LA::Matrix(T)

Create row from start_val...end_val with step of delta between

Create a block diagonal matrix from provided matrices

Given the inputs A, B and C, the output will have these matrices arranged on the diagonal:

Example:

m = Mat.block_diag(a, b, c)

m will have following structure:

[[a, 0, 0],
 [0, b, 0],
 [0, 0, c]]

Construct a Circulant matrix

c - first column of matrix

Example:

a = circulant([1, 2, 3])
a.to_aa # => [[1, 3, 2],[2, 1, 3],[3, 2, 1]])

Behaviour copied from scipy

Returns single column matrix with elements taken from array values

Create a companion matrix associated with the polynomial whose coefficients are given in a

Behaviour copied from scipy

Example:

Mat.companion([1, -10, 31, -30]) # =>
# LA::GeneralMatrix(Float64) (3x3, None):
# [10.0, -31.0, 30.0]
# [1.0, 0.0, 0.0]
# [0.0, 1.0, 0.0]

Returns diagonal matrix of given size with all diagonal elements equal to value

Returns diagonal matrix of given size with diagonal elements taken from array values

Returns square diagonal matrix with diagonal elements taken from array values

Returns diagonal matrix of given size with diagonal elements equal to block value

returns identity matrix of size n

Returns a symmetric Fiedler matrix

Given an sequence of numbers values, Fiedler matrices have the structure f[i, j] = (values[i] - values[j]).abs, and hence zero diagonals and nonnegative entries. A Fiedler matrix has a dominant positive eigenvalue and other eigenvalues are negative.

Behaviour copied from scipy

Example:

Mat.fiedler([1, 4, 12, 45, 77]) # =>
# LA::GeneralMatrix(Float64) (5x5, Symmetric | Hermitian):
# [0.0, 3.0, 11.0, 44.0, 76.0]
# [3.0, 0.0, 8.0, 41.0, 73.0]
# [11.0, 8.0, 0.0, 33.0, 65.0]
# [44.0, 41.0, 33.0, 0.0, 32.0]
# [76.0, 73.0, 65.0, 32.0, 0.0]

Converts a string of given format to matrix Example:

str = "( 1,2,3 | 4,5,6 | 7,8,9 )"
LA::GMat.from_custom(str, "(", ",", "|", ")")

Converts a string of given format to matrix Example:

str = "( 1,2,3 | 4,5,6 | 7,8,9 )"
LA::GMat.from_custom(IO::Memory.new(str), prefix: "(", columns_separator: ",", rows_separator: "|", postfix: ")")

Converts a string from matlab format to matrix Example:

str = "[1,2,3; 4,5,6; 7,8,9]"
LA::GMat.from_matlab(str)

Constructs an n-by-n Hadamard matrix, n must be power of 2

Behaviour copied from scipy

Example:

Mat.hadamard(4) # =>
# LA::GeneralMatrix(Float64) (4x4,  Symmetric | Hermitian):
# [1.0, 1.0, 1.0, 1.0]
# [1.0, -1.0, 1.0, -1.0]
# [1.0, 1.0, -1.0, -1.0]
# [1.0, -1.0, -1.0, 1.0]

Create a Hankel matrix The Hankel matrix has constant anti-diagonals, with column as its first column and row as its last row. If `row is nil, then row with elements equal to zero is assumed. Behaviour copied from scipy Examples:

a = Mat.hankel([1, 17, 99])
a.to_aa # => [
# [ 1, 17, 99],
# [17, 99,  0],
# [99,  0,  0]]
a = Mat.hankel([1, 2, 3, 4], [4, 7, 7, 8, 9])
a.to_aa # => [
# [1, 2, 3, 4, 7],
# [2, 3, 4, 7, 7],
# [3, 4, 7, 7, 8],
# [4, 7, 7, 8, 9]]

Create an Helmert matrix of order n

If full is true the (n x n) matrix will be returned. Otherwise the submatrix that does not include the first row will be returned

Behaviour copied from scipy

Example:

