Matrix¶

class Matrix(*args)¶

Real rectangular matrix.

Parameters:

n_rowsint, $n_r > 0$ , optional: Number of rows. Default is 1.
n_columnsint, $n_c > 0$ , optional: Number of columns. Default is 1.
valuessequence of float with size $n_r \times n_c$ , optional: Values. column-major ordering is used (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.
`computeGram`([transpose])	Compute the associated Gram matrix.
`computeHadamardProduct`(other)	Compute the Hadamard product matrix.
`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.
`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.
`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.
`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 rectangular 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.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 ]]

__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.

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 ]]

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

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]

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

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 rectangular linear system whose the present matrix is the operator.

Parameters:

rhsPoint 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 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 explicitly 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)

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:

MTMatrix: 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 ]]