SquareMatrix

class SquareMatrix(*args)

Real square matrix.

Parameters:
sizeint, n > 0, optional

Matrix size. Default is 1.

valuessequence of float with size n^2, optional

Values. OpenTURNS uses column-major ordering (like Fortran) for reshaping the flat list of values. Default creates a zero matrix.

Examples

Create a matrix

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

Get or set terms

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

Create an openturns matrix from a square 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]])
>>> ot_matrix = ot.SquareMatrix(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.SquareMatrix(np.eye(2))
>>> C = ot.Matrix(3, 2, [1.0] * 3 * 2)
>>> print(A * B - C)
[[ 0 1 ]
 [ 2 3 ]
 [ 4 5 ]]
>>> A = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> print(A ** 2)
[[  7 10 ]
 [ 15 22 ]]

Methods

clean(threshold)

Set elements smaller than a threshold to zero.

computeDeterminant([keepIntact])

Compute the determinant.

computeEV([keepIntact])

Compute the eigenvalues decomposition (EVD).

computeEigenValues([keepIntact])

Compute eigenvalues.

computeGram([transpose])

Compute the associated Gram matrix.

computeLargestEigenValueModule(*args)

Compute the largest eigenvalue module.

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()

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.

reshape(newRowDim, newColDim)

Reshape the matrix.

reshapeInPlace(newRowDim, newColDim)

Reshape the matrix, in place.

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.

computeHadamardProduct

computeSumElements

getDiagonal

setDiagonal

squareElements

__init__(*args)
clean(threshold)

Set elements smaller than a threshold to zero.

Parameters:
thresholdfloat

Threshold for zeroing elements.

Returns:
cleaned_matrixMatrix

Input matrix with elements smaller than the threshold set to zero.

computeDeterminant(keepIntact=True)

Compute the determinant.

Parameters:
keep_intactbool, optional

A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.

Returns:
determinantfloat

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 \mat{M} with size n reads:

\mat{M} = \mat{\Phi} \mat{\Lambda} \mat{\Phi}^{-1}

where \mat{\Lambda} is an n \times n diagonal matrix and \mat{\Phi} is an n \times n orthogonal matrix.

Parameters:
keep_intactbool, optional

A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.

Returns:
eigen_valuesComplexCollection

The vector of eigenvalues with size n that form the diagonal of the n \times n matrix \mat{\Lambda} of the EVD.

PhiSquareComplexMatrix

The left matrix of the EVD.

Notes

This uses LAPACK’S DGEEV.

Examples

>>> import openturns as ot
>>> import numpy as np
>>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> eigen_values, Phi = M.computeEV()
>>> Lambda = ot.SquareComplexMatrix(M.getDimension())
>>> for i in range(eigen_values.getSize()):
...     Lambda[i, i] = eigen_values[i]
>>> # from scipy.linalg import inv # SquareComplexMatrix does not implement solveLinearSystem
>>> # Phi, Lambda = np.matrix(Phi), np.matrix(Lambda)
>>> # np.testing.assert_array_almost_equal(Phi * Lambda * inv(Phi), M)
computeEigenValues(keepIntact=True)

Compute eigenvalues.

Parameters:
keep_intactbool, optional

A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.

Returns:
eigenvaluesComplexCollection

Eigenvalues.

See also

computeEV

Examples

>>> import openturns as ot
>>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> M.computeEigenValues()
[(-0.372281,0),(5.37228,0)]
computeGram(transpose=True)

Compute the associated Gram matrix.

Parameters:
transposedbool

Tells if matrix is to be transposed or not. Default value is True

Returns:
MMTMatrix

The Gram matrix.

Notes

When transposed is set to True, the method computes cM^t \times \cM. Otherwise it computes \cM \ times \cM^t

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 ]]
computeLargestEigenValueModule(*args)

Compute the largest eigenvalue module.

Parameters:
maximumIterationsint, optional

The maximum number of power iterations to perform to get the approximation. Default is given by the ‘Matrix-LargestEigenValueIterations’ key in the ResourceMap.

epsilonfloat, optional

The target relative error. Default is given by the ‘Matrix-LargestEigenValueRelativeError’ key in the ResourceMap.

Returns:
largestEigenvalueModulefloat

The largest eigenvalue module.

Examples

>>> import openturns as ot
>>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]])
>>> M.computeLargestEigenValueModule()
5.3722...
computeLogAbsoluteDeterminant(keepIntact=True)

Compute the logarithm of the absolute value of the determinant.

Parameters:
keep_intactbool, optional

A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.