Mat.helmert(5, full: true) # =>
# LA::GeneralMatrix(Float64) (5x5, Orthogonal):
# [0.4472135954999579, 0.4472135954999579, 0.4472135954999579, 0.4472135954999579, 0.4472135954999579]
# [0.7071067811865476, -0.7071067811865476, 0.0, 0.0, 0.0]
# [0.408248290463863, 0.408248290463863, -0.816496580927726, 0.0, 0.0]
# [0.28867513459481287, 0.28867513459481287, 0.28867513459481287, -0.8660254037844386, 0.0]
# [0.22360679774997896, 0.22360679774997896, 0.22360679774997896, 0.22360679774997896, -0.8944271909999159]

Create a Hilbert matrix of order n.

Returns the n by n matrix with entries h[i,j] = 1 / (i + j + 1).

Behaviour copied from scipy

Example:

Mat.hilbert(3) # =>
# LA::GeneralMatrix(Float64) (3x3, Symmetric | Hermitian | PositiveDefinite):
# [1.0, 0.5, 0.3333333333333333]
# [0.5, 0.3333333333333333, 0.25]
# [0.3333333333333333, 0.25, 0.2]

returns identity matrix of size n

Compute the inverse of the Hilbert matrix of order n.

The entries in the inverse of a Hilbert matrix are integers.

Behaviour copied from scipy

Example:

Mat.invhilbert(4) # => LA::GeneralMatrix(Float64) (4x4, Symmetric | Hermitian | PositiveDefinite):
# [16.0, -120.0, 240.0, -140.0]
# [-120.0, 1200.0, -2700.0, 1680.0]
# [240.0, -2700.0, 6480.0, -4200.0]
# [-140.0, 1680.0, -4200.0, 2800.0]

Mat.invhilbert(16)[7, 7] # => 4.247509952853739e+19

Returns the inverse of the n x n Pascal matrix

The Pascal matrix is a matrix containing the binomial coefficients as its elements.

kind : see PascalKind

Behaviour copied from scipy

Example:

Mat.invpascal(4) # =>
# LA::GeneralMatrix(Float64) (5x5, Symmetric | Hermitian | PositiveDefinite):
# [5.0, -10.0, 10.0, -5.0, 1.0]
# [-10.0, 30.0, -35.0, 19.0, -4.0]
# [10.0, -35.0, 46.0, -27.0, 6.0]
# [-5.0, 19.0, -27.0, 17.0, -4.0]
# [1.0, -4.0, 6.0, -4.0, 1.0]
Mat.invpascal(5, PascalKind::Lower) # =>
# LA::GeneralMatrix(Float64) (5x5, LowerTriangular):
# [1.0, 0.0, 0.0, 0.0, 0.0]
# [-1.0, 1.0, 0.0, 0.0, 0.0]
# [1.0, -2.0, 1.0, 0.0, 0.0]
# [-1.0, 3.0, -3.0, 1.0, 0.0]
# [1.0, -4.0, 6.0, -4.0, 1.0]

Returns kroneker product of matrices

Resulting matrix size is {a.nrows*b.nrows, a.ncolumns*b.ncolumns}

Create a Leslie matrix

Given the length n array of fecundity coefficients f and the length n-1 array of survival coefficients s, return the associated Leslie matrix.

Behaviour copied from scipy

Example:

Mat.leslie([0.1, 2.0, 1.0, 0.1], [0.2, 0.8, 0.7]) # =>
# LA::GeneralMatrix(Float64) (4x4, None):
# [0.1, 2.0, 1.0, 0.1]
# [0.2, 0.0, 0.0, 0.0]
# [0.0, 0.8, 0.0, 0.0]
# [0.0, 0.0, 0.7, 0.0]

Loads a matrix from CSV (comma separated values) file Example:

LA::GMat.rand(30, 30).save_csv("./test.csv")
a = LA::GMat.load_csv("./test.csv")

Generate matrix of given size with all elements equal to one

Returns the n x n Pascal matrix.

