Chi2fit.Matrix (Chi-SquaredFit v2.0.0) View Source

This module provides matrix inverse operations and supporting functions.

It provides 2 types of matrix norms and an iterative approach to calculating the matrix inverse. The implementation is based on the work [1].

References

[1] F. Soleymani, A Rapid Numerical Algorithm to Compute Matrix Inversion, International Journal of Mathematics and Mathematical Sciences, Volume 2012, Article ID 134653, doi:10.1155/2012/134653

Link to this section Summary

Types

A list of vectors (list of lists of numbers)

A list of numbers

Functions

Adds two vectors.

Calculates the determinant of the matrix.

Returns the diagonal elements of the matrix as a vector.

Calculates the inner product of two vectors.

Returns a matrix with the supplied vector as its diagonal elements.

Returns the matrix inverse of the argument.

Calculates the norm of the matrix as the sum of the absolutes value of all elements.

Calculates the norm of the matrix. All absolute values of the elements of each row are summed. The maximum value is returned

Calculates the norm of the matrix as norm_1/1 but over the columns instead of over the rows.

Subtracts two matrices and returns the result.

Returns the tranpose of the matrix

Constructs a unit matrix of size n. All diagonal elements have value 1 and the rest has value 0.

Link to this section Types

Specs

matrix() :: [vector()]

A list of vectors (list of lists of numbers)

Specs

vector() :: [number()]

A list of numbers

Link to this section Functions

Specs

add(vector(), vector()) :: vector()

Adds two vectors.

Examples

iex> add [1,2], [3,4]
[4,6]

iex> add [], []
[]

iex> add [1], [5]
[6]

Specs

det(matrix()) :: number()

Calculates the determinant of the matrix.

Specs

diagonal(matrix()) :: vector()

Returns the diagonal elements of the matrix as a vector.

Example

iex> diagonal [[1,2,3],[4,5,6],[7,8,9]]
[1, 5, 9]
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dotproduct(vector1, vector2, sum \\ 0)

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Calculates the inner product of two vectors.

Examples

iex> dotproduct [1,2], [3,4]
11

iex> dotproduct [], []
0

iex> dotproduct [1,2], []
** (ArgumentError) Vectors of unequal length

iex> dotproduct [1,2], [1]
** (ArgumentError) Vectors of unequal length

Specs

from_diagonal(vector()) :: matrix()

Returns a matrix with the supplied vector as its diagonal elements.

Examples

iex> from_diagonal [1,5,9]
[[1, 0, 0], [0, 5, 0], [0, 0, 9]]
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inverse(matrix, options \\ [])

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Specs

inverse(matrix(), options :: Keyword.t()) ::
  {:ok, inverse :: matrix()}
  | :failed_to_find_v0
  | :no_inverse
  | {:failed_to_reach_tolerance, inverse :: matrix(), error :: float()}

Returns the matrix inverse of the argument.

Options

`:tolerance` - Iterate until the `norm_1/1` of I-AV is less than this value
`:algorithm` - Four algorithms are supported: `:hotelling_bodewig` (second order), `:lie` (third order),
    `:krishnamurthy_sen` (sixth order), and `:soleymani` (seventh order); defaults to `:lie`
`:max_iterations` - Maximum number of iterations to perform; defaults to 500
`:range` - Range of values from -range to +range as a multiple of the unit matrix to try as an estimate
    of the inverse matric; defaults to 100
`:size` - Number of tries to estimate initial inverse; defautls to 100

Examples

iex> inverse [[3]]
{:ok,[[0.3333333333333333]]}

iex> inverse [[1,2],[3,4]]
{:ok,[[-2.0, 1.0], [1.5, -0.5]]}

iex> inverse([[3,2,0],[0,0,1],[2,-2,1]]) |> elem(1) |> Enum.map(fn row -> Enum.map(row, & Float.round(&1,10)) end)
[[0.2, -0.2, 0.2], [0.2, 0.3, -0.3], [0.0, 1.0, 0.0]]

iex> inverse([[3,2,0],[0,0,1],[2,-2,1]], algorithm: :soleymani) |> elem(1) |> Enum.map(fn row -> Enum.map(row, & Float.round(&1,14)) end)
[[0.2, -0.2, 0.2], [0.2, 0.3, -0.3], [0.0, 1.0, 0.0]]

iex> inverse([[3,2,0],[0,0,1],[2,-2,1]], tolerance: 1.0e-15) |> elem(1) |> Enum.map(fn row -> Enum.map(row, & Float.round(&1,14)) end)
[[0.2, -0.2, 0.2], [0.2, 0.3, -0.3], [0.0, 1.0, 0.0]]

For matrices that have no inverse:

iex> try do inverse [[1,2,3],[4,5,6],[7,8,9]] catch x->x end
:no_inverse

Specs

norm(matrix()) :: number()

Calculates the norm of the matrix as the sum of the absolutes value of all elements.

Example

iex> norm [[1,2,3],[4,5,6],[7,8,9]]
45

Specs

norm_1(matrix()) :: number()

Calculates the norm of the matrix. All absolute values of the elements of each row are summed. The maximum value is returned

Example

iex> norm_1 [[1,2,3],[4,5,6],[7,8,9]]
24

Specs

norm_inf(matrix()) :: number()

Calculates the norm of the matrix as norm_1/1 but over the columns instead of over the rows.

Example

iex> norm_inf [[1,2,3],[4,5,6],[7,8,9]]
18
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subtract(matrix1, matrix2)

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Specs

subtract(matrix(), matrix()) :: matrix()

Subtracts two matrices and returns the result.

Specs

transpose(matrix()) :: matrix()

Returns the tranpose of the matrix

Examples:

iex> transpose [ [1] ]
[[1]]

iex> transpose [ [1,2], [3,4] ]
[[1, 3], [2, 4]]

Specs

unit(n :: pos_integer()) :: [[0 | 1]]

Constructs a unit matrix of size n. All diagonal elements have value 1 and the rest has value 0.

Examples

iex> unit(3)
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]

iex> unit(0)
** (ArgumentError) Illegal argument '0'

iex> unit -1
** (ArgumentError) Illegal argument '-1'

iex> unit 1.3
** (ArgumentError) Illegal argument '1.3'