Numeric.LinearAlgebra.Array.Solve

Solution of general multidimensional linear and multilinear systems.

Linear systems

solve

Arguments

:: (Compat i, Coord t)

=> NArray i t

coefficients (a)

-> NArray i t

target (b)

-> NArray i t

result (x)

Solution of the linear system a x = b, where a and b are general multidimensional arrays. The structure and dimension names of the result are inferred from the arguments.

solveHomog

Arguments

:: (Compat i, Coord t)

=> NArray i t

coefficients (a)

-> [Name]

desired dimensions for the result (a subset selected from the target).

-> Either Double Int

Left "numeric zero" (e.g. eps), Right "theoretical" rank

-> [NArray i t]

basis for the solutions (x)

Solution of the homogeneous linear system a x = 0, where a is a general multidimensional array.

If the system is overconstrained we may provide the theoretical rank to get a MSE solution.

solveHomog1 :: (Compat i, Coord t) => NArray i t -> [Name] -> NArray i t

A simpler way to use solveHomog, which returns just one solution. If the system is overconstrained it returns the MSE solution.

solveH :: (Compat i, Coord t) => NArray i t -> [Char] -> NArray i t

solveHomog1 for single letter index names.

solveP

Arguments

:: Tensor Double

coefficients (a)

-> Tensor Double

desired result (b)

-> Name

the homogeneous dimension

-> Tensor Double

result (x)

Solution of the linear system a x = b, where a and b are general multidimensional arrays, with homogeneous equality along a given index.

Multilinear systems

General

data ALSParam i t

optimization parameters for alternating least squares

Constructors

ALSParam

Fields

nMax :: Int

maximum number of iterations

delta :: Double

minimum relative improvement in the optimization (percent, e.g. 0.1)

epsilon :: Double

maximum relative error. For nonhomogeneous problems it is the reconstruction error in percent (e.g. 1E-3), and for homogeneous problems is the frobenius norm of the expected zero structure in the right hand side.

post :: [NArray i t] -> [NArray i t]

post-processing function after each full iteration (e.g. id)

postk :: Int -> NArray i t -> NArray i t

post-processing function for the k-th argument (e.g. const id)

presys :: Matrix t -> Matrix t

preprocessing function for the linear systems (eg. id, or infoRank)

defaultParameters :: ALSParam i t

nMax = 20, epsilon = 1E-3, delta = 1, post = id, postk = const id, presys = id

mlSolve

Arguments

:: (Compat i, Coord t, Num (NArray i t), Show (NArray i t))

=> ALSParam i t

optimization parameters

-> [NArray i t]

coefficients (a), given as a list of factors.

-> [NArray i t]

initial solution [x,y,z...]

-> NArray i t

target (b)

-> ([NArray i t], [Double])

Solution and error history

Solution of a multilinear system a x y z ... = b based on alternating least squares.

mlSolveH

Arguments

:: (Compat i, Coord t, Num (NArray i t), Show (NArray i t))

=> ALSParam i t

optimization parameters

-> [NArray i t]

coefficients (a), given as a list of factors.

-> [NArray i t]

initial solution [x,y,z...]

-> ([NArray i t], [Double])

Solution and error history

Solution of the homogeneous multilinear system a x y z ... = 0 based on alternating least squares.

mlSolveP

Arguments

:: ALSParam Variant Double

optimization parameters

-> [Tensor Double]

coefficients (a), given as a list of factors.

-> [Tensor Double]

initial solution [x,y,z...]

-> Tensor Double

target (b)

-> Name

homogeneous index

-> ([Tensor Double], [Double])

Solution and error history

Solution of a multilinear system a x y z ... = b, with a homogeneous index, based on alternating least squares.

Factorized

solveFactors

Arguments

:: (Coord t, Random t, Compat i, Num (NArray i t), Show (NArray i t))

=> Int

seed for random initialization

-> ALSParam i t

optimization parameters

-> [NArray i t]

source (also factorized)

-> String

index pairs for the factors separated by spaces

-> NArray i t

target

-> ([NArray i t], [Double])

solution and error history

Given two arrays a (source) and b (target), we try to compute linear transformations x,y,z,... for each dimension, such that product [a,x,y,z,...] == b. (We can use eqnorm for post processing, or id.)

solveFactorsH

Arguments

:: (Coord t, Random t, Compat i, Num (NArray i t), Show (NArray i t))

=> Int

seed for random initialization

-> ALSParam i t

optimization parameters

-> [NArray i t]

coefficient array (a), (also factorized)

-> String

index pairs for the factors separated by spaces

-> ([NArray i t], [Double])

solution and error history

Homogeneous factorized system. Given an array a, given as a list of factors as, and a list of pairs of indices ["pi","qj", "rk", etc.], we try to compute linear transformations x!"pi", y!"pi", z!"rk", etc. such that product [a,x,y,z,...] == 0.

Utilities

eps :: Double

The machine precision of a Double: eps = 2.22044604925031e-16 (the value used by GNU-Octave).

eqnorm :: (Compat i, Show (NArray i Double)) => [NArray i Double] -> [NArray i Double]

post processing function that modifies a list of tensors so that they have equal frobenius norm

infoRank :: Field t => Matrix t -> Matrix t

debugging function (e.g. for presys), which shows rows, columns and rank of the coefficient matrix of a linear system.

solve' :: (Compat i, Coord a, Coord t) => (Matrix t -> Matrix a) -> NArray i t -> NArray i t -> NArray i a

solveHomog' :: (Compat i, Coord a, Coord t) => (Matrix t -> Matrix a) -> NArray i t -> [Name] -> Either Double Int -> [NArray i a]

solveHomog1' :: (Compat i, Coord t, Coord a) => (Matrix t -> Matrix a) -> NArray i t -> [Name] -> NArray i a

solveP' :: Coord b => (Matrix Double -> Matrix b) -> NArray Variant Double -> NArray Variant Double -> Name -> NArray Variant b