The Pascal matrix is a matrix containing the binomial coefficients as its elements.

kind : see PascalKind

Behaviour copied from scipy

Example:

Mat.pascal(4) # =>
# LA::GeneralMatrix(Float64) (4x4, Symmetric | Hermitian | PositiveDefinite):
# [1.0, 1.0, 1.0, 1.0]
# [1.0, 2.0, 3.0, 4.0]
# [1.0, 3.0, 6.0, 10.0]
# [1.0, 4.0, 10.0, 20.0]
Mat.pascal(4, PascalKind::Lower) # =>
# LA::GeneralMatrix(Float64) (4x4, LowerTriangular):
# [1.0, 0.0, 0.0, 0.0]
# [1.0, 1.0, 0.0, 0.0]
# [1.0, 2.0, 1.0, 0.0]
# [1.0, 3.0, 3.0, 1.0]

Generate matrix of given size with elements randomly distributed from range 0.0..1.0

Alias for #repmat

Returns single row matrix with elements taken from array values

Create a Toeplitz matrix

column - first column of matrix

row - first row of matrix (if nil, it is assumed that row = column.map(&.conj))

Behaviour copied from scipy

Example:

MatComplex.toeplitz([1, 2, 3], [1, 4, 5, 6]) # =>
# LA::GeneralMatrix(Float64) (3x4, None):
# [1.0, 4.0, 5.0, 6.0]
# [2.0, 1.0, 4.0, 5.0]
# [3.0, 2.0, 1.0, 4.0]
MatComplex.toeplitz([1.0, 2 + 3.i, 4 - 1.i]) # =>
# LA::GeneralMatrix(Complex) (3x3, Hermitian):
# [1.0 + 0.0i, 2.0 - 3.0i, 4.0 + 1.0i]
# [2.0 + 3.0i, 1.0 + 0.0i, 2.0 - 3.0i]
# [4.0 - 1.0i, 2.0 + 3.0i, 1.0 + 0.0i]

Construct (nrows, ncolumns) matrix filled with ones at and below the kth diagonal.

The matrix has A[i,j] == 1 for j <= i + k

k - Number of subdiagonal below which matrix is filled with ones. k = 0 is the main diagonal, k < 0 subdiagonal and k > 0 superdiagonal.

Behaviour copied from scipy

Example:

Mat.tri(3, 5, 2).to_aa # => [[
# [1, 1, 1, 0, 0],
# [1, 1, 1, 1, 0],
# [1, 1, 1, 1, 1]]
Mat.tri(3, 5, -1).to_aa # => [[
# [0, 0, 0, 0, 0],
# [1, 0, 0, 0, 0],
# [1, 1, 0, 0, 0]]

Generate matrix of given size with all elements equal to zero

Multiplies matrix to scalar

Multiplies matrix to scalar

Matrix product to given m

Raises ArgumentError if inner dimensions do not match

This method automatically calls optimal function depending on MatrixFlags.

If one of the matrix is square and triangular - trmm is called

If one of the matrix is symmetric\hermitian - symm/hemm is called

Otherwise - gemm is called

Raises the square matrix to the integer power other

Implementation taken from https://github.com/Exilor/matrix/

Adds scalar value to every element

Adds scalar value to every element

Returns element-wise sum

This method raises if another matrix doesn't have same size

Substracts scalar value from every element

Returns element-wise substract

This method raises if another matrix doesn't have same size

Multiplies matrix to -1

Divides matrix to scalar

Compare with another matrix

Returns True only if all element are exactly equal. Use #almost_eq If you need approximate equality

Return element of matrix on row i and column j

If i or j negative they are counted from end of matrix If i>=nrows or j>=ncolumns exception is raised Use #unsafe_fetch if you need to skip these checks

Return submatrix over given ranges.

See SubMatrix(T) for details on submatrices Example:

m = GMat32.new([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]])
m1 = m[1..3, 2..3] # => [[7.0, 8.0], [11.0, 12.0], [15.0, 16.0]]
m2 = m[..3, 2]     # => [[3.0], [7.0], [11.0], [15.0]]

Return submatrix over given ranges.

