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => HermitianP pack sh0 a -> General sh0 sh1 a -> HermitianP pack sh1 a -> HermitianP pack (sh0 ::+ sh1) a
(*%%%#) :: (Packing pack, C sh0, Eq sh0, C sh1, Eq sh1, Floating a) => (HermitianP pack sh0 a, General sh0 sh1 a) -> HermitianP pack sh1 a -> HermitianP pack (sh0 ::+ sh1) a
infixr 2 *%%%#
split :: (Packing pack, C neg, C zero, C pos, C sh0, C sh1, Floating a) => FlexHermitianP pack neg zero pos (sh0 ::+ sh1) a -> (FlexHermitianP pack neg zero pos sh0 a, General sh0 sh1 a, FlexHermitianP pack neg zero pos sh1 a)

-- | Sub-matrices maintain definiteness of the original matrix. Consider
--   x^* A x &gt; 0. Then y^* (take A) y = x^* A x where some components of
--   x are zero.
takeTopLeft :: (Packing pack, C neg, C zero, C pos, C sh0, C sh1, Floating a) => FlexHermitianP pack neg zero pos (sh0 ::+ sh1) a -> FlexHermitianP pack neg zero pos sh0 a
takeTopRight :: (Packing pack, C neg, C zero, C pos, C sh0, C sh1, Floating a) => FlexHermitianP pack neg zero pos (sh0 ::+ sh1) a -> General sh0 sh1 a
takeBottomRight :: (Packing pack, C neg, C zero, C pos, C sh0, C sh1, Floating a) => FlexHermitianP pack neg zero pos (sh0 ::+ sh1) a -> FlexHermitianP pack neg zero pos sh1 a
toSquare :: (Packing pack, C neg, C zero, C pos, C sh, Floating a) => FlexHermitianP pack neg zero pos sh a -> Square sh a
fromSymmetric :: (Packing pack, C sh, Real a) => SymmetricP pack sh a -> HermitianP pack sh a
negate :: (C neg, C zero, C pos, C sh, Floating a) => AnyHermitianP pack neg zero pos bands sh a -> AnyHermitianP pack pos zero neg bands sh a
multiplyVector :: Packing pack => (C neg, C zero, C pos, C sh, Eq sh, Floating a) => Transposition -> FlexHermitianP pack neg zero pos sh a -> Vector sh a -> Vector sh a
multiplyFull :: Packing pack => (C neg, C zero, C pos, Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Transposition -> FlexHermitianP pack neg zero pos height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
square :: Packing pack => (C neg, C zero, C pos, C sh, Floating a) => FlexHermitianP pack neg zero pos sh a -> FlexHermitianP pack neg zero pos sh a
outer :: (Packing pack, C sh, Floating a) => Order -> Vector sh a -> HermitianPosSemidefP pack sh a
sumRank1 :: (Packing pack, C sh, Eq sh, Floating a) => Order -> sh -> [(RealOf a, Vector sh a)] -> HermitianPosSemidefP pack sh a
sumRank1NonEmpty :: (Packing pack, C sh, Eq sh, Floating a) => Order -> T [] (RealOf a, Vector sh a) -> HermitianPosSemidefP pack sh a
sumRank2 :: (Packing pack, C sh, Eq sh, Floating a) => Order -> sh -> [(a, (Vector sh a, Vector sh a))] -> HermitianP pack sh a
sumRank2NonEmpty :: (Packing pack, C sh, Eq sh, Floating a) => Order -> T [] (a, (Vector sh a, Vector sh a)) -> HermitianP pack sh a

-- | gramian A = A^H * A
gramian :: Packing pack => (C height, C width, Floating a) => General height width a -> HermitianPosSemidefP pack width a

-- | gramianAdjoint A = A * A^H = gramian (A^H)
gramianAdjoint :: Packing pack => (C height, C width, Floating a) => General height width a -> HermitianPosSemidefP pack height a

-- | congruenceDiagonal D A = A^H * D * A
congruenceDiagonal :: Packing pack => (C height, Eq height, C width, Floating a) => Vector height (RealOf a) -> General height width a -> HermitianP pack width a

-- | congruenceDiagonalAdjoint A D = A * D * A^H
congruenceDiagonalAdjoint :: Packing pack => (C height, C width, Eq width, Floating a) => General height width a -> Vector width (RealOf a) -> HermitianP pack height a

-- | congruence B A = A^H * B * A
congruence :: Packing pack => (C neg, C pos, C height, Eq height, C width, Floating a) => FlexHermitianP pack neg True pos height a -> General height width a -> FlexHermitianP pack neg True pos width a

-- | congruenceAdjoint B A = A * B * A^H
congruenceAdjoint :: Packing pack => (C neg, C pos, C height, C width, Eq width, Floating a) => General height width a -> FlexHermitianP pack neg True pos width a -> FlexHermitianP pack neg True pos height a

-- | anticommutator A B = A^H * B + B^H * A
--   
--   Not exactly a matrix anticommutator, thus I like to call it Hermitian
--   anticommutator.
anticommutator :: Packing pack => (Measure meas, C vert, C horiz, C height, Eq height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Full meas vert horiz height width a -> HermitianP pack width a

-- | anticommutatorAdjoint A B = A * B^H + B * A^H = anticommutator
--   (adjoint A) (adjoint B)
anticommutatorAdjoint :: Packing pack => (Measure meas, C vert, C horiz, C height, Eq height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Full meas vert horiz height width a -> HermitianP pack height a

-- | addAdjoint A = A^H + A
addAdjoint :: (Packing pack, C sh, Floating a) => Square sh a -> HermitianP pack sh a
solve :: Packing pack => (C neg, C zero, C pos, Measure meas, C vert, C horiz, C sh, Eq sh, C nrhs, Floating a) => FlexHermitianP pack neg zero pos sh a -> Full meas vert horiz sh nrhs a -> Full meas vert horiz sh nrhs a
inverse :: (Packing pack, C neg, C zero, C pos, C sh, Floating a) => FlexHermitianP pack neg zero pos sh a -> FlexHermitianP pack neg zero pos sh a
determinant :: (Packing pack, C neg, C zero, C pos, C sh, Floating a) => FlexHermitianP pack neg zero pos sh a -> RealOf a
eigenvalues :: (Packing pack, C neg, C zero, C pos, Permutable sh, Floating a) => FlexHermitianP pack neg zero pos sh a -> Vector sh (RealOf a)

-- | For symmetric eigenvalue problems, <tt>eigensystem</tt> and
--   <tt>schur</tt> coincide.
eigensystem :: (Packing pack, C neg, C zero, C pos, Permutable sh, Floating a) => FlexHermitianP pack neg zero pos sh a -> (Square sh a, Vector sh (RealOf a))

module Numeric.LAPACK.Matrix.BandedHermitianPositiveDefinite
class (C neg, C pos) => Semidefinite neg pos
assureFullRank :: (Semidefinite neg pos, C zero) => AnyHermitianP pack neg zero pos bands sh a -> AnyHermitianP pack neg False pos bands sh a
assureAnyRank :: (Semidefinite neg pos, C zero) => AnyHermitianP pack neg True pos bands sh a -> AnyHermitianP pack neg zero pos bands sh a
relaxSemidefinite :: (C neg, C zero, C pos) => AnyHermitianP pack neg False pos bands sh a -> AnyHermitianP pack neg zero pos bands sh a
relaxIndefinite :: (C neg, C zero, C pos) => AnyHermitianP pack neg zero pos bands sh a -> Quadratic pack HermitianUnknownDefiniteness bands bands sh a
assurePositiveDefiniteness :: (C neg, C zero, C pos) => AnyHermitianP pack neg zero pos bands sh a -> Quadratic pack HermitianPositiveDefinite bands bands sh a
relaxDefiniteness :: (C neg, C zero, C pos) => Quadratic pack HermitianPositiveDefinite bands bands sh a -> AnyHermitianP pack neg zero pos bands sh a
solve :: (Natural offDiag, C size, Eq size, Measure meas, C vert, C horiz, C nrhs, Floating a) => HermitianPosDef offDiag size a -> Full meas vert horiz size nrhs a -> Full meas vert horiz size nrhs a

-- | <pre>
--   solve a b == solveDecomposed (decompose a) b
--   solve (gramian u) b == solveDecomposed u b
--   </pre>
solveDecomposed :: (Natural offDiag, C size, Eq size, Measure meas, C vert, C horiz, C nrhs, Floating a) => Upper offDiag size a -> Full meas vert horiz size nrhs a -> Full meas vert horiz size nrhs a

