sparse-linear-algebra-0.3.1: Numerical computing in native Haskell

Copyright (c) Marco Zocca 2017 GPL-3 (see the file LICENSE) zocca marco gmail experimental portable None Haskell2010

Numeric.LinearAlgebra.Class

Description

Typeclasses for linear algebra and related concepts

Synopsis

Matrix and vector elements (optionally Complex)

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

Minimal complete definition

mag

Associated Types

type EltMag e :: * Source #

Methods

conj :: e -> e Source #

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

mag :: e -> EltMag e Source #

Magnitude

Instances

 Source # Associated Typestype EltMag Double :: * Source # Methods Source # Associated Typestype EltMag Float :: * Source # Methods RealFloat e => Elt (Complex e) Source # Associated Typestype EltMag (Complex e) :: * Source # Methodsconj :: Complex e -> Complex e Source #mag :: Complex e -> EltMag (Complex e) Source #

class AdditiveGroup v where Source #

Minimal complete definition

Methods

zeroV :: v Source #

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

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

negateV :: v -> v Source #

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

Group subtraction

Instances

 Source # Methods Source # Instances for builtin types Methods Source # Methods Source # Methods Source # Methods(^+^) :: SpVector a -> SpVector a -> SpVector a Source #(^-^) :: SpVector a -> SpVector a -> SpVector a Source # Source # SpMatrixes form an additive group, in that they can have an invertible associtative operation (matrix sum) Methods(^+^) :: SpMatrix a -> SpMatrix a -> SpMatrix a Source #(^-^) :: SpMatrix a -> SpMatrix a -> SpMatrix a Source #

Vector space v.

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

Minimal complete definition

(.*)

Associated Types

type Scalar v :: * Source #

Methods

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

Scale a vector

Instances

 Source # Associated Typestype Scalar Double :: * Source # Methods Source # Associated Typestype Scalar Float :: * Source # Methods Source # Associated Typestype Scalar (Complex Double) :: * Source # Methods Source # Associated Typestype Scalar (Complex Float) :: * Source # Methods Source # Associated Typestype Scalar (SpVector a) :: * Source # Methods(.*) :: Scalar (SpVector a) -> SpVector a -> SpVector a Source # Source # Associated Typestype Scalar (SpMatrix a) :: * Source # Methods(.*) :: Scalar (SpMatrix a) -> SpMatrix a -> SpMatrix a Source #

class VectorSpace v => InnerSpace v where Source #

Adds inner (dot) products.

Minimal complete definition

(<.>)

Methods

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

Inner/dot product

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods InnerSpace a => InnerSpace (SpVector a) Source # Methods(<.>) :: SpVector a -> SpVector a -> Scalar (SpVector a) Source #

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

Inner product

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

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

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

Vector multiplied by scalar

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

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

Hilbert-space distance function

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

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 Source #

Minimal complete definition

Associated Types

type Magnitude v :: * Source #

type RealScalar v :: * Source #

Methods

norm1 :: v -> Magnitude v Source #

L1 norm

norm2Sq :: v -> Magnitude v Source #

Euclidean (L2) norm squared

normP :: RealScalar v -> v -> Magnitude v Source #

Lp norm (p > 0)

normalize :: RealScalar v -> v -> v Source #

Normalize w.r.t. Lp norm

normalize2 :: v -> v Source #

Normalize w.r.t. L2 norm

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

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

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

Euclidean (L2) norm

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

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

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

Lp norm (p > 0)

Instances

 Source # Associated Typestype Magnitude Double :: * Source #type RealScalar Double :: * Source # Methods Source # Associated Typestype Magnitude Float :: * Source #type RealScalar Float :: * Source # Methods Source # Associated Typestype Magnitude (Complex Double) :: * Source #type RealScalar (Complex Double) :: * Source # Methods Source # Associated Typestype Magnitude (Complex Float) :: * Source #type RealScalar (Complex Float) :: * Source # Methods (Normed a, (~) * (Magnitude a) (RealScalar a), (~) * (RealScalar a) (Scalar a)) => Normed (SpVector a) Source # Associated Typestype Magnitude (SpVector a) :: * Source #type RealScalar (SpVector a) :: * Source # MethodsnormP :: RealScalar (SpVector a) -> SpVector a -> Magnitude (SpVector a) Source #normalize :: RealScalar (SpVector a) -> SpVector a -> SpVector a Source #norm2' :: SpVector a -> Scalar (SpVector a) Source #norm :: RealScalar (SpVector a) -> SpVector a -> Magnitude (SpVector a) Source #

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

Infinity-norm (Real)

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

Infinity-norm (Complex)

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

Lp inner product (p > 0)

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

Reciprocal

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

Scale

Matrix ring

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

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

Associated Types

type MatrixNorm m :: * Source #

Methods

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

Matrix-matrix product

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

Matrix times matrix transpose (A B^T)

