Numeric.LinearAlgebra.Class

Typeclasses for linear algebra and related concepts

Matrix and vector elements (possibly Complex)

class (Eq e, Fractional e, Floating e, Num (EltMag e), Ord (EltMag e)) => Elt e where

Minimal complete definition

mag

Associated Types

type EltMag e :: *

Methods

conj :: e -> e

mag :: e -> EltMag e

Instances

Elt Double

Associated Types

type EltMag Double :: *

Methods

conj :: Double -> Double

mag :: Double -> EltMag Double

Elt Float

Associated Types

type EltMag Float :: *

Methods

conj :: Float -> Float

mag :: Float -> EltMag Float

RealFloat e => Elt (Complex e)

Associated Types

type EltMag (Complex e) :: *

Methods

conj :: Complex e -> Complex e

mag :: Complex e -> EltMag (Complex e)

Vector space

(.*) :: VectorSpace v => Scalar v -> v -> v

Scale a vector

(./) :: (VectorSpace v, Fractional (Scalar v)) => v -> Scalar v -> v

Scale a vector by the reciprocal of a number (e.g. for normalization)

lerp :: (VectorSpace e, Num (Scalar e)) => Scalar e -> e -> e -> e

Convex combination of two vectors (NB: 0 <= a <= 1).

Hilbert space (inner product)

dot :: InnerSpace v => v -> v -> Scalar v

Inner product

Hilbert-space distance function

hilbertDistSq :: InnerSpace v => v -> v -> Scalar v

`hilbertDistSq x y = || x - y ||^2` computes the squared L2 distance between two vectors

Normed vector spaces

class (InnerSpace v, Num (RealScalar v), Eq (RealScalar v), Epsilon (Magnitude v), Show (Magnitude v), Ord (Magnitude v)) => Normed v where

Minimal complete definition

norm1, norm2Sq, normP, normalize, normalize2

Associated Types

type Magnitude v :: *

type RealScalar v :: *

Methods

norm1 :: v -> Magnitude v

norm2Sq :: v -> Magnitude v

normP :: RealScalar v -> v -> Magnitude v

normalize :: RealScalar v -> v -> v

normalize2 :: v -> v

normalize2' :: Floating (Scalar v) => v -> v

norm2 :: Floating (Magnitude v) => v -> Magnitude v

norm2' :: Floating (Scalar v) => v -> Scalar v

norm :: Floating (Magnitude v) => RealScalar v -> v -> Magnitude v

Instances

Normed Double

Associated Types

type Magnitude Double :: *

type RealScalar Double :: *

Methods

norm1 :: Double -> Magnitude Double

norm2Sq :: Double -> Magnitude Double

normP :: RealScalar Double -> Double -> Magnitude Double

normalize :: RealScalar Double -> Double -> Double

normalize2 :: Double -> Double

normalize2' :: Double -> Double

norm2 :: Double -> Magnitude Double

norm2' :: Double -> Scalar Double

norm :: RealScalar Double -> Double -> Magnitude Double

Normed (Complex Double)

Associated Types

type Magnitude (Complex Double) :: *

type RealScalar (Complex Double) :: *

Methods

norm1 :: Complex Double -> Magnitude (Complex Double)

norm2Sq :: Complex Double -> Magnitude (Complex Double)

normP :: RealScalar (Complex Double) -> Complex Double -> Magnitude (Complex Double)

normalize :: RealScalar (Complex Double) -> Complex Double -> Complex Double

normalize2 :: Complex Double -> Complex Double

normalize2' :: Complex Double -> Complex Double

norm2 :: Complex Double -> Magnitude (Complex Double)

norm2' :: Complex Double -> Scalar (Complex Double)

norm :: RealScalar (Complex Double) -> Complex Double -> Magnitude (Complex Double)

Normed (SpVector Double)

Associated Types

type Magnitude (SpVector Double) :: *

type RealScalar (SpVector Double) :: *

Methods

norm1 :: SpVector Double -> Magnitude (SpVector Double)

norm2Sq :: SpVector Double -> Magnitude (SpVector Double)

normP :: RealScalar (SpVector Double) -> SpVector Double -> Magnitude (SpVector Double)

normalize :: RealScalar (SpVector Double) -> SpVector Double -> SpVector Double

normalize2 :: SpVector Double -> SpVector Double

normalize2' :: SpVector Double -> SpVector Double

norm2 :: SpVector Double -> Magnitude (SpVector Double)

norm2' :: SpVector Double -> Scalar (SpVector Double)

norm :: RealScalar (SpVector Double) -> SpVector Double -> Magnitude (SpVector Double)

Normed (SpVector (Complex Double))