-- | Cholesky decomposition
decompose :: (Natural offDiag, C size, Floating a) => HermitianPosDef offDiag size a -> Upper offDiag size a
determinant :: (Natural offDiag, C size, Floating a) => HermitianPosDef offDiag size a -> RealOf a

module Numeric.LAPACK.Matrix.Banded
type Banded sub super meas vert horiz height width = FlexBanded Arbitrary sub super meas vert horiz height width
type FlexBanded prop sub super meas vert horiz height width = ArrayMatrix Packed prop (Bands sub) (Bands super) meas vert horiz height width
type General sub super height width = Banded sub super Size Big Big height width
type Square sub super size = Quadratic Arbitrary sub super size
type Upper super size = Square U0 super size
type UnitUpper super size = UnitTriangular U0 super size
type Lower sub size = Square sub U0 size
type UnitLower sub size = UnitTriangular sub U0 size
type Diagonal size = FlexDiagonal Arbitrary size
type FlexDiagonal diag size = SymmQuadratic diag U0 size
type RectangularDiagonal meas vert horiz height width = Banded U0 U0 meas vert horiz height width
type Hermitian offDiag sh = SymmQuadratic HermitianUnknownDefiniteness offDiag sh
type HermitianPosSemidef offDiag sh = SymmQuadratic HermitianPositiveSemidefinite offDiag sh
type HermitianPosDef offDiag sh = SymmQuadratic HermitianPositiveDefinite offDiag sh
type FlexHermitian neg zero pos offDiag sh = SymmQuadratic (Hermitian neg zero pos) offDiag sh
height :: (Measure meas, C vert, C horiz) => Banded sub super meas vert horiz height width a -> height
width :: (Measure meas, C vert, C horiz) => Banded sub super meas vert horiz height width a -> width
fromList :: (Natural sub, Natural super, C height, C width, Storable a) => (UnaryProxy sub, UnaryProxy super) -> Order -> height -> width -> [a] -> General sub super height width a
squareFromList :: (Natural sub, Natural super, C size, Storable a) => (UnaryProxy sub, UnaryProxy super) -> Order -> size -> [a] -> Square sub super size a
lowerFromList :: (Natural sub, C size, Storable a) => UnaryProxy sub -> Order -> size -> [a] -> Lower sub size a
upperFromList :: (Natural super, C size, Storable a) => UnaryProxy super -> Order -> size -> [a] -> Upper super size a
mapExtent :: (C vertA, C horizA) => (C vertB, C horizB) => (Natural sub, Natural super) => Map measA vertA horizA measB vertB horizB height width -> Banded sub super measA vertA horizA height width a -> Banded sub super measB vertB horizB height width a
diagonal :: (C sh, Floating a) => Order -> Vector sh a -> Diagonal sh a
takeDiagonal :: (Natural sub, Natural super, C sh, Floating a) => Square sub super sh a -> Vector sh a
forceOrder :: (Natural sub, Natural super, Measure meas, C vert, C horiz, C height, C width, Floating a) => Order -> Banded sub super meas vert horiz height width a -> Banded sub super meas vert horiz height width a
noUnit :: UnitTriangular sub super sh a -> Square sub super sh a
toFull :: (Natural sub, Natural super, Measure meas, C vert, C horiz, C height, C width, Floating a) => Banded sub super meas vert horiz height width a -> Full meas vert horiz height width a
toLowerTriangular :: (Natural sub, C sh, Floating a) => Lower sub sh a -> Lower sh a
toUpperTriangular :: (Natural super, C sh, Floating a) => Upper super sh a -> Upper sh a
takeTopLeftSquare :: (Natural sub, Natural super, C sh0, C sh1, Floating a) => Square sub super (sh0 ::+ sh1) a -> Square sub super sh0 a
takeBottomRightSquare :: (Natural sub, Natural super, C sh0, C sh1, Floating a) => Square sub super (sh0 ::+ sh1) a -> Square sub super sh1 a
transpose :: (TriDiag diag, Natural sub, Natural super, Measure meas, C vert, C horiz) => FlexBanded diag sub super meas vert horiz height width a -> FlexBanded diag super sub meas horiz vert width height a
adjoint :: (Natural sub, Natural super, Measure meas, C vert, C horiz, C height, C width, Floating a) => FlexBanded diag sub super meas vert horiz height width a -> FlexBanded diag super sub meas horiz vert width height a
multiplyVector :: (Natural sub, Natural super, Measure meas, C vert, C horiz, C height, C width, Eq width, Floating a) => Banded sub super meas vert horiz height width a -> Vector width a -> Vector height a
multiply :: (Natural subA, Natural superA, Natural subB, Natural superB, (subA :+: subB) ~ subC, (superA :+: superB) ~ superC, Measure meas, C vert, C horiz, C height, C width, C fuse, Eq fuse, Floating a) => Banded subA superA meas vert horiz height fuse a -> Banded subB superB meas vert horiz fuse width a -> Banded subC superC meas vert horiz height width a
multiplyFull :: (Natural sub, Natural super, Measure meas, C vert, C horiz, C height, C width, C fuse, Eq fuse, Floating a) => Banded sub super meas vert horiz height fuse a -> Full meas vert horiz fuse width a -> Full meas vert horiz height width a
solve :: (TriDiag diag, Natural sub, Natural super, Measure meas, C vert, C horiz, C sh, Eq sh, C nrhs, Floating a) => Quadratic diag sub super sh a -> Full meas vert horiz sh nrhs a -> Full meas vert horiz sh nrhs a
determinant :: (TriDiag diag, Natural sub, Natural super, C sh, Floating a) => Quadratic diag sub super sh a -> a

module Numeric.LAPACK.Matrix.Diagonal
type Diagonal size = FlexDiagonal Arbitrary size
type FlexDiagonal diag size = SymmQuadratic diag U0 size
fromList :: (C sh, Storable a) => Order -> sh -> [a] -> Diagonal sh a
autoFromList :: Storable a => Order -> [a] -> Diagonal ShapeInt a
lift :: (Packing pack, C sha, C shb, Floating a, Floating b) => (Array sha a -> Array shb b) -> FlexDiagonalP pack Arbitrary sha a -> FlexDiagonalP pack Arbitrary shb b
stack :: Packing pack => (TriDiag diag, C sh0, C sh1, Floating a) => FlexDiagonalP pack diag sh0 a -> FlexDiagonalP pack diag sh1 a -> FlexDiagonalP pack diag (sh0 ::+ sh1) a
(%%%) :: Packing pack => (TriDiag diag, C sh0, C sh1, Floating a) => FlexDiagonalP pack diag sh0 a -> FlexDiagonalP pack diag sh1 a -> FlexDiagonalP pack diag (sh0 ::+ sh1) a
infixr 2 %%%
split :: (TriDiag diag, C sh0, C sh1, Floating a) => FlexDiagonalP pack diag (sh0 ::+ sh1) a -> (FlexDiagonalP pack diag sh0 a, FlexDiagonalP pack diag sh1 a)
multiply :: (TriDiag diag, C sh, Eq sh, Floating a) => FlexDiagonal diag sh a -> FlexDiagonal diag sh a -> FlexDiagonal diag sh a
solve :: (TriDiag diag, Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => FlexDiagonal diag height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
inverse :: (TriDiag diag, C sh, Floating a) => FlexDiagonal diag sh a -> FlexDiagonal diag sh a
determinant :: (TriDiag diag, C sh, Floating a) => FlexDiagonal diag sh a -> a

module Numeric.LAPACK.Orthogonal.Householder
type Householder = HouseholderFlex Filled Filled
type General height width = Householder Size Big Big height width
type Tall height width = Householder Size Big Small height width
type Wide height width = Householder Size Small Big height width
type Square sh = SquareMeas Shape sh sh
type LiberalSquare height width = SquareMeas Size height width
mapExtent :: (Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => Map measA vertA horizA measB vertB horizB height width -> Householder measA vertA horizA height width a -> Householder measB vertB horizB height width a
fromMatrix :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> Householder meas vert horiz height width a
determinant :: (C sh, Floating a) => HouseholderFlex lower upper Shape Small Small sh sh a -> a
determinantAbsolute :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Householder meas vert horiz height width a -> RealOf a
leastSquares :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Eq width, C nrhs, Floating a) => Householder meas horiz Small height width a -> Full meas vert horiz height nrhs a -> Full meas vert horiz width nrhs a

-- | <pre>
--   HH.minimumNorm (HH.fromMatrix a) b
--   ==
--   Ortho.minimumNorm (adjoint a) b
--   </pre>
minimumNorm :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Eq width, C nrhs, Floating a) => Householder meas vert Small width height a -> Full meas vert horiz height nrhs a -> Full meas vert horiz width nrhs a
data Transposition
NonTransposed :: Transposition
Transposed :: Transposition
data Conjugation
NonConjugated :: Conjugation
Conjugated :: Conjugation
extractQ :: (Measure meas, C vert, C horiz, Permutable height, C width, Floating a) => Householder meas vert horiz height width a -> Square height a
extractR :: (Measure meas, C vert, C horiz, Permutable height, C width, Floating a) => Householder meas vert horiz height width a -> UpperTrapezoid meas vert horiz height width a
multiplyQ :: (Measure measA, C vertA, C horizA, C widthA, Measure measB, C vertB, C horizB, C widthB, Permutable height, Eq height, Floating a) => Transposition -> Conjugation -> Householder measA vertA horizA height widthA a -> Full measB vertB horizB height widthB a -> Full measB vertB horizB height widthB a
tallExtractQ :: (Measure meas, C vert, C height, Permutable width, Floating a) => Householder meas vert Small height width a -> Full meas vert Small height width a
tallExtractR :: (Measure meas, C vert, C height, Permutable width, Floating a) => Householder meas vert Small height width a -> Upper width a
tallMultiplyQ :: (Measure meas, C vert, C horiz, Permutable height, Eq height, C width, C fuse, Eq fuse, Floating a) => Householder meas vert Small height fuse a -> Full meas vert horiz fuse width a -> Full meas vert horiz height width a
tallMultiplyQAdjoint :: (Measure meas, C vert, C horiz, Permutable height, C width, C fuse, Eq fuse, Floating a) => Householder meas horiz Small fuse height a -> Full meas vert horiz fuse width a -> Full meas vert horiz height width a
tallMultiplyR :: (Measure measA, C vertA, Measure meas, C vert, C horiz, Permutable height, Eq height, C heightA, C widthB, Floating a) => Transposition -> Householder measA vertA Small heightA height a -> Full meas vert horiz height widthB a -> Full meas vert horiz height widthB a
tallSolveR :: (Measure measA, C vertA, Measure meas, C vert, C horiz, C height, Permutable width, Eq width, C nrhs, Floating a) => Transposition -> Conjugation -> Householder measA vertA Small height width a -> Full meas vert horiz width nrhs a -> Full meas vert horiz width nrhs a

module Numeric.LAPACK.Orthogonal

-- | If <tt>x = leastSquares a b</tt> then <tt>x</tt> minimizes
--   <tt>Vector.norm2 (multiply a x <tt>sub</tt> b)</tt>.
--   
--   Precondition: <tt>a</tt> must have full rank and <tt>height a &gt;=
--   width a</tt>.
leastSquares :: (Measure meas, C vert, C horiz, C height, Eq height, C width, C nrhs, Floating a) => Full meas horiz Small height width a -> Full meas vert horiz height nrhs a -> Full meas vert horiz width nrhs a

