Numeric.LinearAlgebra.Class

Typeclasses for linear algebra and related concepts

Matrix and vector elements (optionally 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

Complex conjugate, or identity function if its input is real-valued

mag :: e -> EltMag e

Magnitude

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)

Additive group

class AdditiveGroup v where

Minimal complete definition

zeroV, (^+^), negateV

Methods

zeroV :: v

The zero element: identity for '(^+^)'

(^+^) :: v -> v -> v infixl 6

Add vectors

negateV :: v -> v

Additive inverse

(^-^) :: v -> v -> v infixl 6

Group subtraction

Instances

AdditiveGroup Double

Methods

zeroV :: Double

(^+^) :: Double -> Double -> Double

negateV :: Double -> Double

(^-^) :: Double -> Double -> Double

AdditiveGroup Float

Instances for builtin types

Methods

zeroV :: Float

(^+^) :: Float -> Float -> Float

negateV :: Float -> Float

(^-^) :: Float -> Float -> Float

AdditiveGroup (Complex Double)

Methods

zeroV :: Complex Double

(^+^) :: Complex Double -> Complex Double -> Complex Double

negateV :: Complex Double -> Complex Double

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

AdditiveGroup (Complex Float)

Methods

zeroV :: Complex Float

(^+^) :: Complex Float -> Complex Float -> Complex Float

negateV :: Complex Float -> Complex Float

(^-^) :: Complex Float -> Complex Float -> Complex Float

AdditiveGroup a => AdditiveGroup (SpVector a)

Methods

zeroV :: SpVector a

(^+^) :: SpVector a -> SpVector a -> SpVector a

negateV :: SpVector a -> SpVector a

(^-^) :: SpVector a -> SpVector a -> SpVector a

AdditiveGroup a => AdditiveGroup (SpMatrix a)

SpMatrixes form an additive group, in that they can have an invertible associtative operation (matrix sum)

Methods

zeroV :: SpMatrix a

(^+^) :: SpMatrix a -> SpMatrix a -> SpMatrix a

negateV :: SpMatrix a -> SpMatrix a

(^-^) :: SpMatrix a -> SpMatrix a -> SpMatrix a

Vector space v.

class (AdditiveGroup v, Num (Scalar v)) => VectorSpace v where

Minimal complete definition

(.*)

Associated Types

type Scalar v :: *

Methods

(.*) :: Scalar v -> v -> v infixr 7

Scale a vector

Instances

VectorSpace Double

Associated Types

type Scalar Double :: *

Methods

(.*) :: Scalar Double -> Double -> Double

VectorSpace Float

Associated Types

type Scalar Float :: *

Methods

(.*) :: Scalar Float -> Float -> Float

VectorSpace (Complex Double)

Associated Types

type Scalar (Complex Double) :: *

Methods

(.*) :: Scalar (Complex Double) -> Complex Double -> Complex Double

VectorSpace (Complex Float)

Associated Types

type Scalar (Complex Float) :: *

Methods

(.*) :: Scalar (Complex Float) -> Complex Float -> Complex Float

VectorSpace a => VectorSpace (SpVector a)

Associated Types

type Scalar (SpVector a) :: *

Methods

(.*) :: Scalar (SpVector a) -> SpVector a -> SpVector a

VectorSpace a => VectorSpace (SpMatrix a)

Associated Types

type Scalar (SpMatrix a) :: *

Methods

(.*) :: Scalar (SpMatrix a) -> SpMatrix a -> SpMatrix a

class VectorSpace v => InnerSpace v where

Adds inner (dot) products.

Minimal complete definition

(<.>)

Methods

(<.>) :: v -> v -> Scalar v

Inner/dot product

Instances

InnerSpace Double

Methods

(<.>) :: Double -> Double -> Scalar Double

InnerSpace Float

Methods

(<.>) :: Float -> Float -> Scalar Float

InnerSpace (Complex Double)

Methods

(<.>) :: Complex Double -> Complex Double -> Scalar (Complex Double)

InnerSpace (Complex Float)

Methods

(<.>) :: Complex Float -> Complex Float -> Scalar (Complex Float)

InnerSpace a => InnerSpace (SpVector a)

Methods

(<.>) :: SpVector a -> SpVector a -> Scalar (SpVector a)

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

Inner product

(./) :: (VectorSpace v, s ~ Scalar v, Fractional s) => v -> s -> v infixr 7

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

(*.) :: (VectorSpace v, s ~ Scalar v) => v -> s -> v infixl 7

Vector multiplied by scalar

cvx :: VectorSpace v => Scalar v -> v -> v -> v

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

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

L1 norm

norm2Sq :: v -> Magnitude v

Euclidean (L2) norm squared

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

Lp norm (p > 0)

normalize :: RealScalar v -> v -> v

Normalize w.r.t. Lp norm

normalize2 :: v -> v

Normalize w.r.t. L2 norm

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

Normalize w.r.t. norm2' instead of norm2

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

Euclidean (L2) norm

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

Euclidean (L2) norm; returns a Complex (norm :+ 0) for Complex-valued vectors

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

Lp norm (p > 0)