Associated Types

type Magnitude (SpVector (Complex Double)) :: *

type RealScalar (SpVector (Complex Double)) :: *

Methods

norm1 :: SpVector (Complex Double) -> Magnitude (SpVector (Complex Double))

norm2Sq :: SpVector (Complex Double) -> Magnitude (SpVector (Complex Double))

normP :: RealScalar (SpVector (Complex Double)) -> SpVector (Complex Double) -> Magnitude (SpVector (Complex Double))

normalize :: RealScalar (SpVector (Complex Double)) -> SpVector (Complex Double) -> SpVector (Complex Double)

normalize2 :: SpVector (Complex Double) -> SpVector (Complex Double)

normalize2' :: SpVector (Complex Double) -> SpVector (Complex Double)

norm2 :: SpVector (Complex Double) -> Magnitude (SpVector (Complex Double))

norm2' :: SpVector (Complex Double) -> Scalar (SpVector (Complex Double))

norm :: RealScalar (SpVector (Complex Double)) -> SpVector (Complex Double) -> Magnitude (SpVector (Complex Double))

normInftyR :: (Foldable t, Ord a) => t a -> a

Infinity-norm (Real)

normInftyC :: (Foldable t, RealFloat a, Functor t) => t (Complex a) -> a

Infinity-norm (Complex)

dotLp :: (Set t, Foldable t, Floating a) => a -> t a -> t a -> a

Lp inner product (p > 0)

reciprocal :: (Functor f, Fractional b) => f b -> f b

Reciprocal

scale :: (Num b, Functor f) => b -> f b -> f b

Scale

Matrix ring

class (AdditiveGroup m, Epsilon (MatrixNorm m)) => MatrixRing m where

A matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication

Minimal complete definition

(##), (##^), transpose, normFrobenius

Associated Types

type MatrixNorm m :: *

Methods

(##) :: m -> m -> m

(##^) :: m -> m -> m

(#^#) :: m -> m -> m

transpose :: m -> m

normFrobenius :: m -> MatrixNorm m

Instances

MatrixRing (SpMatrix Double)

Associated Types

type MatrixNorm (SpMatrix Double) :: *

Methods

(##) :: SpMatrix Double -> SpMatrix Double -> SpMatrix Double

(##^) :: SpMatrix Double -> SpMatrix Double -> SpMatrix Double

(#^#) :: SpMatrix Double -> SpMatrix Double -> SpMatrix Double

transpose :: SpMatrix Double -> SpMatrix Double

normFrobenius :: SpMatrix Double -> MatrixNorm (SpMatrix Double)

MatrixRing (SpMatrix (Complex Double))

Associated Types

type MatrixNorm (SpMatrix (Complex Double)) :: *

Methods

(##) :: SpMatrix (Complex Double) -> SpMatrix (Complex Double) -> SpMatrix (Complex Double)

(##^) :: SpMatrix (Complex Double) -> SpMatrix (Complex Double) -> SpMatrix (Complex Double)

(#^#) :: SpMatrix (Complex Double) -> SpMatrix (Complex Double) -> SpMatrix (Complex Double)

transpose :: SpMatrix (Complex Double) -> SpMatrix (Complex Double)

normFrobenius :: SpMatrix (Complex Double) -> MatrixNorm (SpMatrix (Complex Double))

Linear vector space

class (VectorSpace v, MatrixRing (MatrixType v)) => LinearVectorSpace v where

Minimal complete definition

(#>), (<#)

Associated Types

type MatrixType v :: *

Methods

(#>) :: MatrixType v -> v -> v

(<#) :: v -> MatrixType v -> v

LinearVectorSpace + Normed

type V v = (LinearVectorSpace v, Normed v)

Linear systems

class LinearVectorSpace v => LinearSystem v where

Minimal complete definition

(<\>)

Methods

(<\>) :: (MonadIO m, MonadThrow m) => MatrixType v -> v -> m v

FiniteDim : finite-dimensional objects

class Functor f => FiniteDim f where

Minimal complete definition

dim

Associated Types

type FDSize f :: *

Methods

dim :: f a -> FDSize f

Instances

FiniteDim SpVector

SpVectors form a vector space because they can be multiplied by a scalar

SpVectors are finite-dimensional vectors

Associated Types

type FDSize (SpVector :: * -> *) :: *

Methods

dim :: SpVector a -> FDSize SpVector

FiniteDim SpMatrix

SpMatrixes are maps between finite-dimensional spaces

Associated Types

type FDSize (SpMatrix :: * -> *) :: *

Methods

dim :: SpMatrix a -> FDSize SpMatrix

class FiniteDim' f where

Minimal complete definition

dim'