-- | The vector <tt>x</tt> with <tt>x = minimumNorm a b</tt> is the vector
--   with minimal <tt>Vector.norm2 x</tt> that satisfies <tt>multiply a x
--   == b</tt>.
--   
--   Precondition: <tt>a</tt> must have full rank and <tt>height a &lt;=
--   width a</tt>.
minimumNorm :: (Measure meas, C vert, C horiz, C height, Eq height, C width, C nrhs, Floating a) => Full meas Small vert height width a -> Full meas vert horiz height nrhs a -> Full meas vert horiz width nrhs a

-- | If <tt>(rank,x) = leastSquaresMinimumNormRCond rcond a b</tt> then
--   <tt>x</tt> is the vector with minimum <tt>Vector.norm2 x</tt> that
--   minimizes <tt>Vector.norm2 (a #*| x <tt>sub</tt> b)</tt>.
--   
--   Matrix <tt>a</tt> can have any rank but you must specify the
--   reciprocal condition of the rank-truncated matrix.
leastSquaresMinimumNormRCond :: (Measure meas, C vert, C horiz, C height, Eq height, C width, C nrhs, Floating a) => RealOf a -> Full meas horiz vert height width a -> Full meas vert horiz height nrhs a -> (Int, Full meas vert horiz width nrhs a)
pseudoInverseRCond :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => RealOf a -> Full meas vert horiz height width a -> (Int, Full meas horiz vert width height a)

-- | <tt>project b d x</tt> projects <tt>x</tt> on the plane described by
--   <tt>B*x = d</tt>.
--   
--   <tt>b</tt> must have full rank.
project :: (C height, Eq height, C width, Eq width, Floating a) => Wide height width a -> Vector height a -> Vector width a -> Vector width a

-- | <tt>leastSquaresConstraint a c b d</tt> computes <tt>x</tt> with
--   minimal <tt>|| c - A*x ||_2</tt> and constraint <tt>B*x = d</tt>.
--   
--   <tt>b</tt> must be wide and <tt>a===b</tt> must be tall and both
--   matrices must have full rank.
leastSquaresConstraint :: (C height, Eq height, C width, Eq width, C constraints, Eq constraints, Floating a) => General height width a -> Vector height a -> Wide constraints width a -> Vector constraints a -> Vector width a

-- | <tt>gaussMarkovLinearModel a b d</tt> computes <tt>(x,y)</tt> with
--   minimal <tt>|| y ||_2</tt> and constraint <tt>d = A*x + B*y</tt>.
--   
--   <tt>a</tt> must be tall and <tt>a|||b</tt> must be wide and both
--   matrices must have full rank.
gaussMarkovLinearModel :: (C height, Eq height, C width, Eq width, C opt, Eq opt, Floating a) => Tall height width a -> General height opt a -> Vector height a -> (Vector width a, Vector opt a)
determinant :: (C sh, Floating a) => Square sh a -> a

-- | Gramian determinant - works also for non-square matrices, but is
--   sensitive to transposition.
--   
--   <pre>
--   determinantAbsolute a = sqrt (Herm.determinant (Herm.gramian a))
--   </pre>
determinantAbsolute :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> RealOf a

-- | For an m-by-n-matrix <tt>a</tt> with m&gt;=n this function computes an
--   m-by-(m-n)-matrix <tt>b</tt> such that <tt>Matrix.multiply (adjoint b)
--   a</tt> is a zero matrix. The function does not try to compensate a
--   rank deficiency of <tt>a</tt>. That is, <tt>a|||b</tt> has full rank
--   if and only if <tt>a</tt> has full rank.
--   
--   For full-rank matrices you might also call this <tt>kernel</tt> or
--   <tt>nullspace</tt>.
complement :: (C height, C width, Floating a) => Tall height width a -> Tall height ShapeInt a

-- | <pre>
--   affineFrameFromFiber a b == (c,d)
--   </pre>
--   
--   Means: An affine subspace is given implicitly by {x : a#*|x == b}.
--   Convert this into an explicit representation {c#*|y|+|d : y}. Matrix
--   <tt>a</tt> must have full rank, otherwise the explicit representation
--   will miss dimensions and we cannot easily determine the origin
--   <tt>d</tt> as a minimum norm solution.
--   
--   The computation is like
--   
--   <pre>
--   c = complement $ adjoint a
--   d = minimumNorm a b
--   </pre>
--   
--   but the QR decomposition of <tt>a</tt> is computed only once.
affineFrameFromFiber :: (C width, Eq width, C height, Eq height, Floating a) => Wide height width a -> Vector height a -> (Tall width ShapeInt a, Vector width a)

-- | This conversion is somehow inverse to <a>affineFrameFromFiber</a>.
--   However, it is not precisely inverse in either direction. This is
--   because both <a>affineFrameFromFiber</a> and
--   <a>affineFiberFromFrame</a> accept non-orthogonal matrices but always
--   return orthogonal ones.
--   
--   In <tt>affineFiberFromFrame c d</tt>, matrix <tt>c</tt> should have
--   full rank, otherwise the implicit representation will miss dimensions.
affineFiberFromFrame :: (C width, Eq width, C height, Eq height, Floating a) => Tall height width a -> Vector height a -> (Wide ShapeInt height a, Vector ShapeInt a)
householder :: (Measure meas, C vert, C horiz, Permutable height, C width, Floating a) => Full meas vert horiz height width a -> (Square height a, UpperTrapezoid meas vert horiz height width a)
householderTall :: (Measure meas, C vert, C height, Permutable width, Floating a) => Full meas vert Small height width a -> (Full meas vert Small height width a, Upper width a)

module Numeric.LAPACK.Matrix.Special
data family Matrix typ extraLower extraUpper lower upper meas vert horiz height width a
type Scale sh = Quadratic Scale () () Empty Empty sh
type Inverse typ lower upper sh = Quadratic (Inverse typ) lower upper Filled Filled sh
inverse :: (Strip lower, Fill lower ~ lowerf, Strip upper, Fill upper ~ upperf) => QuadraticMeas typ xl xu upper lower meas width height a -> QuadraticMeas (Inverse typ) (xl, lower) (xu, upper) lowerf upperf meas height width a

module Numeric.LAPACK.Matrix
data family Matrix typ extraLower extraUpper lower upper meas vert horiz height width a
type Full meas vert horiz height width = UnpackedMatrix Arbitrary Filled Filled meas vert horiz height width
type General height width = Full Size Big Big height width
type Tall height width = Full Size Big Small height width
type Wide height width = Full Size Small Big height width
type Square sh = SquareMeas Shape sh sh
type LiberalSquare height width = SquareMeas Size height width
type Quadratic typ extraLower extraUpper lower upper sh = QuadraticMeas typ extraLower extraUpper lower upper Shape sh sh
type Triangular lo diag up sh = TriangularP Packed lo diag up sh
type Upper sh = FlexUpper Arbitrary sh
type Lower sh = FlexLower Arbitrary sh
type Symmetric sh = SymmetricP Packed sh
type Hermitian sh = HermitianP Packed sh
type HermitianPosDef sh = HermitianPosDefP Packed sh
type HermitianPosSemidef sh = HermitianPosSemidefP Packed sh

