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.

Methods

`clean`(threshold)	Set elements smaller than a threshold to zero.
`computeDeterminant`()	Compute the determinant.
`computeDeterminantInPlace`()	Compute the determinant in place.
`computeEV`()	Compute the eigenvalues decomposition (EVD).
`computeEVInPlace`()	Compute the eigenvalues decomposition (EVD) in place.
`computeEigenValues`()	Compute eigenvalues.
`computeEigenValuesInPlace`()	Compute eigenvalues in place.
`computeGram`([transpose])	Compute the associated Gram matrix.
`computeHadamardProduct`(other)	Compute the Hadamard product matrix.
`computeLargestEigenValueModule`(*args)	Compute the largest eigenvalue module.
`computeLogAbsoluteDeterminant`()	Compute the logarithm of the absolute value of the determinant.
`computeLogAbsoluteDeterminantInPlace`()	Compute the determinant in place.
`computeQR`([fullQR])	Compute the QR factorization.
`computeQRInPlace`([fullQR])	Compute the QR factorization in place.
`computeSVD`([fullSVD])	Compute the singular values decomposition (SVD).
`computeSVDInPlace`([fullSVD])	Compute the singular values decomposition (SVD).
`computeSingularValues`()	Compute the singular values.
`computeSingularValuesInPlace`()	Compute the singular values in place.
`computeSumElements`()	Compute the sum of the matrix elements.
`computeTrace`()	Compute the trace of the matrix.
`frobeniusNorm`()	Frobenius norm accessor.
`getClassName`()	Accessor to the object's name.
`getDiagonal`([k])	Get the k-th diagonal of the matrix.
`getDiagonalAsPoint`([k])	Get the k-th diagonal of the matrix.
`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.
`inverse`()	Compute the inverse of the matrix.
`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.
`setDiagonal`(*args)	Set the k-th diagonal of the matrix.
`setName`(name)	Accessor to the object's name.
`solveLinearSystem`(*args)	Solve a square linear system whose the present matrix is the operator.
`solveLinearSystemInPlace`(*args)	Solve a rectangular linear system whose the present matrix is the operator.
`squareElements`()	Square the Matrix, ie each element of the matrix is squared.
`transpose`()	Transpose the 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 a 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 ]]

__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()¶

Compute the determinant.

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

computeDeterminantInPlace()¶

Compute the determinant in place.

Similar to computeDeterminant() but modifies the matrix in place to avoid copy.

computeEV()¶

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.

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)

computeEVInPlace()¶

Compute the eigenvalues decomposition (EVD) in place.

Similar to computeEVInPlace() but the matrix is modified in place to avoid copy.

computeEigenValues()¶

Compute eigenvalues.

Returns:

eigenvaluesComplexCollection: Eigenvalues.

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())
[9.52552,0.514301]

computeSingularValuesInPlace()¶

Compute the singular values in place.

Similar to computeSingularValues() but the matrix is modified in place to avoid copy.

computeSumElements()¶

Compute the sum of the matrix elements.

Returns:

suma float: The sum of the elements.

Notes

Compute the sum of elements of the matrix $\mat{A} \in \Rset^{m \times n}$ :

$s = \sum_{i=1}^{m} \sum_{j=1}^{n} a_{i,j}$

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> s = M.computeSumElements()
>>> print(s)
21.0

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

frobeniusNorm()¶

Frobenius norm accessor.

Returns:

normfloat: The Frobenius norm $||\mat{A}||_F = \sqrt{\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2}$ .

getClassName()¶

Accessor to the object’s name.

Returns:

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

getDiagonal(k=0)¶

Get the k-th diagonal of the matrix.

Parameters:

kint: The k-th diagonal to extract Default value is 0

Returns:

D: Matrix: The k-th diagonal.

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])
>>> diag = M.getDiagonal()
>>> print(diag)
[[ 1 ]
 [ 5 ]
 [ 9 ]]
>>> print(M.getDiagonal(1))
[[ 2 ]
 [ 6 ]]

getDiagonalAsPoint(k=0)¶

Get the k-th diagonal of the matrix.

Parameters:

kint: The k-th diagonal to extract Default value is 0

Returns:

ptPoint: The k-th digonal.

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])
>>> pt = M.getDiagonalAsPoint()
>>> print(pt)
[1,5,9]

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

inverse()¶

Compute the inverse of the matrix.

Returns:

inverseMatrixSquareMatrix: The inverse of the matrix.

Examples

>>> import openturns as ot
>>> M = ot.SquareMatrix([[1.0, 2.0, 3.0], [3.0, 2.0, 1.0], [2.0, 1.0, 3.0]])
>>> print(12.0 * M.inverse())
[[ -5  3  4 ]
 [  7  3 -8 ]
 [  1 -3  4 ]]

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 ]]

setDiagonal(*args)¶

Set the k-th diagonal of the matrix.

Parameters:

diagPoint or Matrix or a float: Value(s) used to fill the diagonal of the matrix
kint: The k-th diagonal to fill Default value is 0

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])
>>> M.setDiagonal([-1, 23, 9])
>>> print(M)
[[ -1  2  3 ]
 [  4 23  6 ]
 [  7  8  9 ]]
>>> M.setDiagonal(1.0)
>>> print(M)
[[ 1 2 3 ]
 [ 4 1 6 ]
 [ 7 8 1 ]]
>>> M.setDiagonal([2, 6, 9])
>>> print(M)
[[ 2 2 3 ]
 [ 4 6 6 ]
 [ 7 8 9 ]]

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.

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 = [1.0] * 2
>>> x = M.solveLinearSystem(b)
>>> np.testing.assert_array_almost_equal(M * x, b)

solveLinearSystemInPlace(*args)¶

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

Similar to solveLinearSystem() except the matrix is modified in-place during the resolution avoiding the need to allocate an extra copy if the original copy is not re-used.

squareElements()¶

Square the Matrix, ie each element of the matrix is squared.

Examples

>>> import openturns as ot
>>> M = ot.Matrix([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
>>> M.squareElements()
>>> print(M)
[[  1  4 ]
 [  9 16 ]
 [ 25 36 ]]

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 ]]