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 Float

Associated Types

type Magnitude Float :: *

type RealScalar Float :: *

Methods

norm1 :: Float -> Magnitude Float

norm2Sq :: Float -> Magnitude Float

normP :: RealScalar Float -> Float -> Magnitude Float

normalize :: RealScalar Float -> Float -> Float

normalize2 :: Float -> Float

normalize2' :: Float -> Float

norm2 :: Float -> Magnitude Float

norm2' :: Float -> Scalar Float

norm :: RealScalar Float -> Float -> Magnitude Float

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 (Complex Float)

Associated Types

type Magnitude (Complex Float) :: *

type RealScalar (Complex Float) :: *

Methods

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

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

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

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

normalize2 :: Complex Float -> Complex Float

normalize2' :: Complex Float -> Complex Float

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

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

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

(Normed a, (~) * (Magnitude a) (RealScalar a), (~) * (RealScalar a) (Scalar a)) => Normed (SpVector a)

Associated Types

type Magnitude (SpVector a) :: *

type RealScalar (SpVector a) :: *

Methods

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

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

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

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

normalize2 :: SpVector a -> SpVector a

normalize2' :: SpVector a -> SpVector a

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

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

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

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

Matrix-matrix product

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

Matrix times matrix transpose (A B^T)

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

Matrix transpose times matrix (A^T B)

transpose :: m -> m

Matrix transpose (Hermitian conjugate in the Complex case)

normFrobenius :: m -> MatrixNorm m

Frobenius norm

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 => LinearVectorSpace v where

Minimal complete definition

(#>), (<#)

Associated Types

type MatrixType v :: *

Methods

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

Matrix-vector action

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

Dual matrix-vector action

LinearVectorSpace + Normed

type V v = (LinearVectorSpace v, Normed v)

Linear systems

class LinearVectorSpace v => LinearSystem v where

Minimal complete definition

(<\>)

Methods

(<\>)

Arguments

:: (MonadIO m, MonadThrow m)

=> MatrixType v

System matrix

-> v

Right-hand side

-> m v

Result

Solve a linear system; uses GMRES internally as default method

FiniteDim : finite-dimensional objects

class FiniteDim f where

Minimal complete definition

dim

Associated Types

type FDSize f

Methods

dim :: f -> FDSize f

Dimension (i.e. Int for SpVector, (Int, Int) for SpMatrix)

Instances

FiniteDim (SpVector a)

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

SpVectors are finite-dimensional vectors

Associated Types

type FDSize (SpVector a) :: *

Methods

dim :: SpVector a -> FDSize (SpVector a)

FiniteDim (SpMatrix a)

SpMatrixes are maps between finite-dimensional spaces

Associated Types

type FDSize (SpMatrix a) :: *

Methods

dim :: SpMatrix a -> FDSize (SpMatrix a)

HasData : accessing inner data (do not export)

class HasData f where

Minimal complete definition

nnz, dat

Associated Types

type HDData f

Methods

nnz :: f -> Int

Number of nonzeros

dat :: f -> HDData f

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)

Sparse : sparse datastructures

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

Minimal complete definition

spy

Methods

spy :: Fractional b => f -> b

Sparsity (fraction of nonzero elements)

Instances

Sparse (SpVector a)

Methods

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

Sparse (SpMatrix a)

Methods

spy :: Fractional b => SpMatrix a -> 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 => SpContainer c where

Minimal complete definition

scInsert, scLookup, scToList, (@@)

Associated Types

type ScIx c :: *

type ScElem c

Methods

scInsert :: ScIx c -> ScElem c -> c -> c

scLookup :: c -> ScIx c -> Maybe (ScElem c)

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

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

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 a) :: *

type ScElem (SpVector a) :: *

Methods

scInsert :: ScIx (SpVector a) -> ScElem (SpVector a) -> SpVector a -> SpVector a

scLookup :: SpVector a -> ScIx (SpVector a) -> Maybe (ScElem (SpVector a))

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

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

type ScElem (SpMatrix a) :: *

Methods

scInsert :: ScIx (SpMatrix a) -> ScElem (SpMatrix a) -> SpMatrix a -> SpMatrix a

scLookup :: SpMatrix a -> ScIx (SpMatrix a) -> Maybe (ScElem (SpMatrix a))

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

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

SparseVector

class SpContainer v => SparseVector v where

Minimal complete definition

svFromList, svFromListDense, svConcat

Associated Types

type SpvIx v :: *

Methods

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

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

svConcat :: Foldable t => t v -> v

SparseMatrix

class SpContainer m => SparseMatrix m where

Minimal complete definition

smFromVector, smTranspose, encodeIx, decodeIx

Methods

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

smTranspose :: m -> m

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

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

SparseMatVec

Utilities

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

Lift a real number onto the complex plane