# CovarianceMatrix¶

class CovarianceMatrix(*args)

Covariance (real symmetric positive definite) matrix.

Parameters: size : int, , optional Matrix size. Default is 1. values : sequence of float with size , optional Values. OpenTURNS uses column-major ordering (like Fortran) for reshaping the flat list of values. Default creates an identity matrix. TypeError : If the matrix is not symmetric.

Examples

Create a matrix

>>> import openturns as ot
>>> C = ot.CovarianceMatrix(2, [1.0, 0.5, 0.5, 1.0])
>>> print(C)
[[ 1   0.5 ]
[ 0.5 1   ]]


Get or set terms

>>> print(C[0, 1])
0.5
>>> C[0, 1] = 0.6
>>> print(C[0, 1])
0.6
>>> print(C[:, 0])
[[ 1   ]
[ 0.6 ]]


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

>>> import numpy as np
>>> np_2d_array = np.array([[1.0, 0.5], [0.5, 1.0]])
>>> ot_matrix = ot.CovarianceMatrix(np_2d_array)


and back

>>> np_matrix = np.matrix(ot_matrix)

Attributes: thisown The membership flag

Methods

 checkSymmetry() Check if the internal representation is really symmetric. clean(threshold) Set elements smaller than a threshold to zero. computeCholesky([keepIntact]) Compute the Cholesky factor. computeDeterminant([keepIntact]) Compute the determinant. computeEV([keepIntact]) Compute the eigenvalues decomposition (EVD). computeEigenValues([keepIntact]) Compute eigenvalues. computeGram([transpose]) Compute the associated Gram matrix. computeLogAbsoluteDeterminant([keepIntact]) Compute the logarithm of the absolute value of the determinant. computeQR([fullQR, keepIntact]) Compute the QR factorization. computeSVD([fullSVD, keepIntact]) Compute the singular values decomposition (SVD). computeSingularValues([keepIntact]) Compute the singular values. computeTrace() Compute the trace of the matrix. getClassName() Accessor to the object’s name. getDimension() Accessor to the dimension (the number of rows). 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. isDiagonal() Test whether the matrix is diagonal or not. isEmpty() Tell if the matrix is empty. isPositiveDefinite() Test whether the matrix is positive definite or not. setName(name) Accessor to the object’s name. solveLinearSystem(*args) Solve a square linear system whose the present matrix is the operator. transpose() Transpose the matrix.
__init__(*args)

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

checkSymmetry()

Check if the internal representation is really symmetric.

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.
computeCholesky(keepIntact=True)

Compute the Cholesky factor.

The Cholesky factor of a covariance (real symmetric positive definite) matrix is the lower triangular matrix such that:

Parameters: 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. cholesky_factor : SquareMatrix The left (lower) Cholesky factor.

Notes

This uses LAPACK’s DPOTRF.

computeDeterminant(keepIntact=True)

Compute the determinant.

Parameters: 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. determinant : float The square matrix determinant.

Examples

>>> import openturns as ot
>>> A = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> A.computeDeterminant()
-2.0

computeEV(keepIntact=True)

Compute the eigenvalues decomposition (EVD).

The eigenvalues decomposition of a square matrix with size reads:

where is an diagonal matrix and is an orthogonal matrix.

Parameters: 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. eigenvalues : Point The vector of eigenvalues with size that form the diagonal of the matrix of the EVD. Phi : SquareComplexMatrix The left matrix of the EVD.

Notes

This uses LAPACK’S DSYEV.

Examples

>>> import openturns as ot
>>> import numpy as np
>>> M = ot.SymmetricMatrix([[1.0, 2.0], [2.0, -4.0]])
>>> eigen_values, Phi = M.computeEV()
>>> Lambda = ot.SquareMatrix(M.getDimension())
>>> for i in range(eigen_values.getSize()):
...     Lambda[i, i] = eigen_values[i]
>>> np.testing.assert_array_almost_equal(Phi * Lambda * Phi.transpose(), M)

computeEigenValues(keepIntact=True)

Compute eigenvalues.

Parameters: 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. eigenvalues : Point Eigenvalues.

Examples

>>> import openturns as ot
>>> M = ot.SymmetricMatrix([[1.0, 2.0], [2.0, -4.0]])
>>> print(M.computeEigenValues())
[-4.70156,1.70156]

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 ]]

computeLogAbsoluteDeterminant(keepIntact=True)

Compute the logarithm of the absolute value of the determinant.

Parameters: 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. determinant : float The logarithm of the absolute value of the square matrix determinant. sign : float The sign of the determinant.

Examples

>>> import openturns as ot
>>> A = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> A.computeLogAbsoluteDeterminant()
[0.693147..., -1.0]

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]

computeTrace()

Compute the trace of the matrix.

Returns: trace : float The trace of the matrix.

Examples

>>> import openturns as ot
>>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> M.computeTrace()
5.0

getClassName()

Accessor to the object’s name.

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

Accessor to the dimension (the number of rows).

Returns: dimension : int
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
isDiagonal()

Test whether the matrix is diagonal or not.

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

isPositiveDefinite()

Test whether the matrix is positive definite or not.

A matrix is positive definite if is positive for every compatible non-zero column vector .

Notes

This uses LAPACK’s DPOTRF.

setName(name)

Accessor to the object’s name.

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

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

Parameters: rhs : sequence of float 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 square linear system.

Notes

This will handle both matrices and vectors. Note that you’d better type explicitely the matrix if it has some properties that could simplify the resolution (see TriangularMatrix).

This uses LAPACK’S DGESV for matrices and DGELSY for vectors.

Examples

>>> import openturns as ot
>>> import numpy as np
>>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> b = ot.Point([1.0] * 2)
>>> x = M.solveLinearSystem(b)
>>> np.testing.assert_array_almost_equal(M * x, b)

thisown

The membership flag

transpose()

Transpose the matrix.

Returns: MT : SquareMatrix The transposed matrix.

Examples

>>> import openturns as ot
>>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> print(M)
[[ 1 2 ]
[ 3 4 ]]
>>> print(M.transpose())
[[ 1 3 ]
[ 2 4 ]]