-- | The definiteness tags mean:
--   
--   <ul>
--   <li><tt>neg == False</tt>: There is no <tt>x</tt> with <tt>x^T * A * x
--   &lt; 0</tt>.</li>
--   <li><tt>zero == False</tt>: There is no <tt>x</tt> with <tt>x^T * A *
--   x = 0</tt>.</li>
--   <li><tt>pos == False</tt>: There is no <tt>x</tt> with <tt>x^T * A * x
--   &gt; 0</tt>.</li>
--   </ul>
--   
--   If a tag is <tt>True</tt> then this imposes no further restriction on
--   the matrix.
type FlexHermitian neg zero pos sh = FlexHermitianP Packed neg zero pos sh
type Diagonal size = FlexDiagonal Arbitrary size
type FlexDiagonal diag size = SymmQuadratic diag U0 size
type RectangularDiagonal meas vert horiz height width = Banded U0 U0 meas vert horiz height width
type Banded sub super meas vert horiz height width = FlexBanded Arbitrary sub super meas vert horiz height width
type BandedHermitian offDiag sh = Hermitian offDiag sh
type Permutation sh = FlexPermutation Filled Filled sh
type ShapeInt = ZeroBased Int
shapeInt :: Int -> ShapeInt
class (Box typ) => Transpose typ
transpose :: (Transpose typ, Measure meas, C vert, C horiz) => (C width, C height, Floating a) => Matrix typ xl xu lower upper meas vert horiz height width a -> Matrix typ xu xl upper lower meas horiz vert width height a
adjoint :: (Transpose typ, Complex typ) => (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => Matrix typ xl xu lower upper meas vert horiz height width a -> Matrix typ xu xl upper lower meas horiz vert width height a
height :: (Box typ, Measure meas, C vert, C horiz) => Matrix typ xl xu lower upper meas vert horiz height width a -> height
width :: (Box typ, Measure meas, C vert, C horiz) => Matrix typ xl xu lower upper meas vert horiz height width a -> width
class Box typ
indices :: (Box typ, Measure meas, C vert, C horiz) => (Indexed height, Indexed width) => Matrix typ xl xu lower upper meas vert horiz height width a -> [(Index height, Index width)]
reshape :: (Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => (C heightA, C widthA) => (C heightB, C widthB) => Omni packB propB lowerB upperB measB vertB horizB heightB widthB -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB a
mapShape :: (Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => (C heightA, C widthA) => (C heightB, C widthB) => (Omni packA propA lowerA upperA measA vertA horizA heightA widthA -> Omni packB propB lowerB upperB measB vertB horizB heightB widthB) -> ArrayMatrix packA propA lowerA upperA measA vertA horizA heightA widthA a -> ArrayMatrix packB propB lowerB upperB measB vertB horizB heightB widthB a

-- | Square matrices will be classified as <a>Tall</a>.
caseTallWide :: (Measure meas, C vert, C horiz, C height, C width) => Full meas vert horiz height width a -> Either (Tall height width a) (Wide height width a)
fromScalar :: Storable a => a -> General () () a
toScalar :: Storable a => General () () a -> a
fromList :: (C height, C width, Storable a) => height -> width -> [a] -> General height width a
mapExtent :: (MapExtent typ xl xu lower upper, Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => Map measA vertA horizA measB vertB horizB height width -> Matrix typ xl xu lower upper measA vertA horizA height width a -> Matrix typ xl xu lower upper measB vertB horizB height width a
fromFull :: (Measure meas, C vert, C horiz) => Full meas vert horiz height width a -> General height width a
toFull :: (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> Full meas vert horiz height width a
unpack :: (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix Unpacked property lower upper meas vert horiz height width a
asGeneral :: Id (General height width a)
asTall :: Id (Tall height width a)
asWide :: Id (Wide height width a)
tallFromGeneral :: (C height, C width, Storable a) => General height width a -> Tall height width a
wideFromGeneral :: (C height, C width, Storable a) => General height width a -> Wide height width a
generalizeTall :: (Measure meas, C vert, C horiz) => Full meas vert Small height width a -> Full Size vert horiz height width a
generalizeWide :: (Measure meas, C vert, C horiz) => Full meas Small horiz height width a -> Full Size vert horiz height width a

-- | The number of rows and columns must be maintained by the shape mapping
--   function.
mapHeight :: (MapSize typ, C vert, C horiz, C heightA, C heightB, C width) => (heightA -> heightB) -> Matrix typ extraLower extraUpper lower upper Size vert horiz heightA width a -> Matrix typ extraLower extraUpper lower upper Size vert horiz heightB width a
mapWidth :: (MapSize typ, C vert, C horiz, C height, C widthA, C widthB) => (widthA -> widthB) -> Matrix typ extraLower extraUpper lower upper Size vert horiz height widthA a -> Matrix typ extraLower extraUpper lower upper Size vert horiz height widthB a

-- | The number of rows and columns must be maintained by the shape mapping
--   function.
mapSquareSize :: (SquareShape typ, C shA, C shB) => (shA -> shB) -> Quadratic typ xl xu lower upper shA a -> Quadratic typ xl xu lower upper shB a
identity :: (Quadratic pack property lower upper, C sh, Floating a) => Order -> sh -> Quadratic pack property lower upper sh a
diagonal :: (Diagonal property, Quadratic pack property lower upper, C sh, Floating a) => Order -> Vector sh a -> Quadratic pack property lower upper sh a
fromRowsNonEmpty :: (C width, Eq width, Storable a) => T [] (Vector width a) -> General ShapeInt width a
fromRowArray :: (C height, C width, Eq width, Storable a) => width -> Array height (Vector width a) -> General height width a
fromRows :: (C width, Eq width, Storable a) => width -> [Vector width a] -> General ShapeInt width a
fromRowsNonEmptyContainer :: (f ~ T g, C g, C width, Eq width, Storable a) => f (Vector width a) -> General (Shape f) width a
fromRowContainer :: (C f, C width, Eq width, Storable a) => width -> f (Vector width a) -> General (Shape f) width a
fromColumnsNonEmpty :: (C height, Eq height, Storable a) => T [] (Vector height a) -> General height ShapeInt a
fromColumnArray :: (C height, Eq height, C width, Storable a) => height -> Array width (Vector height a) -> General height width a
fromColumns :: (C height, Eq height, Storable a) => height -> [Vector height a] -> General height ShapeInt a
fromColumnsNonEmptyContainer :: (f ~ T g, C g, C height, Eq height, Storable a) => f (Vector height a) -> General height (Shape f) a
fromColumnContainer :: (C f, C height, Eq height, Storable a) => height -> f (Vector height a) -> General height (Shape f) a
singleRow :: Order -> Vector width a -> General () width a
singleColumn :: Order -> Vector height a -> General height () a
flattenRow :: General () width a -> Vector width a
flattenColumn :: General height () a -> Vector height a
liftRow :: Order -> (Vector height0 a -> Vector height1 b) -> General () height0 a -> General () height1 b
liftColumn :: Order -> (Vector height0 a -> Vector height1 b) -> General height0 () a -> General height1 () b
unliftRow :: Order -> (General () height0 a -> General () height1 b) -> Vector height0 a -> Vector height1 b
unliftColumn :: Order -> (General height0 () a -> General height1 () b) -> Vector height0 a -> Vector height1 b
toRows :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> [Vector width a]
toColumns :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> [Vector height a]
toRowArray :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> Array height (Vector width a)
toColumnArray :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> Array width (Vector height a)
toRowContainer :: (Measure meas, C vert, C horiz, C f, C width, Floating a) => Full meas vert horiz (Shape f) width a -> f (Vector width a)
toColumnContainer :: (Measure meas, C vert, C horiz, C height, C f, Floating a) => Full meas vert horiz height (Shape f) a -> f (Vector height a)
takeRow :: (Measure meas, C vert, C horiz, Indexed height, C width, Index height ~ ix, Floating a) => Full meas vert horiz height width a -> ix -> Vector width a
takeColumn :: (Measure meas, C vert, C horiz, C height, Indexed width, Index width ~ ix, Floating a) => Full meas vert horiz height width a -> ix -> Vector height a
takeRows :: (C vert, C width, Floating a) => Int -> Full Size vert Big ShapeInt width a -> Full Size vert Big ShapeInt width a
takeColumns :: (C horiz, C height, Floating a) => Int -> Full Size Big horiz height ShapeInt a -> Full Size Big horiz height ShapeInt a

-- | Take a left-top aligned square or as much as possible of it. The
--   advantange of this function is that it maintains the matrix size
--   relation, e.g. Square remains Square, Tall remains Tall.
takeEqually :: (Measure meas, C vert, C horiz, Floating a) => Int -> Full meas vert horiz ShapeInt ShapeInt a -> Full meas vert horiz ShapeInt ShapeInt a
dropRows :: (C vert, C width, Floating a) => Int -> Full Size vert Big ShapeInt width a -> Full Size vert Big ShapeInt width a
dropColumns :: (C horiz, C height, Floating a) => Int -> Full Size Big horiz height ShapeInt a -> Full Size Big horiz height ShapeInt a

-- | Drop the same number of top-most rows and left-most columns. The
--   advantange of this function is that it maintains the matrix size
--   relation, e.g. Square remains Square, Tall remains Tall.
dropEqually :: (Measure meas, C vert, C horiz, Floating a) => Int -> Full meas vert horiz ShapeInt ShapeInt a -> Full meas vert horiz ShapeInt ShapeInt a
takeTop :: (C vert, C height0, C height1, C width, Floating a) => Full Size vert Big (height0 ::+ height1) width a -> Full Size vert Big height0 width a
takeBottom :: (C vert, C height0, C height1, C width, Floating a) => Full Size vert Big (height0 ::+ height1) width a -> Full Size vert Big height1 width a
takeLeft :: (C vert, C height, C width0, C width1, Floating a) => Full Size Big vert height (width0 ::+ width1) a -> Full Size Big vert height width0 a
takeRight :: (C vert, C height, C width0, C width1, Floating a) => Full Size Big vert height (width0 ::+ width1) a -> Full Size Big vert height width1 a