See SubMatrix(T) for details on submatrices Example:

m = GMat32.new([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]])
m1 = m[1..3, 2..3] # => [[7.0, 8.0], [11.0, 12.0], [15.0, 16.0]]
m2 = m[..3, 2]     # => [[3.0], [7.0], [11.0], [15.0]]

Return submatrix over given ranges.

See SubMatrix(T) for details on submatrices Example:

m = GMat32.new([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]])
m1 = m[1..3, 2..3] # => [[7.0, 8.0], [11.0, 12.0], [15.0, 16.0]]
m2 = m[..3, 2]     # => [[3.0], [7.0], [11.0], [15.0]]

Assign element of matrix on row i and column j

If i or j negative they are counted from end of matrix If i>=nrows or j>=ncolumns exception is raised Use #unsafe_set if you need to skip these checks Note this method reset all matrix flags

Assign value to a given submatrix

value can be a scalar or a matrix of same size as affected submatrix

Assign value to a given submatrix

value can be a scalar or a matrix of same size as affected submatrix

Assign value to a given submatrix

value can be a scalar or a matrix of same size as affected submatrix

Alias for #norm

Perform inplace addition with matrix m multiplied to scalar k

a.add!(k, b) is equal to a = a + k * b, but faster as no new matrix is allocated

Performs inplace addition with matrix m

a.add!(b) is equal to a = a + b, but faster as no new matrix is allocated

performs c = alphaab + beta*c (BLAS routines gemm/symm/hemm)

Approximately compare with other matrix

Returns true if all elements are within eps from corresponding elements of other matrix

Approximately compare with other matrix

Returns true if all elements are within eps from corresponding elements of other matrix

Uses #tolerance as an eps by default

Directly set or reset matrix flag without check See MatrixFlags for description of flags

Returns concatenation with another matrix by Axis::Rows (horizontal) or Axis::Columns (vertical)

Converts complex matrix to real one if all imaginary parts are less then eps, returns nil otherwise

Returns just matrix if it is already real

Reset matrix flags to None Usually is done automatically, but this method could be needed if internal content was changed using to_unsafe

Returns Indexable(SubMatrix(T)) that allows iterating over columns

alias to #conjtranspose

alias to #conjtranspose!

Returns conjurgate transposed matrix

result is same as #transpose for real matrices

optimization idea for noncomplex matrix is from scipy

Calculates determinant for a square matrix

if overwrite_a is true, a will be overriden in process of calculation

Uses getrf LAPACK routine

Detect if given aflags are true or flase for a matrix with tolerance eps Update flags property See MatrixFlags for description of matrix flags Returns self for method chaining

Detect if given aflags are true or flase for a matrix with tolerance eps Update flags property See MatrixFlags for description of matrix flags Returns True if all given flags are set

Returns Indexable(T) that allow iterating over k-th diagonal of matrix

Example:

m = GMat32.new([[-1, 2, 3, 4],
                [5, -6, 7, 8],
                [9, 10, -11, 12]])
m.diag(0).to_a.should eq [-1, -6, -11]
m.diag(1).to_a.should eq [2, 7, 12]
m.diag(2).to_a.should eq [3, 8]
m.diag(3).to_a.should eq [4]
expect_raises(ArgumentError) { m.diag(4) }
m.diag(-1).to_a.should eq [5, 10]
m.diag(-2).to_a.should eq [9]
expect_raises(ArgumentError) { m.diag(-3) }
expect_raises(ArgumentError) { m.diag(-4) }

Yields every element of matrix

all argument controls whether to yield all or non-empty elements for banded\sparse matrices Example: m.each { |v| raise "negative element found" if v < 0 }

Yields every index

all argument controls whether to yield all or non-empty elements for banded\sparse matrices Example: m.each_index { |i, j| m[i, j] = -m[i, j] }

For every element of matrix that is below main diagonal it yields a block with value and with corresponding row and column