Associated Types

type FDSize' f :: *

Methods

dim' :: f -> FDSize' f

HasData : accessing inner data (do not export)

class HasData f a where

Minimal complete definition

nnz, dat

Associated Types

type HDData f a :: *

Methods

nnz :: f a -> Int

dat :: f a -> HDData f a

Instances

HasData SpVector a

Associated Types

type HDData (SpVector :: * -> *) a :: *

Methods

nnz :: SpVector a -> Int

dat :: SpVector a -> HDData SpVector a

HasData SpMatrix a

Associated Types

type HDData (SpMatrix :: * -> *) a :: *

Methods

nnz :: SpMatrix a -> Int

dat :: SpMatrix a -> HDData SpMatrix a

class HasData' f where

Minimal complete definition

nnz', dat'

Associated Types

type HDD f :: *

Methods

nnz' :: f -> Int

dat' :: f -> HDD f

Sparse : sparse datastructures

class (FiniteDim f, HasData f a) => Sparse f a where

Minimal complete definition

spy

Methods

spy :: Fractional b => f a -> b

Instances

Sparse SpVector a

Methods

spy :: Fractional b => SpVector a -> b

Sparse SpMatrix a

Methods

spy :: Fractional b => SpMatrix a -> b

class (FiniteDim' f, HasData' f) => Sparse' f where

Minimal complete definition

spy'

Methods

spy' :: Fractional b => f -> b

Set : types that behave as sets

class Functor f => Set f where

Minimal complete definition

liftU2, liftI2

Methods

liftU2 :: (a -> a -> a) -> f a -> f a -> f a

union binary lift : apply function on _union_ of two "sets"

liftI2 :: (a -> a -> b) -> f a -> f a -> f b

intersection binary lift : apply function on _intersection_ of two "sets"

Instances

Set SpVector

Methods

liftU2 :: (a -> a -> a) -> SpVector a -> SpVector a -> SpVector a

liftI2 :: (a -> a -> b) -> SpVector a -> SpVector a -> SpVector b

Set SpMatrix

Methods

liftU2 :: (a -> a -> a) -> SpMatrix a -> SpMatrix a -> SpMatrix a

liftI2 :: (a -> a -> b) -> SpMatrix a -> SpMatrix a -> SpMatrix b

SpContainer : sparse container datastructures. Insertion, lookup, toList, lookup with 0 default

class Sparse c a => SpContainer c a where

Minimal complete definition

scInsert, scLookup, scToList, (@@)

Associated Types

type ScIx c :: *

Methods

scInsert :: ScIx c -> a -> c a -> c a

scLookup :: c a -> ScIx c -> Maybe a

scToList :: c a -> [(ScIx c, a)]

(@@) :: c a -> ScIx c -> a

Instances

Elt a => SpContainer SpVector a

SpVectors are sparse containers too, i.e. any specific component may be missing (so it is assumed to be 0)

Associated Types

type ScIx (SpVector :: * -> *) :: *

Methods

scInsert :: ScIx SpVector -> a -> SpVector a -> SpVector a

scLookup :: SpVector a -> ScIx SpVector -> Maybe a

scToList :: SpVector a -> [(ScIx SpVector, a)]

(@@) :: SpVector a -> ScIx SpVector -> a

Num a => SpContainer SpMatrix a

SpMatrixes are sparse containers too, i.e. any specific component may be missing (so it is assumed to be 0)

Associated Types

type ScIx (SpMatrix :: * -> *) :: *

Methods

scInsert :: ScIx SpMatrix -> a -> SpMatrix a -> SpMatrix a

scLookup :: SpMatrix a -> ScIx SpMatrix -> Maybe a

scToList :: SpMatrix a -> [(ScIx SpMatrix, a)]

(@@) :: SpMatrix a -> ScIx SpMatrix -> a

class SpContainer' c where

Minimal complete definition

scInsert', scLookup', scToList'

Associated Types

type ScIx' c :: *

Methods

scInsert' :: ScIx' c -> a -> c -> c

scLookup' :: c -> ScIx' c -> Maybe a

scToList' :: c -> [a]

SparseVector

class SpContainer v e => SparseVector v e where

Minimal complete definition

svFromList, svFromListDense, svConcat

Associated Types

type SpvIx v :: *

Methods

svFromList :: Int -> [(SpvIx v, e)] -> v e

svFromListDense :: Int -> [e] -> v e

svConcat :: Foldable t => t (v e) -> v e

SparseMatrix

class SpContainer m e => SparseMatrix m e where

Minimal complete definition

smFromVector, smTranspose, encodeIx, decodeIx

Methods

smFromVector :: LexOrd -> (Int, Int) -> Vector (IxRow, IxCol, e) -> m e

smTranspose :: m e -> m e

encodeIx :: m e -> LexOrd -> (IxRow, IxCol) -> LexIx

decodeIx :: m e -> LexOrd -> LexIx -> (IxRow, IxCol)

SparseMatVec

Utilities

toC :: Num a => a -> Complex a

Lift a real number onto the complex plane