-- | The function is optimized for blocks of consecutive rows. For
--   scattered rows in column major order the function has quite ugly
--   memory access patterns.
takeRowArray :: (Indexed height, C width, C sh, Floating a) => Array sh (Index height) -> General height width a -> General sh width a
takeColumnArray :: (C height, Indexed width, C sh, Floating a) => Array sh (Index width) -> General height width a -> General height sh a
swapRows :: (Measure meas, C vert, C horiz, Indexed height, C width, Floating a) => Index height -> Index height -> Full meas vert horiz height width a -> Full meas vert horiz height width a
swapColumns :: (Measure meas, C vert, C horiz, C height, Indexed width, Floating a) => Index width -> Index width -> Full meas vert horiz height width a -> Full meas vert horiz height width a
reverseRows :: (Measure meas, C vert, C horiz, Permutable height, C width, Floating a) => Full meas vert horiz height width a -> Full meas vert horiz height width a
reverseColumns :: (Measure meas, C vert, C horiz, C height, Permutable width, Floating a) => Full meas vert horiz height width a -> Full meas vert horiz height width a
fromRowMajor :: (C height, C width, Floating a) => Array (height, width) a -> General height width a
toRowMajor :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> Array (height, width) a
forceOrder :: (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => Order -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a

-- | <tt>adaptOrder x y</tt> contains the data of <tt>y</tt> with the
--   layout of <tt>x</tt>.
adaptOrder :: (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
data OrderBias

-- | Use the element order of the first operand.
leftBias :: OrderBias

-- | Use the element order of the second operand.
rightBias :: OrderBias

-- | Choose element order such that, if possible, one part can be copied as
--   one block. For <a>above</a> this means that <tt>RowMajor</tt> is
--   chosen whenever at least one operand is <tt>RowMajor</tt> and
--   <tt>ColumnMajor</tt> is chosen when both operands are
--   <tt>ColumnMajor</tt>.
contiguousBias :: OrderBias
(|||) :: (C vertA, C vertB, C vertC, Append vertA vertB ~ vertC, C height, Eq height, C widthA, C widthB, Floating a) => Full Size vertA Big height widthA a -> Full Size vertB Big height widthB a -> Full Size vertC Big height (widthA ::+ widthB) a
infixr 3 |||
beside :: (C vertA, C vertB, C vertC, C height, Eq height, C widthA, C widthB, Floating a) => OrderBias -> AppendMode vertA vertB vertC height widthA widthB -> Full Size vertA Big height widthA a -> Full Size vertB Big height widthB a -> Full Size vertC Big height (widthA ::+ widthB) a
(===) :: (C horizA, C horizB, C horizC, Append horizA horizB ~ horizC, C width, Eq width, C heightA, C heightB, Floating a) => Full Size Big horizA heightA width a -> Full Size Big horizB heightB width a -> Full Size Big horizC (heightA ::+ heightB) width a
infixr 2 ===
above :: (C horizA, C horizB, C horizC, C width, Eq width, C heightA, C heightB, Floating a) => OrderBias -> AppendMode horizA horizB horizC width heightA heightB -> Full Size Big horizA heightA width a -> Full Size Big horizB heightB width a -> Full Size Big horizC (heightA ::+ heightB) width a
stack :: (Measure meas, C vert, C horiz, C heightA, Eq heightA, C heightB, Eq heightB, C widthA, Eq widthA, C widthB, Eq widthB, Floating a) => Full meas vert horiz heightA widthA a -> General heightA widthB a -> General heightB widthA a -> Full meas vert horiz heightB widthB a -> Full meas vert horiz (heightA ::+ heightB) (widthA ::+ widthB) a
(|*-) :: (C height, Eq height, C width, Eq width, Floating a) => Vector height a -> Vector width a -> General height width a
infixl 7 |*-

-- | <pre>
--   tensorProduct order x y = singleColumn order x #*# singleRow order y
--   </pre>
tensorProduct :: (C height, Eq height, C width, Eq width, Floating a) => Order -> Vector height a -> Vector width a -> General height width a

