TriangularMatrix¶

class TriangularMatrix(*args)¶

Triangular Matrix.

Parameters:

dimint: The dimension of the triangular matrix (square matrix with dim rows and dim columns).
isLowerbool, optional: Flag telling if the matrix is triangular lower (True) or upper (False). Default is True.

Methods

`checkTriangularity`()	Check if the internal representation is really triangular.
`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.
`isLowerTriangular`()	Test whether the matrix is lower triangular or upper triangular.
`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.

See also

Matrix

Notes

The triangular matrix is filled with 0.

__init__(*args)¶

checkTriangularity()¶: Check if the internal representation is really triangular.

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

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)]

computeEigenValuesInPlace()¶

Compute eigenvalues in place.

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

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 True, compute $\Tr{M} M$ . Otherwise, compute $M \Tr{M}$ .

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 ]]

computeHadamardProduct(other)¶

Compute the Hadamard product matrix.

Parameters:

MatMatrix: The right hand matrix

Returns:

CMatrix: The Hadamard product matrix.

Notes

The matrix $\mat{C} \in \Rset^{m \times n}$ resulting from the Hadamard product ( also known as the elementwise product) of the matrices $\mat{A} \in \Rset^{m \times n}$ and $\mat{B} \in \Rset^{m \times n}$ is:

$c_{i,j} = a_{i,j} b_{i,j}$

for any $i = 1, \cdots, m$ and $j = 1, \cdots, n$ .

Examples

>>> import openturns as ot
>>> A = ot.Matrix([[1.0, 2.0], [3.0, 4.0]])
>>> B = ot.Matrix([[1.0, 2.0], [3.0, 4.0]])
>>> C = A.computeHadamardProduct(B)
>>> print(C)
[[  1  4 ]
 [  9 16 ]]
>>> print(B.computeHadamardProduct(A))
[[  1  4 ]
 [  9 16 ]]

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.

See also

computeEV, computeEigenValues

Examples

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

computeLogAbsoluteDeterminant()¶

Compute the logarithm of the absolute value of the determinant.

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]

computeLogAbsoluteDeterminantInPlace()¶

Compute the determinant in place.

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

computeQR(fullQR=False)¶

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.

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)

computeQRInPlace(fullQR=False)¶

Compute the QR factorization in place.

Similar to computeQR()

computeSVD(fullSVD=False)¶

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.

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)

computeSVDInPlace(fullSVD=False)¶

Compute the singular values decomposition (SVD).

Unlike computeSVD, this modifies the matrix in place and avoids a copy.

computeSingularValues()¶

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.

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())
[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

isLowerTriangular()¶

Test whether the matrix is lower triangular or upper triangular.

Returns:

isLowerbool: Flag telling if the matrix is triangular lower (True) or upper (False).

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 ]]

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