Returns:
determinantfloat

The logarithm of the absolute value of the square matrix determinant.

signfloat

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 \mat{M} with n_r \geq n_c (more rows than columns) is defined as follows:

\mat{M} = \mat{Q} \mat{R}
        = \mat{Q} \begin{bmatrix} \mat{R_1} \\ \mat{0} \end{bmatrix}
        = \begin{bmatrix} \mat{Q_1}, \mat{Q_2} \end{bmatrix}
          \begin{bmatrix} \mat{R_1} \\ \mat{0} \end{bmatrix}
        = \mat{Q_1} \mat{R_1}

where \mat{R_1} is an n_c \times n_c upper triangular matrix, \mat{Q_1} is n_r \times n_c, \mat{Q_2} is n_r \times (n_r - n_c), and \mat{Q_1} and \mat{Q_2} both have orthogonal columns.

Parameters:
full_qrbool, optional

A flag telling whether Q, R or Q1, R1 are returned. Default is False and returns Q1, R1.

keep_intactbool, 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.

Returns:
Q1Matrix

The orthogonal matrix of the economic QR factorization.

R1TriangularMatrix

The right (upper) triangular matrix of the economic QR factorization.

QMatrix

The orthogonal matrix of the full QR factorization.

RTriangularMatrix

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 \mat{M} has n_r \geq n_c) in the least-square sense because it implies solving a (simple) triangular system:

\vect{\hat{x}} = \arg\min\limits_{\vect{x} \in \Rset^{n_r}} \|\mat{M} \vect{x} - \vect{b}\|
               = \mat{R_1}^{-1} (\Tr{\mat{Q_1}} \vect{b})

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 \mat{M} with size n_r > n_c reads:

\mat{M} = \mat{U} \mat{\Sigma} \Tr{\mat{V}}

where \mat{U} is an n_r \times n_r orthogonal matrix, \mat{\Sigma} is an n_r \times n_c diagonal matrix and \mat{V} is an n_c \times n_c orthogonal matrix.

Parameters:
fullSVDbool, optional

Whether the null parts of the orthogonal factors are explicitly stored or not. Default is False and computes a reduced SVD.

keep_intactbool, optional

A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.

Returns:
singular_valuesPoint

The vector of singular values with size n = \min(n_r, n_c) that form the diagonal of the n_r \times n_c matrix \mat{\Sigma} of the SVD.

USquareMatrix

The left orthogonal matrix of the SVD.

VTSquareMatrix

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:
fullSVDbool, optional

Whether the null parts of the orthogonal factors are explicitly stored or not. Default is False and computes a reduced SVD.

keep_intactbool, optional

A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.

Returns:
singular_valuesPoint

The vector of singular values with size n = \min(n_r, n_c) that form the diagonal of the n_r \times n_c matrix \mat{\Sigma} of the SVD decomposition.

See also

computeSVD

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:
tracefloat

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_namestr

The object class name (object.__class__.__name__).

getDimension()

Accessor to the dimension (the number of rows).

Returns:
dimensionint
getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getImplementation()

Accessor to the underlying implementation.

Returns:
implImplementation

A copy of the underlying implementation object.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getNbColumns()

Accessor to the number of columns.

Returns:
n_columnsint
getNbRows()

Accessor to the number of rows.

Returns:
n_rowsint
isDiagonal()

Test whether the matrix is diagonal or not.

Returns:
testbool

Answer.

isEmpty()

Tell if the matrix is empty.

Returns:
is_emptybool

True if the matrix contains no element.

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[]])
>>> M.isEmpty()
True
reshape(newRowDim, newColDim)

Reshape the matrix.

Parameters:
newRowDimint

The row dimension of the reshaped matrix.

newColDimint

The column dimension of the reshaped matrix.

Returns:
MTMatrix

The reshaped matrix.

Notes

If the size of the reshaped matrix is smaller than the size of the matrix to be reshaped, only the newRowDim\times newColDim first elements are kept (in a column-major storage sense). If the size is greater, the new elements are set to zero.

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.reshape(1, 6))
1x6
[[ 1 3 5 2 4 6 ]]
>>> print(M.reshape(2, 2))
[[ 1 5 ]
 [ 3 2 ]]
>>> print(M.reshape(2, 6))
2x6
[[ 1 5 4 0 0 0 ]
 [ 3 2 6 0 0 0 ]]
reshapeInPlace(newRowDim, newColDim)

Reshape the matrix, in place.

Parameters:
newRowDimint

The row dimension of the reshaped matrix.

newColDimint

The column dimension of the reshaped matrix.

Notes

If the size of the reshaped matrix is smaller than the size of the matrix to be reshaped, only the newRowDim\times newColDim first elements are kept (in a column-major storage sense). If the size is greater, the new elements are set to zero. If the size is unchanged, no copy of data is done.

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 ]]
>>> M.reshapeInPlace(1, 6)
>>> print(M)
1x6
[[ 1 3 5 2 4 6 ]]
>>> M.reshapeInPlace(2, 2)
>>> print(M)
[[ 1 5 ]
 [ 3 2 ]]
>>> M.reshapeInPlace(2, 6)
>>> print(M)
2x6
[[ 1 5 0 0 0 0 ]
 [ 3 2 0 0 0 0 ]]
setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

solveLinearSystem(*args)

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

Parameters:
rhssequence of float or Matrix with n_r values or rows, respectively

The right hand side member of the linear system.

keep_intactbool, optional

A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.