-- | <pre>
--   outer order x y = tensorProduct order x (Vector.conjugate y)
--   </pre>
outer :: (C height, Eq height, C width, Eq width, Floating a) => Order -> Vector height a -> Vector width a -> General height width a
kronecker :: (Measure meas, C vert, C horiz, C heightA, C widthA, C heightB, C widthB, Floating a) => Full meas vert horiz heightA widthA a -> Full meas vert horiz heightB widthB a -> Full meas vert horiz (heightA, heightB) (widthA, widthB) a
sumRank1 :: (C height, Eq height, C width, Eq width, Floating a) => (height, width) -> [(a, (Vector height a, Vector width a))] -> General height width a
map :: (Measure meas, C vert, C horiz, C height, C width, Storable a, Storable b) => (a -> b) -> Full meas vert horiz height width a -> Full meas vert horiz height width b
class Complex typ
conjugate :: (Complex typ, Matrix typ xl xu lower upper meas vert horiz height width ~ matrix, Measure meas, C vert, C horiz, C height, C width, Floating a) => matrix a -> matrix a
fromReal :: (Complex typ, Matrix typ xl xu lower upper meas vert horiz height width ~ matrix, Measure meas, C vert, C horiz, C height, C width, Floating a) => matrix (RealOf a) -> matrix a
toComplex :: (Complex typ, Matrix typ xl xu lower upper meas vert horiz height width ~ matrix, Measure meas, C vert, C horiz, C height, C width, Floating a) => matrix a -> matrix (ComplexOf a)
class (Box typ) => SquareShape typ
toSquare :: (SquareShape typ, C sh, Floating a) => Quadratic typ xl xu lower upper sh a -> Square sh a
identityFromShape :: (C sh, Floating a) => Quadratic pack property lower upper sh -> Quadratic pack property lower upper sh a
identityFrom :: (SquareShape typ, C sh, Floating a) => Quadratic typ xl xu lower upper sh a -> Quadratic typ xl xu lower upper sh a
takeDiagonal :: (SquareShape typ, C sh, Floating a) => Quadratic typ xl xu lower upper sh a -> Vector sh a
trace :: (SquareShape typ, C sh, Floating a) => Quadratic typ xl xu lower upper sh a -> a
type family RealOf x
rowSums :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> Vector height a
columnSums :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> Vector width a
rowArgAbsMaximums :: (Measure meas, C vert, C horiz, C height, InvIndexed width, Index width ~ ix, Storable ix, Floating a) => Full meas vert horiz height width a -> (Vector height ix, Vector height a)
columnArgAbsMaximums :: (Measure meas, C vert, C horiz, InvIndexed height, C width, Index height ~ ix, Storable ix, Floating a) => Full meas vert horiz height width a -> (Vector width ix, Vector width a)
scaleRows :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Vector height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
scaleColumns :: (Measure meas, C vert, C horiz, C height, C width, Eq width, Floating a) => Vector width a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
scaleRowsReal :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Vector height (RealOf a) -> Full meas vert horiz height width a -> Full meas vert horiz height width a
scaleColumnsReal :: (Measure meas, C vert, C horiz, C height, C width, Eq width, Floating a) => Vector width (RealOf a) -> Full meas vert horiz height width a -> Full meas vert horiz height width a
(\*#) :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Vector height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
infixr 7 \*#
(#*\) :: (Measure meas, C vert, C horiz, C height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Vector width a -> Full meas vert horiz height width a
infixl 7 #*\
(\\#) :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Vector height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
infixr 7 \\#
(#/\) :: (Measure meas, C vert, C horiz, C height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Vector width a -> Full meas vert horiz height width a
infixl 7 #/\
multiply :: (Measure meas, C vert, C horiz, C height, C fuse, Eq fuse, C width, Floating a) => Full meas vert horiz height fuse a -> Full meas vert horiz fuse width a -> Full meas vert horiz height width a
multiplyVector :: (Measure meas, C vert, C horiz, C height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Vector width a -> Vector height a
class (Box typ) => ToQuadratic typ
zero :: Homogeneous property => (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => Omni pack property lower upper meas vert horiz height width -> ArrayMatrix pack property lower upper meas vert horiz height width a
negate :: Homogeneous property => (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
scale :: (Measure meas, C vert, C horiz) => (Scale property, C height, C width, Floating a) => a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
scaleReal :: Homogeneous property => (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => RealOf a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
scaleRealReal :: Homogeneous property => (Measure meas, C vert, C horiz) => (C height, C width, Real a) => a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
(.*#) :: (Measure meas, C vert, C horiz) => (Scale property, C height, C width, Floating a) => a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 7 .*#
add :: (Measure meas, C vert, C horiz) => (Additive property, C height, Eq height, C width, Eq width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 6 `add`
sub :: (Measure meas, C vert, C horiz) => (Subtractive property, C height, Eq height, C width, Eq width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 6 `sub`
(#+#) :: (Measure meas, C vert, C horiz) => (Additive property, C height, Eq height, C width, Eq width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 6 #+#
(#-#) :: (Measure meas, C vert, C horiz) => (Subtractive property, C height, Eq height, C width, Eq width, Floating a) => ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a -> ArrayMatrix pack property lower upper meas vert horiz height width a
infixl 6 #-#
class (Box typA, Box typB) => Multiply typA xlA xuA typB xlB xuB lowerC upperC measC
(#*#) :: (Box typA, Strip lowerA, Strip upperA) => (Box typB, Strip lowerB, Strip upperB) => (Box typC, Strip lowerC, Strip upperC) => Multiply typA xlA xuA typB xlB xuB lowerC upperC measC => Multiplied typA xlA xuA typB xlB xuB lowerC upperC measC ~ typC => MultipliedExtra typA xlA xuA typB xlB xuB ~ xlC => MultipliedExtra typA xuA xlA typB xuB xlB ~ xuC => MultipliedStrip lowerA lowerB ~ lowerC => MultipliedStrip lowerB lowerA ~ lowerC => MultipliedStrip upperA upperB ~ upperC => MultipliedStrip upperB upperA ~ upperC => MultipliedBands lowerA lowerB ~ lowerC => MultipliedBands lowerB lowerA ~ lowerC => MultipliedBands upperA upperB ~ upperC => MultipliedBands upperB upperA ~ upperC => (Measure measA, C vertA, C horizA) => (Measure measB, C vertB, C horizB) => MultiplyMeasure measA measB ~ measC => MultiplyMeasure measB measA ~ measC => Multiply vertA vertB ~ vertC => Multiply vertB vertA ~ vertC => Multiply horizA horizB ~ horizC => Multiply horizB horizA ~ horizC => (C height, C fuse, Eq fuse, C width) => Floating a => Matrix typA xlA xuA lowerA upperA measA vertA horizA height fuse a -> Matrix typB xlB xuB lowerB upperB measB vertB horizB fuse width a -> Matrix typC xlC xuC lowerC upperC measC vertC horizC height width a
infixl 7 #*#
class (Box typ) => MultiplyVector typ xl xu
(#*|) :: (MultiplyVector typ xl xu, Strip lower, Strip upper) => (Measure meas, C vert, C horiz, C height, C width, Eq width, Floating a) => Matrix typ xl xu lower upper meas vert horiz height width a -> Vector width a -> Vector height a
infixr 7 #*|
(-*#) :: (MultiplyVector typ xl xu, Strip lower, Strip upper) => (Measure meas, C vert, C horiz, C height, C width, Eq height, Floating a) => Vector height a -> Matrix typ xl xu lower upper meas vert horiz height width a -> Vector width a
infixl 7 -*#
class (Box typ) => MultiplySquare typ xl xu
multiplySquare :: MultiplySquare typ xl xu => (Strip lower, Strip upper) => (Measure meas, C vert, C horiz, C height, Eq height, C width, Floating a) => Transposition -> Quadratic typ xl xu lower upper height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
class (Box typ) => Power typ xl xu
square :: (Power typ xl xu, PowerStrip lower, PowerStrip upper) => (C sh, Floating a) => Quadratic typ xl xu lower upper sh a -> Quadratic typ xl xu lower upper sh a
power :: (Power typ xl xu, PowerStrip lower, PowerStrip upper) => (C sh, Floating a) => Integer -> Quadratic typ xl xu lower upper sh a -> Quadratic typ xl xu lower upper sh a
powers :: (Power typ xl xu, SquareShape typ) => (PowerStrip lower, PowerStrip upper) => (C sh, Floating a) => Quadratic typ xl xu lower upper sh a -> Stream (Quadratic typ xl xu lower upper sh a)
powers1 :: (Power typ xl xu, PowerStrip lower, PowerStrip upper) => (C sh, Floating a) => Quadratic typ xl xu lower upper sh a -> Stream (Quadratic typ xl xu lower upper sh a)
(##*#) :: (MultiplySquare typ xl xu, ToQuadratic typ) => (Strip lowerA, Strip upperA) => (Strip lowerB, Strip upperB) => (Strip lowerC, Strip upperC) => MultipliedBands lowerA lowerB ~ lowerC => MultipliedBands lowerB lowerA ~ lowerC => MultipliedBands upperA upperB ~ upperC => MultipliedBands upperB upperA ~ upperC => (Measure measA, Measure measB, Measure measC, MultiplyMeasure measA measB ~ measC) => (C vert, C horiz) => (C height, C fuse, Eq fuse, C width, Floating a) => Unpacked lowerB upperB measB vert horiz height fuse a -> QuadraticMeas typ xl xu lowerA upperA measA fuse width a -> Unpacked lowerC upperC measC vert horiz height width a
infixl 7 ##*#
(#*##) :: (MultiplySquare typ xl xu, ToQuadratic typ) => (Strip lowerA, Strip upperA) => (Strip lowerB, Strip upperB) => (Strip lowerC, Strip upperC) => MultipliedBands lowerA lowerB ~ lowerC => MultipliedBands lowerB lowerA ~ lowerC => MultipliedBands upperA upperB ~ upperC => MultipliedBands upperB upperA ~ upperC => (Measure measA, Measure measB, Measure measC, MultiplyMeasure measA measB ~ measC) => (C vert, C horiz) => (C height, C fuse, Eq fuse, C width, Floating a) => QuadraticMeas typ xl xu lowerA upperA measA height fuse a -> Unpacked lowerB upperB measB vert horiz fuse width a -> Unpacked lowerC upperC measC vert horiz height width a
infixr 7 #*##
class (Box typ) => Indexed typ
(#!) :: (Indexed typ, Measure meas, C vert, C horiz, Indexed height, Indexed width, Floating a) => Matrix typ xl xu lower upper meas vert horiz height width a -> (Index height, Index width) -> a
infixl 9 #!
class (Box typ) => Determinant typ xl xu
determinant :: (Determinant typ xl xu, Strip lower, Strip upper) => (C sh, Floating a) => Quadratic typ xl xu lower upper sh a -> a
class (Box typ) => Solve typ xl xu
solve :: (Solve typ xl xu, Strip lower, Strip upper) => (Measure meas, C vert, C horiz) => (C height, C width, Eq height, Floating a) => Transposition -> Quadratic typ xl xu lower upper height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
solveLeft :: (Solve typ xl xu, Strip lower, Strip upper) => (Measure meas, C vert, C horiz) => (C height, C width, Eq width, Floating a) => Full meas vert horiz height width a -> Quadratic typ xl xu lower upper width a -> Full meas vert horiz height width a
solveRight :: (Solve typ xl xu, Strip lower, Strip upper) => (Measure meas, C vert, C horiz) => (C height, C width, Eq height, Floating a) => Quadratic typ xl xu lower upper height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
(##/#) :: (Solve typ xl xu, ToQuadratic typ, Strip lower, Strip upper, C height, C width, Eq width, C nrhs, Measure measA, Measure measB, Measure measC, MultiplyMeasure measA measB ~ measC, C vert, C horiz, Floating a) => Full measB vert horiz nrhs width a -> QuadraticMeas typ xl xu lower upper measA height width a -> Full measC vert horiz nrhs height a
infixl 7 ##/#
(#\##) :: (Solve typ xl xu, ToQuadratic typ, Strip lower, Strip upper, C height, Eq height, C width, C nrhs, Measure measA, Measure measB, Measure measC, MultiplyMeasure measA measB ~ measC, C vert, C horiz, Floating a) => QuadraticMeas typ xl xu lower upper measA height width a -> Full measB vert horiz height nrhs a -> Full measC vert horiz width nrhs a
infixr 7 #\##
solveVector :: (Solve typ xl xu, Strip lower, Strip upper, C sh, Eq sh, Floating a) => Transposition -> Quadratic typ xl xu lower upper sh a -> Vector sh a -> Vector sh a
(-/#) :: (Solve typ xl xu, ToQuadratic typ, Strip lower, Strip upper, Measure meas, C height, C width, Eq width, Floating a) => Vector width a -> QuadraticMeas typ xl xu lower upper meas height width a -> Vector height a
infixl 7 -/#
(#\|) :: (Solve typ xl xu, ToQuadratic typ, Strip lower, Strip upper, Measure meas, C height, C width, Eq height, Floating a) => QuadraticMeas typ xl xu lower upper meas height width a -> Vector height a -> Vector width a
infixr 7 #\|
class (Solve typ xl xu) => Inverse typ xl xu
inverse :: (Inverse typ xl xu, PowerStrip lower, PowerStrip upper, Measure meas, C height, C width, Floating a) => QuadraticMeas typ xl xu lower upper meas height width a -> QuadraticMeas typ xl xu lower upper meas width height a
data Transposition
NonTransposed :: Transposition
Transposed :: Transposition

module Numeric.LAPACK.Singular
values :: (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => Full meas vert horiz height width a -> RectangularDiagonal meas vert horiz height width (RealOf a)
valuesTall :: (Measure meas, C vert, C height, C width, Floating a) => Full meas vert Small height width a -> RealVector width a
valuesWide :: (Measure meas, C horiz, C height, C width, Floating a) => Full meas Small horiz height width a -> RealVector height a
decompose :: (Measure meas, C vert, C horiz) => (C height, C width, Floating a) => Full meas vert horiz height width a -> (Square height a, RectangularDiagonal meas vert horiz height width (RealOf a), Square width a)

-- | <pre>
--   let (u,s,vt) = Singular.decomposeTall a
--   in a  ==  u ##*# Matrix.scaleRowsReal s vt
--   </pre>
decomposeTall :: (Measure meas, C vert, C height, C width, Floating a) => Full meas vert Small height width a -> (Full meas vert Small height width a, RealVector width a, Square width a)

-- | <pre>
--   let (u,s,vt) = Singular.decomposeWide a
--   in a  ==  u #*## Matrix.scaleRowsReal s vt
--   </pre>
decomposeWide :: (Measure meas, C horiz, C height, C width, Floating a) => Full meas Small horiz height width a -> (Square height a, RealVector height a, Full meas Small horiz height width a)
determinantAbsolute :: (C height, C width, Floating a) => General height width a -> RealOf a
leastSquaresMinimumNormRCond :: (Measure meas, C vert, C horiz, C height, Eq height, C width, C nrhs, Floating a) => RealOf a -> Full meas horiz vert height width a -> Full meas vert horiz height nrhs a -> (Int, Full meas vert horiz width nrhs a)
pseudoInverseRCond :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => RealOf a -> Full meas vert horiz height width a -> (Int, Full meas horiz vert width height a)

