Data.Sparse.Matrix

newtype Matrix a

Constructors

• Matrix { coefficients :: Poly2 a, height :: Int, width :: Int }

Instances

• (Show a, Semiring a, Eq a) => Show (Matrix a)
• (Eq a) => Eq (Matrix a)
• (Eq a, Semiring a) => Semiring (Matrix a)
• Functor Matrix
• (Eq a, Ring a) => Ring (Matrix a)

type Poly2 a = Polynomial (Polynomial a)

Representation of a matrix without storage of the zero coefficients.

> -- Working with Sparse Matrices
> import Data.Sparse.Matrix
> import Data.Sparse.Polynomial ((^))
> -- Define a matrix with sum of terms (see Data.Sparse.Polynomial)
> a = Matrix { height:3, width:2, coefficients: 5.0^2^0+7.0^0^1 }
> a
 0.0 7.0
 0.0 0.0
 5.0 0.0

> -- Modify all the non-zero elements with the functor
> b = (_ + 2.0) <$> a
> b
 0.0 9.0
 0.0 0.0
 7.0 0.0

> -- Set a specific element
> c = 2.0^1^0 ~ b
> c
 0.0 9.0
 2.0 0.0
 7.0 0.0

> -- Coefficient extraction
> c ?? [1,0]
2.0

> -- Column(s) extraction
> c * Matrix { height: 3, width: 1, coefficients: 1.0^1^0 }
 9.0
 0.0
 0.0

> -- Row(s) extraction
> Matrix { height: 2, width: 3, coefficients: 1.0^0^0 + 1.0^1^2 } * c
 0.0 9.0
 7.0 0.0

> -- transpose, multiply
> d = a * transpose c
> d
 63.0 0.0 0.0
 0.0 0.0 0.0
 0.0 10.0 35.0

> -- Weight a specific element
> e = 0.5^0^0 ~* d
> e
 31.5 0.0 0.0
 0.0 0.0 0.0
 0.0 10.0 35.0

> -- Increment a specific element
> f = 4.0^1^1 ~+ e
> f
 31.5 0.0 0.0
 0.0 4.0 0.0
 0.0 10.0 35.0

> -- Identity, subtraction
> g = f - eye 3
> g
 30.5 0.0 0.0
 0.0 3.0 0.0
 0.0 10.0 34.0

> -- Vectors are just column matrices
> v = Matrix { height:3, width:1, coefficients: 1.0^0^0+2.0^1^0+3.0^2^0 }
> v
 1.0
 2.0
 3.0

> -- Solve the linear system
> pluSolve g v
 0.03278688524590164
 0.6666666666666666
 -0.10784313725490192

type Square a = Matrix a

Instances

• (Eq a, DivisionRing a, Ord a, Divisible a, Leadable a) => DivisionRing (Square a)

type Symmetric = Square Number

type Vector a = Matrix a

applyPoly :: forall a. Eq a => Semiring a => Polynomial (Square a) -> Square a -> Square a

Polynomial application

coefficients :: forall a. Matrix a -> Poly2 a

determinant :: forall a. Eq a => Ord a => Ring a => DivisionRing a => Divisible a => Leadable a => Square a -> a

Determinant of a square matrix in a field

diagonalize :: Symmetric -> { val :: Symmetric, vec :: Square Number }

Square real matrix diagonalization such that m = vec * val * recip vec

eigenValues :: Square Number -> Array (Cartesian Number)

Eigen values of a real square matrix

extract :: forall a. Eq a => Semiring a => Matrix a -> Array Int -> a

Coefficient extraction

eye :: forall a. Eq a => Semiring a => Int -> Square a

Identity matrix

faddeev :: Square Number -> Polynomial Number

Characteristic polynomial of a real square matrix

height :: forall a. Matrix a -> Int

increment :: forall a. Eq a => Semiring a => Ring a => Poly2 a -> Matrix a -> Matrix a

Element increment

internalMap :: forall a. Polynomial a -> Map Int a

Returns the polynomial internal structure

luSolve :: forall a. Eq a => Ring a => DivisionRing a => Square a -> Square a -> Matrix a -> Matrix a

Solves the system LUx=B for x, for L,U square inversible and any B

parseMonom :: forall a. Eq a => Semiring a => Poly2 a -> Maybe { i :: Int, j :: Int, v :: a }

pivot :: forall a. Eq a => Ord a => Ring a => Int -> Matrix a -> Int

Returns the row index of the maximum element (in magnitude) among all the elements of the column whose index is provided below (possibly as high as) the diagonal element.

pivot' :: forall a. Eq a => Ord a => Ring a => Int -> Int -> Matrix a -> Maybe Int

plu :: forall a. Eq a => Ring a => Ord a => DivisionRing a => Divisible a => Leadable a => Square a -> { l :: Square a, p :: Square a, parity :: Boolean, u :: Square a }

Returns the 3 square matrices P, L and U and a Boolean such that

• LU = PA where A is the given square matrix with not null determinant
• P is a permutation matrix with parity given by the Boolean (true: 1, false: -1)
• L is triangular inferior with unitary diagonal elements
• U is triangular superior

pluSolve :: forall a. Eq a => Ring a => Ord a => Divisible a => Leadable a => DivisionRing a => Square a -> Matrix a -> Matrix a

Solves the system Ax=B for x, for A square inversible and any B

pow :: forall a. Eq a => Semiring a => Square a -> Int -> Square a

Integer power of a square matrix

replace :: forall a. Eq a => Semiring a => Poly2 a -> Matrix a -> Matrix a

Element replacement

replace' :: forall a. Eq a => Semiring a => Int -> Int -> a -> Poly2 a -> Poly2 a

ringDeterminant :: forall a. Ring a => Eq a => Ord a => Divisible a => Leadable a => EuclideanRing a => Matrix a -> a

Determinant of a square matrix in a ring

rowCombination :: forall a. Eq a => Divisible a => Ring a => Leadable a => Int -> a -> Int -> a -> Matrix a -> Matrix a

rowCombination orig ko dest kd performs : dest <- ko * orig + kd * dest which only alters dest

scale :: forall a. Eq a => Semiring a => Ring a => Poly2 a -> Matrix a -> Matrix a

Element weighting

swapRows :: forall a. Eq a => Divisible a => Ring a => Leadable a => Int -> Int -> Matrix a -> Matrix a

As the name suggests. Note that the two rows are provided as 0-indices

trace :: forall a. Eq a => Semiring a => Square a -> a

transpose :: forall a. Eq a => Semiring a => Matrix a -> Matrix a

Matrix transposition

width :: forall a. Matrix a -> Int

zeros :: forall a. Eq a => Semiring a => Int -> Int -> Matrix a

Zero matrix

Operator alias for Data.Sparse.Matrix.extract (left-associative / precedence 8)

Coefficient extraction infix notation

Operator alias for Data.Sparse.Matrix.replace (non-associative / precedence 7)

Element replacement infix notation

Operator alias for Data.Sparse.Matrix.scale (non-associative / precedence 7)

Element weighting infix notation

Operator alias for Data.Sparse.Matrix.increment (non-associative / precedence 7)

Element increment infix notation