# Matrix¶

class Matrix(*args)

Real rectangular matrix.

Parameters: n_rows : int, , optional Number of rows. Default is 1. n_columns : int, , optional Number of columns. Default is 1. values : sequence of float with size , optional Values. column-major ordering is used (like Fortran) for reshaping the flat list of values. Default creates a zero matrix.

Examples

Create a matrix

>>> import openturns as ot
>>> M = ot.Matrix(2, 2, range(2 * 2))
>>> print(M)
[[ 0 2 ]
[ 1 3 ]]


Get or set terms

>>> print(M[0, 0])
0.0
>>> M[0, 0] = 1.
>>> print(M[0, 0])
1.0
>>> print(M[:, 0])
[[ 1 ]
[ 1 ]]


Create an openturns matrix from a numpy 2d-array (or matrix, or 2d-list)…

>>> import numpy as np
>>> np_2d_array = np.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
>>> ot_matrix = ot.Matrix(np_2d_array)


and back

>>> np_matrix = np.matrix(ot_matrix)


Basic linear algebra operations (provided the dimensions are compatible)

>>> A = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> B = ot.Matrix(np.eye(2))
>>> C = ot.Matrix(3, 2, [1.] * 3 * 2)
>>> print(A * B - C)
[[ 0 1 ]
[ 2 3 ]
[ 4 5 ]]


Methods

 clean(threshold) Set elements smaller than a threshold to zero. computeGram([transpose]) Compute the associated Gram matrix. computeQR([fullQR, keepIntact]) Compute the QR factorization. computeSVD([fullSVD, keepIntact]) Compute the singular values decomposition (SVD). computeSingularValues([keepIntact]) Compute the singular values. getClassName() Accessor to the object’s name. getId() Accessor to the object’s id. getImplementation(*args) Accessor to the underlying implementation. getName() Accessor to the object’s name. getNbColumns() Accessor to the number of columns. getNbRows() Accessor to the number of rows. isEmpty() Tell if the matrix is empty. setName(name) Accessor to the object’s name. solveLinearSystem(*args) Solve a rectangular linear system whose the present matrix is the operator. transpose() Transpose the matrix.
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

clean(threshold)

Set elements smaller than a threshold to zero.

Parameters: threshold : float Threshold for zeroing elements. cleaned_matrix : Matrix Input matrix with elements smaller than the threshold set to zero.
computeGram(transpose=True)

Compute the associated Gram matrix.

Parameters: transposed : bool Tells if matrix is to be transposed or not. Default value is True MMT : Matrix The Gram matrix.

Notes

When transposed is set to True, the method computes . Otherwise it computes

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> MtM = M.computeGram()
>>> print(MtM)
[[ 35 44 ]
[ 44 56 ]]
>>> MMt = M.computeGram(False)
>>> print(MMt)
[[  5 11 17 ]
[ 11 25 39 ]
[ 17 39 61 ]]

computeQR(fullQR=False, keepIntact=True)

Compute the QR factorization. By default, it is the economic decomposition which is computed.

The economic QR factorization of a rectangular matrix with (more rows than columns) is defined as follows:

where is an upper triangular matrix, is , is , and and both have orthogonal columns.

Parameters: full_qr : bool, optional A flag telling whether Q, R or Q1, R1 are returned. Default is False and returns Q1, R1. keep_intact : bool, optional A flag telling whether the present matrix is preserved or not in the computation of the decomposition. Default is True and leaves the present matrix unchanged. Q1 : Matrix The orthogonal matrix of the economic QR factorization. R1 : TriangularMatrix The right (upper) triangular matrix of the economic QR factorization. Q : Matrix The orthogonal matrix of the full QR factorization. R : TriangularMatrix The right (upper) triangular matrix of the full QR factorization.

Notes

The economic QR factorization is often used for solving overdetermined linear systems (where the operator has ) in the least-square sense because it implies solving a (simple) triangular system:

This uses LAPACK’s DGEQRF and DORGQR.

Examples

>>> import openturns as ot
>>> import numpy as np
>>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> Q1, R1 = M.computeQR()
>>> np.testing.assert_array_almost_equal(Q1 * R1, M)

computeSVD(fullSVD=False, keepIntact=True)

Compute the singular values decomposition (SVD).

The singular values decomposition of a rectangular matrix with size reads:

where is an orthogonal matrix, is an diagonal matrix and is an orthogonal matrix.

Parameters: fullSVD : bool, optional Whether the null parts of the orthogonal factors are explicitely stored or not. Default is False and computes a reduced SVD. keep_intact : bool, optional A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged. singular_values : Point The vector of singular values with size that form the diagonal of the matrix of the SVD. U : SquareMatrix The left orthogonal matrix of the SVD. VT : SquareMatrix The transposed right orthogonal matrix of the SVD.

Notes

This uses LAPACK’s DGESDD.

Examples

>>> import openturns as ot
>>> import numpy as np
>>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> singular_values, U, VT = M.computeSVD(True)
>>> Sigma = ot.Matrix(M.getNbRows(), M.getNbColumns())
>>> for i in range(singular_values.getSize()):
...     Sigma[i, i] = singular_values[i]
>>> np.testing.assert_array_almost_equal(U * Sigma * VT, M)

computeSingularValues(keepIntact=True)

Compute the singular values.

Parameters: fullSVD : bool, optional Whether the null parts of the orthogonal factors are explicitely stored or not. Default is False and computes a reduced SVD. keep_intact : bool, optional A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged. singular_values : Point The vector of singular values with size that form the diagonal of the matrix of the SVD decomposition.

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> print(M.computeSingularValues(True))
[9.52552,0.514301]

getClassName()

Accessor to the object’s name.

Returns: class_name : str The object class name (object.__class__.__name__).
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getImplementation(*args)

Accessor to the underlying implementation.

Returns: impl : Implementation The implementation class.
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getNbColumns()

Accessor to the number of columns.

Returns: n_columns : int
getNbRows()

Accessor to the number of rows.

Returns: n_rows : int
isEmpty()

Tell if the matrix is empty.

Returns: is_empty : bool True if the matrix contains no element.

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[]])
>>> M.isEmpty()
True

setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
solveLinearSystem(*args)

Solve a rectangular linear system whose the present matrix is the operator.

Parameters: rhs : Point or Matrix with values or rows, respectively The right hand side member of the linear system. keep_intact : bool, optional A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged. solution : The solution of the rectangular linear system.

Notes

This will handle both matrices and vectors, as well as underdetermined, square or overdetermined linear systems although you’d better type explicitely your matrix if it has some properties that could simplify the resolution (see TriangularMatrix, SquareMatrix).

This uses LAPACK’s DGELSY. The RCOND parameter of this routine can be changed through the MatrixImplementation-DefaultSmallPivot key of the ResourceMap.

Examples

>>> import openturns as ot
>>> import numpy as np
>>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> b = ot.Point([1.0] * 3)
>>> x = M.solveLinearSystem(b)
>>> np.testing.assert_array_almost_equal(M * x, b)

transpose()

Transpose the matrix.

Returns: MT : Matrix The transposed matrix.

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> print(M)
[[ 1 2 ]
[ 3 4 ]
[ 5 6 ]]
>>> print(M.transpose())
[[ 1 3 5 ]
[ 2 4 6 ]]