-- | In <tt>decomposePolar a = (u,h)</tt>, <tt>u</tt> is the orthogonal
--   matrix closest to <tt>a</tt> with respect to the 2- and the Frobenius
--   norm. (Higham: Functions of Matrices - Theory and Computation.)
decomposePolar :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> (Full meas vert horiz height width a, Hermitian width a)
type family RealOf x


-- | This module provides an infix operator and some data constructors that
--   shall resemble mathematical notation. E.g. you can write <tt>a#^T</tt>
--   for <tt>Matrix.transpose a</tt>. It is clearly abuse of the type
--   system because I suspect you will never write algorithms that are
--   generic in the superscript type.
module Numeric.LAPACK.Matrix.Superscript

-- | left associative in contrast to ^, ^^ etc. because there is no law
--   analogous to power law <tt>(x^a)^b = x^(a*b)</tt>.
class Superscript sup where {
    data family Exponent sup typA xlA xuA lowerA upperA measA vertA horizA heightA widthA typB xlB xuB lowerB upperB measB vertB horizB heightB widthB a;
}
(#^) :: (Superscript sup, Measure measA, C vertA, C horizA) => (C widthA, C heightA) => (Measure measB, C vertB, C horizB) => (C widthB, C heightB) => Floating a => Matrix typA xlA xuA lowerA upperA measA vertA horizA heightA widthA a -> Exponent sup typA xlA xuA lowerA upperA measA vertA horizA heightA widthA typB xlB xuB lowerB upperB measB vertB horizB heightB widthB a -> Matrix typB xlB xuB lowerB upperB measB vertB horizB heightB widthB a
infixl 8 #^
data None
data Transpose
data Adjoint
data Conjugate
data Inverse
data PseudoInverse
data Power
instance Numeric.LAPACK.Matrix.Superscript.Superscript Numeric.LAPACK.Matrix.Superscript.Power
instance Numeric.LAPACK.Matrix.Superscript.Superscript Numeric.LAPACK.Matrix.Superscript.PseudoInverse
instance Numeric.LAPACK.Matrix.Superscript.Superscript Numeric.LAPACK.Matrix.Superscript.Inverse
instance Numeric.LAPACK.Matrix.Superscript.Superscript Numeric.LAPACK.Matrix.Superscript.Conjugate
instance Numeric.LAPACK.Matrix.Superscript.Superscript Numeric.LAPACK.Matrix.Superscript.Adjoint
instance Numeric.LAPACK.Matrix.Superscript.Superscript Numeric.LAPACK.Matrix.Superscript.Transpose
instance Numeric.LAPACK.Matrix.Superscript.Superscript Numeric.LAPACK.Matrix.Superscript.None

module Numeric.LAPACK.Matrix.Function
class (Property property) => SqRt property
sqrt :: (SqRt property, Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack property lower upper sh a -> Quadratic pack property lower upper sh a

-- | Square root solver that works on the Schur decomposition.
--   
--   Schur decomposition enables computing the square root of (some)
--   singular matrices like <tt>((1,0),(0,0))</tt>. However, the Schur
--   decomposition might emit small negative values on the diagonal, where
--   exact computation would yield zeros. This would let the square root
--   solver fail. And there are singular matrices that have no square root,
--   at all, e.g. <tt>((0,1),(0,0))</tt>.
--   
--   The solver is restricted to a real triangular Schur matrix. The check
--   for non-real eigenvalues may exclude matrices that actually have a
--   real-valued square root. E.g. <tt>sqrt ((0,-2),(2,0)) =
--   ((1,-1),(1,-1))</tt>
--   
--   In the future we might fix this by solving 2x2 blocks at the diagonal
--   using <a>sqrt2</a>.
sqrtSchur :: (C sh, Real a) => Square sh a -> Square sh a

-- | Iterative square root solver, similar to Newton iteration.
--   
--   Eigenvalues must all be positive, otherwise, the iteration might loop
--   forever, or if an eigenvalue is zero, the computation of matrix
--   inverse will fail.
sqrtDenmanBeavers :: (Homogeneous prop, Additive prop) => (Packing pack, PowerStrip lower, PowerStrip upper) => (C sh, Floating a, RealOf a ~ ar, Real ar) => Quadratic pack prop lower upper sh a -> Quadratic pack prop lower upper sh a
class (Property property) => Exp property
exp :: (Exp property, Packing pack, PowerStrip lower, PowerStrip upper, C sh, Floating a) => Quadratic pack property lower upper sh a -> Quadratic pack property lower upper sh a

-- | Mathematically the name <tt>expRealSymmetric</tt> would be more
--   common, but we support definiteness tags only for the
--   <tt>Hermitian</tt> type. Formally the result is always positive
--   definite, but negative eigenvalues easily yield numerically singular
--   matrices as result.
expRealHermitian :: (Packing pack, C sh, Real a) => HermitianP pack sh a -> HermitianPosSemidefP pack sh a
class (Property property) => Log property
log :: (Log property, Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack property lower upper sh a -> Quadratic pack property lower upper sh a
logUnipotentUpper :: (Packing pack, C sh, Floating a) => UnitUpperP pack sh a -> UpperP pack sh a
class (Property property) => LiftReal property

-- | Lift any function with a Taylor expansion to a diagonalizable matrix.
liftReal :: (LiftReal property, Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => (a -> a) -> Quadratic pack property lower upper sh a -> Quadratic pack property lower upper sh a
instance Numeric.LAPACK.Matrix.Function.Log Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance (neg GHC.Types.~ Type.Data.Bool.True, zero GHC.Types.~ Type.Data.Bool.True, pos GHC.Types.~ Type.Data.Bool.True) => Numeric.LAPACK.Matrix.Function.Log (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos)
instance Numeric.LAPACK.Matrix.Function.Log Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance Numeric.LAPACK.Matrix.Function.LiftReal Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance (neg GHC.Types.~ Type.Data.Bool.True, zero GHC.Types.~ Type.Data.Bool.True, pos GHC.Types.~ Type.Data.Bool.True) => Numeric.LAPACK.Matrix.Function.LiftReal (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos)
instance Numeric.LAPACK.Matrix.Function.LiftReal Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance Numeric.LAPACK.Matrix.Function.Exp Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance (neg GHC.Types.~ Type.Data.Bool.True, zero GHC.Types.~ Type.Data.Bool.True, pos GHC.Types.~ Type.Data.Bool.True) => Numeric.LAPACK.Matrix.Function.Exp (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos)
instance Numeric.LAPACK.Matrix.Function.Exp Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance Numeric.LAPACK.Matrix.Function.SqRt Numeric.LAPACK.Matrix.Shape.Omni.Unit
instance Numeric.LAPACK.Matrix.Function.SqRt Numeric.LAPACK.Matrix.Shape.Omni.Arbitrary
instance Numeric.LAPACK.Matrix.Function.SqRt Numeric.LAPACK.Matrix.Shape.Omni.Symmetric
instance (neg GHC.Types.~ Type.Data.Bool.False, Type.Data.Bool.C zero, Type.Data.Bool.C pos) => Numeric.LAPACK.Matrix.Function.SqRt (Numeric.LAPACK.Matrix.Shape.Omni.Hermitian neg zero pos)

module Numeric.LAPACK.Linear.LowerUpper
type LowerUpper = LowerUpperFlex Filled Filled
type Tall height width = LowerUpper Size Big Small height width
type Wide height width = LowerUpper Size Small Big height width
type Square sh = SquareMeas Shape sh sh
type LiberalSquare height width = SquareMeas Size height width
data Transposition
NonTransposed :: Transposition
Transposed :: Transposition
data Conjugation
NonConjugated :: Conjugation
Conjugated :: Conjugation
data Inversion
NonInverted :: Inversion
Inverted :: Inversion
mapExtent :: (C vertA, C horizA) => (C vertB, C horizB) => Map measA vertA horizA measB vertB horizB height width -> LowerUpperFlex lower upper measA vertA horizA height width a -> LowerUpperFlex lower upper measB vertB horizB height width a

-- | <tt>LowerUpper.fromMatrix a</tt> computes the LU decomposition of
--   matrix <tt>a</tt> with row pivotisation.
--   
--   You can reconstruct <tt>a</tt> from <tt>lu</tt> depending on whether
--   <tt>a</tt> is tall or wide.
--   
--   <pre>
--   LU.multiplyP NonInverted lu $ LU.extractL lu ##*# LU.tallExtractU lu
--   LU.multiplyP NonInverted lu $ LU.wideExtractL lu #*## LU.extractU lu
--   </pre>
fromMatrix :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> LowerUpper meas vert horiz height width a
toMatrix :: (Measure meas, C vert, C horiz, C height, Eq height, C width, Eq width, Floating a) => LowerUpper meas vert horiz height width a -> Full meas vert horiz height width a
solve :: (Measure meas, C vert, C horiz, Eq height, C height, C width, Floating a) => Square height a -> Full meas vert horiz height width a -> Full meas vert horiz height width a
multiplyFull :: (Measure meas, C vert, C horiz, C height, Eq height, C width, C fuse, Eq fuse, Floating a) => LowerUpper meas vert horiz height fuse a -> Full meas vert horiz fuse width a -> Full meas vert horiz height width a