This method is useful for symmetric matrices and similar cases include_diagonal argument controls whether to include elements on main diagonal all argument controls whether to yield all or non-empty elements for banded\sparse matrices

For every element of matrix that is above main diagonal it yields a block with value and with corresponding row and column

This method is useful for symmetric matrices and similar cases include_diagonal argument controls whether to include elements on main diagonal all argument controls whether to yield all or non-empty elements for banded\sparse matrices

Yields every element of matrix with corresponding row and column

all argument controls whether to yield all or non-empty elements for banded\sparse matrices Example: m.each_with_index { |v, i, j| m2[i, j] = v }

generalized eigenvalues problem

Computes matrix exponential

It exploits triangularity (if any) of A. if schur_fact is true, function uses an initial transformation to Schur form.

Reference: A. H. Al-Mohy and N. J. Higham, A New Scaling and Squaring Algorithm for the Matrix Exponential, SIAM J. Matrix Anal. Appl. 31(3): 970-989, 2009. Awad H. Al-Mohy and Nicholas J. Higham, April 20, 2010.

Returns flags of matrix (see MatrixFlags)

Returns horizontal concatenation with another matrix

Converts matrix to string for inspection

Output looks like:

GeneralMatrix(Float64) (10x10, MatrixFlags::None)
[1, 2, 3, .... 10]
[11, 12, 13, .... 20]
...
[91, 92, 93, .... 100]

Calculate matrix inversion See #inv! for details on algorithm

Calculate matrix inversion inplace Method selects optimal algorithm depending on MatrixFlags transpose returned if matrix is orthogonal trtri is used if matrix is triangular potrf, potri are used if matrix is positive definite hetrf, hetri are used if matrix is hermitian sytrf, sytri are used if matrix is symmetric getrf, getri are used otherwise

Returns kroneker product with matrix b

Compute pivoted LU decomposition of a matrix

If you call p,l,u = a.lu for a matrix m*n a then

• p will be m*m permutation matrix
• l will be m*k lower triangular matrix with unit diagonal. K = min(M, N)
• u will be k*n upper triangular matrix
• (p*l*u) will be equal to a (within calculation tolerance)

if overwrite_a is true, a will be overriden in process of calculation

Uses getrf LAPACK routine

Compute pivoted LU decomposition of a matrix and returns it in a compact form, useful for solving linear equation systems.

Uses getrf LAPACK routine

Compute pivoted LU decomposition of a matrix and returns it in a compact form, useful for solving linear equation systems.

Overrides source matrix in a process of calculation

Uses getrf LAPACK routine

Returns result of appliyng block to every element (without index)

Yields each element (without index) and replace it with returned value

Returns result of appliyng block to every element with index

Yields each element with index and replace it with returned value

Calculate maximum over given axis

Calculate minimum over given axis

Returns number of columns in matrix

returns matrix norm

kind defines type of norm

Uses LAPACK routines lantr, lanhe, lansy, lange depending on matrix flags

Returns number of rows in matrix

Calculate Moore–Penrose inverse of a matrix

https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse Implemented using an svd decomposition

Perform product over given axis

determine effective rank either by SVD method or QR-factorization with pivoting

if overwrite_a is true, a will be overriden in process of calculation

QR method is faster, but could fail to determine rank in some cases

Perform reduce from initial value over given axis

Return matrix repeated arows times by vertical and acolumns times by horizontal

Returns Indexable(SubMatrix(T)) that allows iterating over rows

Save a matrix to CSV (comma separated values) file Example:

LA::GMat.rand(30, 30).save_csv("./test.csv")
a = LA::GMat.load_csv("./test.csv")

