Data.Tensor.LinearAlgebra

Documentation

class VectorSpace v where

Methods

zero :: Num e => v e

(*.) :: Num e => e -> v e -> v e

(.+.) :: Num e => v e -> v e -> v e

Instances

MultiIndex i => VectorSpace (Tensor i)

class Cardinal n => Product n t1 t2 where

A general form of product between two tensors, in which the last n dimensions of t1 are contracted with the first n dimensions of t2. The resulting tensor belongs to the space ProdSpace n t1 t2. The operators .*. and below are particular cases where n is equal to 1 and 0 respectively.

Associated Types

type ProdSpace n t1 t2

Methods

prod :: n -> t1 -> t2 -> ProdSpace n t1 t2

Instances

(Num e, Cardinal n, MultiIndex i, MultiIndex j, JoinList n i j) => Product n (Tensor i e) (Tensor j e)

type MatrixProductSpace t1 t2 = ProdSpace (Succ Zero) t1 t2

(.*.) :: Product (Succ Zero) t1 t2 => t1 -> t2 -> MatrixProductSpace t1 t2

It is the product of the last dimension of t1 with the first dimension of t2. In the case where t1 and t2 are matrices this coincide with the ordinary matrix product.

type :⊗: t1 t2 = ProdSpace C0 t1 t2

(⊗) :: Product C0 t1 t2 => t1 -> t2 -> t1 :⊗: t2

Tensor product of t1 and t2.

class DotProduct t where

Methods

dot :: Num e => t e -> t e -> e

Instances

DotProduct (Tensor i)

class (Tensor t, Index t ~ (i :|: (j :|: Nil))) => Matrix i j t where

A matrix with i rows and j columns.

Methods

rowSwitch :: i -> i -> t -> t

Switch two rows.

rowMult :: (Num e, Elem t ~ e) => i -> Elem t -> t -> t

Multiply a row by a number.

rowAdd :: (Num e, Elem t ~ e) => i -> Elem t -> i -> t -> t

rowAdd i1 a i2 t adds a times the row i2 to the row i1 ot t.

colSwitch :: j -> j -> t -> t

Switch two columns.

colMult :: (Num e, Elem t ~ e) => j -> Elem t -> t -> t

Multiply a column by a number.

colAdd :: (Num e, Elem t ~ e) => j -> Elem t -> j -> t -> t

colAdd j1 a j2 t adds a times the column j2 to the column j1 ot t.

rowEchelonForm :: (Eq e, Fractional e, Elem t ~ e) => t -> t

Reduced row echelon form of the matrix.

Instances

(Ordinal i, Ordinal j) => Matrix i j (Tensor (:|: i (:|: j Nil)) e)

class LinearSystem t1 t2 where

Solves linear systems AX=B; t1 is the type of A, t2 is the type of B, and SolSpace t1 t2 is the type of the solution X.

Associated Types

type SolSpace t1 t2

Methods

triangularSolve :: t1 -> t2 -> (t1, t2)

Performs row operations on the augmented matrix [t1,t2] until t1 is in reduced row echelon form, then slits the result.

parametricSolve :: t1 -> t2 -> Maybe (SolSpace t1 t2, [SolSpace t1 t2])

Returns Nothing if the system AX=B has no solution, otherwise returns a solution for the system and a list of basis vectors for the kernel of A.

Instances

(Eq e, Fractional e, Ordinal i, Ordinal j) => LinearSystem (Tensor (:|: i (:|: j Nil)) e) (Tensor (:|: i Nil) e)

(Eq e, Fractional e, Ordinal i, Ordinal j, Ordinal k, Sum j k) => LinearSystem (Tensor (:|: i (:|: j Nil)) e) (Tensor (:|: i (:|: k Nil)) e)

class SquareMatrix t where

Methods

unit :: Num e => t e

Indentity matrix.

inverse :: (Eq e, Fractional e) => t e -> Maybe (t e)

Inverts, if the matrix is invertible, otherwise Nothing.

tr :: Num e => t e -> e

Trace of the matrix.

charPoly :: Num e => t e -> [e]

Computes the coefficient of the polynomial p(z)=det(A+zI) using the method of closed ordered walks (clow) illustrated in the paper of G. Rote http://page.mi.fu-berlin.de/rote/Papers/pdf/Division-free+algorithms.pdf. The number of operations for the whole process is O(n^4), where n is the number of rows of the matrix. The first coefficient is the known term and equals the determinant, while the last one is the coefficient of z^(n-1) and equals the trace. The coefficient of z^n equals 1 and is not included in the resulting list. The k-th coefficient is the sum of all principal minors of order n-k+1.

minPoly :: (Eq e, Fractional e) => t e -> [e]

Minimal polynomial. The algorithm follows the paper of S. Bialas and M. Bialas http://bulletin.pan.pl/(56-4)391.pdf. The number of operations is O(n^4), where n is the number of rows of the matrix.

det :: Num e => t e -> e

Determinant of the matrix.

polyEval :: Num e => t e -> [e] -> t e

Evaluate a polynomial on a matrix.

Instances

(Ordinal i, Sum i i) => SquareMatrix (Tensor (:|: i (:|: i Nil)))