-- | Caution: <tt>LU.determinant . LU.fromMatrix</tt> will fail for
--   singular matrices.
determinant :: (Measure meas, C height, C width, Floating a) => LowerUpperFlex lower upper meas Small Small height width a -> a
extractP :: (Measure meas, C vert, C horiz, Permutable height, C width) => Inversion -> LowerUpper meas vert horiz height width a -> Permutation height a
multiplyP :: (Measure measA, C vertA, C horizA, Measure measB, C vertB, C horizB, Eq height, Permutable height, C widthA, C widthB, Floating a) => Inversion -> LowerUpper measA vertA horizA height widthA a -> Full measB vertB horizB height widthB a -> Full measB vertB horizB height widthB a
extractL :: (Measure meas, C vert, C horiz, Permutable height, Permutable width, Floating a) => LowerUpper meas vert horiz height width a -> LowerTrapezoid meas vert horiz height width a
wideExtractL :: (Measure meas, C horiz, Permutable height, C width, Floating a) => LowerUpper meas Small horiz height width a -> UnitLower height a

-- | <tt>wideMultiplyL transposed lu a</tt> multiplies the square part of
--   <tt>lu</tt> containing the lower triangular matrix with <tt>a</tt>.
--   
--   <pre>
--   wideMultiplyL NonTransposed lu a == wideExtractL lu #*## a
--   wideMultiplyL Transposed lu a == Tri.transpose (wideExtractL lu) #*## a
--   </pre>
wideMultiplyL :: (Measure measA, C horizA, Measure meas, C vert, C horiz, Permutable height, Eq height, C widthA, C widthB, Floating a) => Transposition -> LowerUpper measA Small horizA height widthA a -> Full meas vert horiz height widthB a -> Full meas vert horiz height widthB a
wideSolveL :: (Measure measA, C horizA, Measure meas, C vert, C horiz, Permutable height, Eq height, C width, C nrhs, Floating a) => Transposition -> Conjugation -> LowerUpper measA Small horizA height width a -> Full meas vert horiz height nrhs a -> Full meas vert horiz height nrhs a
extractU :: (Measure meas, C vert, C horiz, Permutable height, Permutable width, Floating a) => LowerUpper meas vert horiz height width a -> UpperTrapezoid meas vert horiz height width a
tallExtractU :: (Measure meas, C vert, C height, Permutable width, Floating a) => LowerUpper meas vert Small height width a -> Upper width a

-- | <tt>tallMultiplyU transposed lu a</tt> multiplies the square part of
--   <tt>lu</tt> containing the upper triangular matrix with <tt>a</tt>.
--   
--   <pre>
--   tallMultiplyU NonTransposed lu a == tallExtractU lu #*## a
--   tallMultiplyU Transposed lu a == Tri.transpose (tallExtractU lu) #*## a
--   </pre>
tallMultiplyU :: (Measure measA, C vertA, Measure meas, C vert, C horiz, Permutable height, Eq height, C heightA, C widthB, Floating a) => Transposition -> LowerUpper measA vertA Small heightA height a -> Full meas vert horiz height widthB a -> Full meas vert horiz height widthB a
tallSolveU :: (Measure measA, C vertA, Measure meas, C vert, C horiz, C height, Permutable width, Eq width, C nrhs, Floating a) => Transposition -> Conjugation -> LowerUpper measA vertA Small height width a -> Full meas vert horiz width nrhs a -> Full meas vert horiz width nrhs a
caseTallWide :: (Measure meas, C vert, C horiz, C height, C width) => LowerUpperFlex lower upper meas vert horiz height width a -> Either (Tall height width a) (Wide height width a)

module Numeric.LAPACK.Format
(##) :: Format a => a -> String -> IO ()
infix 0 ##
print :: Format a => String -> a -> IO ()
hyper :: Format a => String -> a -> Graphic
class Format a
format :: (Format a, Output out) => String -> a -> out
class (C sh) => FormatArray sh
formatArray :: (FormatArray sh, Floating a, Output out) => String -> Array sh a -> out
class FormatMatrix typ
formatMatrix :: (FormatMatrix typ, Measure meas, C vert, C horiz, C width, C height, Floating a, Output out) => String -> Matrix typ xl xu lower upper meas vert horiz height width a -> out
deflt :: String
instance Numeric.LAPACK.Format.Format GHC.Types.Int
instance Numeric.LAPACK.Format.Format GHC.Types.Float
instance Numeric.LAPACK.Format.Format GHC.Types.Double
instance Numeric.Netlib.Class.Real a => Numeric.LAPACK.Format.Format (Data.Complex.Complex a)
instance Numeric.LAPACK.Format.Format a => Numeric.LAPACK.Format.Format [a]
instance (Numeric.LAPACK.Format.Format a, Numeric.LAPACK.Format.Format b) => Numeric.LAPACK.Format.Format (a, b)
instance (Numeric.LAPACK.Format.Format a, Numeric.LAPACK.Format.Format b, Numeric.LAPACK.Format.Format c) => Numeric.LAPACK.Format.Format (a, b, c)
instance Data.Array.Comfort.Shape.C sh => Numeric.LAPACK.Format.Format (Numeric.LAPACK.Permutation.Private.Permutation sh)
instance (Numeric.LAPACK.Matrix.Plain.Format.FormatArray sh, Numeric.Netlib.Class.Floating a) => Numeric.LAPACK.Format.Format (Data.Array.Comfort.Storable.Private.Array sh a)
instance (Numeric.LAPACK.Matrix.Type.FormatMatrix typ, Numeric.LAPACK.Matrix.Extent.Private.Measure meas, Numeric.LAPACK.Matrix.Extent.Private.C vert, Numeric.LAPACK.Matrix.Extent.Private.C horiz, Data.Array.Comfort.Shape.C width, Data.Array.Comfort.Shape.C height, Numeric.Netlib.Class.Floating a) => Numeric.LAPACK.Format.Format (Numeric.LAPACK.Matrix.Type.Matrix typ xl xu lower upper meas vert horiz height width a)


-- | Do not import this module. It is only for demonstration purposes.
module Numeric.LAPACK.Example.EconomicAllocation
type ZeroInt2 = ShapeInt ::+ ShapeInt
type Vector sh = Vector sh Double
type Matrix height width = General height width Double
type SquareMatrix size = Square size Double
balances0 :: Vector ZeroInt2
expenses0 :: Matrix ShapeInt ZeroInt2
normalize :: (Eq height, C height, C width) => Matrix height width -> Matrix height width
normalizeSplit :: (C sh0, C sh1, Eq sh1) => Matrix sh1 (sh0 ::+ sh1) -> (Matrix sh0 sh1, SquareMatrix sh1)
completeIdSquare :: (C sh0, Eq sh0, C sh1, Eq sh1) => Matrix sh1 (sh0 ::+ sh1) -> SquareMatrix (sh0 ::+ sh1)
iterated :: (C sh0, Eq sh0, C sh1, Eq sh1) => Matrix sh1 (sh0 ::+ sh1) -> Vector (sh0 ::+ sh1) -> Vector (sh0 ::+ sh1)
compensated :: (C sh0, Eq sh0, C sh1, Eq sh1) => Matrix sh1 (sh0 ::+ sh1) -> Vector (sh0 ::+ sh1) -> Vector sh0

-- | <pre>
--   let result = iterated expenses0 balances0 in approxVector result $ compensated expenses0 balances0 +++ Vector.zero (Array.shape $ Vector.takeRight result)
--   </pre>
main :: IO ()


-- | This module demonstrates triangular matrices.
--   
--   It verifies that the divided difference scheme nicely fits into a
--   triangular matrix, where function addition is mapped to matrix
--   addition and function multiplication is mapped to matrix
--   multiplication.
--   
--   <a>http://en.wikipedia.org/wiki/Divided_difference</a>
module Numeric.LAPACK.Example.DividedDifference
size :: Vector ShapeInt a -> Int
subSlices :: Int -> Vector ShapeInt Float -> Vector ShapeInt Float
parameterDifferences :: Vector ShapeInt Float -> [Vector ShapeInt Float]
dividedDifferences :: Vector ShapeInt Float -> Vector ShapeInt Float -> [Vector ShapeInt Float]
upperFromPyramid :: (C sh, Storable a) => sh -> [Vector sh a] -> Upper sh a

-- | <pre>
--   QC.forAll genDD $ \(xs, (ys0,ys1)) -&gt; approxArray (dividedDifferencesMatrix xs (ys0|+|ys1)) (dividedDifferencesMatrix xs ys0 #+# dividedDifferencesMatrix xs ys1)
--   </pre>
--   
--   <pre>
--   QC.forAll genDD $ \(xs, (ys0,ys1)) -&gt; approxArray (dividedDifferencesMatrix xs (Vector.mul ys0 ys1)) (dividedDifferencesMatrix xs ys0 &lt;&gt; dividedDifferencesMatrix xs ys1)
--   </pre>
dividedDifferencesMatrix :: Vector ShapeInt Float -> Vector ShapeInt Float -> Upper ShapeInt Float

-- | <pre>
--   QC.forAll (QC.choose (0,10)) $ \n -&gt; let sh = shapeInt n in QC.forAll (Util.genDistinct [-10..10] [-10..10] sh) $ \xs -&gt; approxArray (DD.parameterDifferencesMatrix xs) (DD.upperFromPyramid sh (Vector.zero sh : DD.parameterDifferences xs))
--   </pre>
parameterDifferencesMatrix :: Vector ShapeInt Float -> Upper ShapeInt Float
main :: IO ()