Perform inplace multiplication to scalar k

Returns shape of matrix in a form of tuple {nrows, ncolumns}

Returns shape of matrix in a form of tuple {nrows, ncolumns}

Solves matrix equation self*x = b and returns x

Method returns matrix of same size as b

Matrix must be square, number of rows must match b

if overwrite_a is true, a will be overriden in process of calculation

if overwrite_b is true, b will be overriden in process of calculation

Uses LAPACK routines trtrs, posv, hesv, sysv, gesv depending on matrix flags

Returns True if matrix is square and False otherwise

Perform sum over given axis

Calculates singular value decomposition for a matrix

If you call u,s,vt = a.svd for a matrix m*n a then

• u will be m*m matrix
• s will be Array(T) with size equal to {m, n}.min
• vt will be n*n matrix
• (u*Mat.diag(a.nrows, a.ncolumns, s)*vt) will be equal to a (within calculation tolerance)

if overwrite_a is true, a will be overriden in process of calculation

Uses gesdd LAPACK routine

Calculates array of singular values for a matrix

if overwrite_a is true, a will be overriden in process of calculation

Uses gesdd LAPACK routine

alias to #transpose

alias to #transpose!

to_custom(io, "[", ",", "],[", "]") Converts a matrix to string with custom format. Example:

a = LA::GMat[[1, 2, 3], [4, 5, 6], [7, 8, 9]]
str = String.build do |io|
  a.to_custom(io, prefix: "(", columns_separator: ",", rows_separator: "|", postfix: ")")
end
str # => "(1,2,3|4,5,6|7,8,9)"

to_custom(io, "[", ",", "],[", "]") Converts a matrix to string with custom format. Example:

a = LA::GMat[[1, 2, 3], [4, 5, 6], [7, 8, 9]]
str = a.to_custom(prefix: "(", columns_separator: ",", rows_separator: "|", postfix: ")")
str # => "(1,2,3|4,5,6|7,8,9)"

Creates general matrix with same content. Useful for banded\sparse matrices

Converts complex matrix to imaginary part

Converts a matrix to matlab format

Converts a matrix to matlab format string Example:

LA::GMat[[1, 2, 3], [4, 5, 6], [7, 8, 9]].to_matlab # => "[1,2,3; 4,5,6; 7,8,9]"

Converts complex matrix to real part

Converts matrix to string, with linefeeds before and after matrix:

Output looks like:

[1, 2, 3, .... 10]
[11, 12, 13, .... 20]
...
[91, 92, 93, .... 100]

Returns pointer to underlying data

Storage format depends of matrix type This method raises at runtime if matrix doesn't have raw pointer

Returns estimated tolerance of equality\inequality

This value is used by default in #almost_eq compare

performs b = alphaab or b = alphaba (BLAS routine trmm)

a must be a square triangular GeneralMatrix(T)

if left is true, alpha*a*b is calculated, otherwise alpha*b*a

Returns sum of diagonal elements (trace of matrix)

Returns transposed matrix

Same as tril in scipy - returns lower triangular or trapezoidal part of matrix

Returns a matrix with all elements above k-th diagonal zeroed

Works like a tril in scipy - remove all elements above k-diagonal

Same as triu in scipy - returns upper triangular or trapezoidal part of matrix

Returns a matrix with all elements below k-th diagonal zeroed

Works like a triu in scipy - remove all elements below k-diagonal

Returns vertical concatenation with another matrix

Complex utility macros that simplifies calling certain LAPACK functions It substitute first letter, allocate workareas, raise exception if return value is negative Check source to see supported functions Example:

# Calling *geev to calculate eigenvalues
lapack(geev, 'N'.ord.to_u8, 'N'.ord.to_u8, nrows, a, nrows,
  vals.to_unsafe.as(LibCBLAS::ComplexDouble*),
  Pointer(LibCBLAS::ComplexDouble).null, nrows,
  Pointer(LibCBLAS::ComplexDouble).null, nrows, worksize: [2*nrows])

Note that it should be called only from methods of Matrix or its descendants

Utility macros that simplifies calling certain LAPACK functions Arguments that point to matrix should be passed as matrix(arg) Example:

# Calling *lange to calculate infinity-norm
lapack_util(lange, worksize, 'I', m.nrows, m.ncolumns, matrix(m), m.nrows)

Note that it should be called only from methods of Matrix or its descendants