Returns:
solutionPoint or Matrix

The solution of the square linear system.

Notes

This will handle both matrices and vectors. Note that you’d better type explicitly 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)
transpose()

Transpose the matrix.

Returns:
MTSquareMatrix

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 ]]

Examples using the class

Estimate Wilks and empirical quantile

Estimate Wilks and empirical quantile

Estimate Wilks and empirical quantile
Test independence

Test independence

Test independence
Fit a parametric copula

Fit a parametric copula

Fit a parametric copula
Fit a non parametric copula

Fit a non parametric copula

Fit a non parametric copula
Estimate a multivariate ARMA process

Estimate a multivariate ARMA process

Estimate a multivariate ARMA process
Estimate a non stationary covariance function

Estimate a non stationary covariance function

Estimate a non stationary covariance function
Estimate a stationary covariance function

Estimate a stationary covariance function

Estimate a stationary covariance function
Visualize pairs

Visualize pairs

Visualize pairs
Visualize clouds

Visualize clouds

Visualize clouds
Create and draw multivariate distributions

Create and draw multivariate distributions

Create and draw multivariate distributions
Quick start guide

Quick start guide

Quick start guide
Draw minimum volume level sets

Draw minimum volume level sets

Draw minimum volume level sets
Create a copula

Create a copula

Create a copula
Extract the copula from a distribution

Extract the copula from a distribution

Extract the copula from a distribution
Assemble copulas

Assemble copulas

Assemble copulas
Create a functional basis process

Create a functional basis process

Create a functional basis process
Create a parametric spectral density function

Create a parametric spectral density function

Create a parametric spectral density function
Export a field to VTK

Export a field to VTK

Export a field to VTK
Create a stationary covariance model

Create a stationary covariance model

Create a stationary covariance model
Create a custom covariance model

Create a custom covariance model

Create a custom covariance model
Create a discrete Markov chain process

Create a discrete Markov chain process

Create a discrete Markov chain process
Create a stationary covariance model

Create a stationary covariance model

Create a stationary covariance model
Create a normal process

Create a normal process

Create a normal process
Draw a field

Draw a field

Draw a field
Sample trajectories from a Gaussian Process with correlated outputs

Sample trajectories from a Gaussian Process with correlated outputs

Sample trajectories from a Gaussian Process with correlated outputs
Draw fields

Draw fields

Draw fields
Mixture of experts

Mixture of experts

Mixture of experts
Kriging :configure the optimization solver

Kriging :configure the optimization solver

Kriging :configure the optimization solver
Metamodel of a field function

Metamodel of a field function

Metamodel of a field function
Analyse the central tendency of a cantilever beam

Analyse the central tendency of a cantilever beam

Analyse the central tendency of a cantilever beam
Estimate moments from Taylor expansions

Estimate moments from Taylor expansions

Estimate moments from Taylor expansions
Use the post-analytical importance sampling algorithm

Use the post-analytical importance sampling algorithm

Use the post-analytical importance sampling algorithm
Subset Sampling

Subset Sampling

Subset Sampling
Time variant system reliability problem

Time variant system reliability problem

Time variant system reliability problem
Create unions or intersections of events

Create unions or intersections of events

Create unions or intersections of events
Create an event based on a process

Create an event based on a process

Create an event based on a process
Estimate Sobol indices on a field to point function

Estimate Sobol indices on a field to point function

Estimate Sobol indices on a field to point function
Parallel coordinates graph as sensitivity tool

Parallel coordinates graph as sensitivity tool

Parallel coordinates graph as sensitivity tool
Use the ANCOVA indices

Use the ANCOVA indices

Use the ANCOVA indices
Create mixed deterministic and probabilistic designs of experiments

Create mixed deterministic and probabilistic designs of experiments

Create mixed deterministic and probabilistic designs of experiments
Calibration without observed inputs

Calibration without observed inputs

Calibration without observed inputs
Calibration of the deflection of a tube

Calibration of the deflection of a tube

Calibration of the deflection of a tube
Calibration of the Chaboche mechanical model

Calibration of the Chaboche mechanical model

Calibration of the Chaboche mechanical model
Calibration of the flooding model

Calibration of the flooding model

Calibration of the flooding model
Bayesian calibration of a computer code

Bayesian calibration of a computer code

Bayesian calibration of a computer code
Bayesian calibration of the flooding model

Bayesian calibration of the flooding model

Bayesian calibration of the flooding model
Linear Regression with interval-censored observations

Linear Regression with interval-censored observations

Linear Regression with interval-censored observations
Iterated Functions System

Iterated Functions System

Iterated Functions System
Control algorithm termination

Control algorithm termination

Control algorithm termination
A quick start guide to graphs

A quick start guide to graphs

A quick start guide to graphs