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

Matrix transpose times matrix (A^T B)

transpose :: m -> m Source #

Matrix transpose (Hermitian conjugate in the Complex case)

normFrobenius :: m -> MatrixNorm m Source #

Frobenius norm

Instances

 Source # Associated Typestype MatrixNorm (SpMatrix Double) :: * Source # Methods Source # Associated Typestype MatrixNorm (SpMatrix (Complex Double)) :: * Source # Methods

Linear vector space

class VectorSpace v => LinearVectorSpace v where Source #

Minimal complete definition

Associated Types

type MatrixType v :: * Source #

Methods

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

Matrix-vector action

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

Dual matrix-vector action

Linear systems

class LinearVectorSpace v => LinearSystem v where Source #

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 Source #

Minimal complete definition

dim

Associated Types

type FDSize f Source #

Methods

dim :: f -> FDSize f Source #

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

Instances

 Source # SpVectors form a vector space because they can be multiplied by a scalarSpVectors are finite-dimensional vectors Associated Typestype FDSize (SpVector a) :: * Source # Methodsdim :: SpVector a -> FDSize (SpVector a) Source # Source # SpMatrixes are maps between finite-dimensional spaces Associated Typestype FDSize (SpMatrix a) :: * Source # Methodsdim :: SpMatrix a -> FDSize (SpMatrix a) Source #

HasData : accessing inner data (do not export)

class HasData f where Source #

Minimal complete definition

Associated Types

type HDData f Source #

Methods

nnz :: f -> Int Source #

Number of nonzeros

dat :: f -> HDData f Source #

Instances

 Source # Associated Typestype HDData (SpVector a) :: * Source # Methodsnnz :: SpVector a -> Int Source #dat :: SpVector a -> HDData (SpVector a) Source # Source # Associated Typestype HDData (SpMatrix a) :: * Source # Methodsnnz :: SpMatrix a -> Int Source #dat :: SpMatrix a -> HDData (SpMatrix a) Source #

Sparse : sparse datastructures

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

Minimal complete definition

spy

Methods

spy :: Fractional b => f -> b Source #

Sparsity (fraction of nonzero elements)

Instances

 Source # Methodsspy :: Fractional b => SpVector a -> b Source # Source # Methodsspy :: Fractional b => SpMatrix a -> b Source #

Set : types that behave as sets

class Functor f => Set f where Source #

Minimal complete definition

Methods

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

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

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

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

Instances

 Source # MethodsliftU2 :: (a -> a -> a) -> SpVector a -> SpVector a -> SpVector a Source #liftI2 :: (a -> a -> b) -> SpVector a -> SpVector a -> SpVector b Source # Source # MethodsliftU2 :: (a -> a -> a) -> SpMatrix a -> SpMatrix a -> SpMatrix a Source #liftI2 :: (a -> a -> b) -> SpMatrix a -> SpMatrix a -> SpMatrix b Source #

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

class Sparse c => SpContainer c where Source #

Minimal complete definition

Associated Types

type ScIx c :: * Source #

type ScElem c Source #

Methods

scInsert :: ScIx c -> ScElem c -> c -> c Source #

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

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

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

Instances

 Elt a => SpContainer (SpVector a) Source # SpVectors are sparse containers too, i.e. any specific component may be missing (so it is assumed to be 0) Associated Typestype ScIx (SpVector a) :: * Source #type ScElem (SpVector a) :: * Source # MethodsscInsert :: ScIx (SpVector a) -> ScElem (SpVector a) -> SpVector a -> SpVector a Source #scLookup :: SpVector a -> ScIx (SpVector a) -> Maybe (ScElem (SpVector a)) Source #scToList :: SpVector a -> [(ScIx (SpVector a), ScElem (SpVector a))] Source #(@@) :: SpVector a -> ScIx (SpVector a) -> ScElem (SpVector a) Source # Num a => SpContainer (SpMatrix a) Source # SpMatrixes are sparse containers too, i.e. any specific component may be missing (so it is assumed to be 0) Associated Typestype ScIx (SpMatrix a) :: * Source #type ScElem (SpMatrix a) :: * Source # MethodsscInsert :: ScIx (SpMatrix a) -> ScElem (SpMatrix a) -> SpMatrix a -> SpMatrix a Source #scLookup :: SpMatrix a -> ScIx (SpMatrix a) -> Maybe (ScElem (SpMatrix a)) Source #scToList :: SpMatrix a -> [(ScIx (SpMatrix a), ScElem (SpMatrix a))] Source #(@@) :: SpMatrix a -> ScIx (SpMatrix a) -> ScElem (SpMatrix a) Source #

SparseVector

class SpContainer v => SparseVector v where Source #

Minimal complete definition

Associated Types

type SpvIx v :: * Source #

Methods

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

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

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

SparseMatrix

class SpContainer m => SparseMatrix m where Source #

Minimal complete definition

Methods

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

smTranspose :: m -> m Source #

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

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

Utilities

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

Lift a real number onto the complex plane