Matrix¶

class
Matrix
(*args)¶ Real rectangular matrix.
Parameters: n_rows : int, , optional
Number of rows. Default is 1.
n_columns : int, , optional
Number of columns. Default is 1.
values : sequence of float with size , optional
Values. OpenTURNS uses columnmajor 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.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 2darray (or matrix, or 2dlist)...
>>> 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 ]]
Methods
clean
(threshold)Set elements smaller than a threshold to zero. computeGram
([transpose])Compute the associated Gram matrix. computeQR
([fullQR, keepIntact])Compute the QR factorization. computeSVD
([fullSVD, keepIntact])Compute the singular values decomposition (SVD). computeSingularValues
([keepIntact])Compute the singular values. getClassName
()Accessor to the object’s name. 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. isEmpty
()Tell if the matrix is empty. setName
(name)Accessor to the object’s name. solveLinearSystem
(*args)Solve a rectangular linear system whose the present matrix is the operator. transpose
()Transpose the matrix. 
__init__
(*args)¶

clean
(threshold)¶ Set elements smaller than a threshold to zero.
Parameters: threshold : float
Threshold for zeroing elements.
Returns: cleaned_matrix :
Matrix
Input matrix with elements smaller than the threshold set to zero.

computeGram
(transpose=True)¶ Compute the associated Gram matrix.
Parameters: transposed : bool
Tells if matrix is to be transposed or not. Default value is True
Returns: 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 ]]

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.
Returns: 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 leastsquare 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.
Returns: 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.
Returns: singular_values :
Point
The vector of singular values with size that form the diagonal of the matrix of the SVD decomposition.
See also
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]

getClassName
()¶ Accessor to the object’s name.
Returns: class_name : str
The object class name (object.__class__.__name__).

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

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

setName
(name)¶ Accessor to the object’s name.
Parameters: name : str
The name of the object.

solveLinearSystem
(*args)¶ Solve a rectangular linear system whose the present matrix is the operator.
Parameters: rhs :
Point
orMatrix
with values or rows, respectivelyThe 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.
Returns: 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 explicitely 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 MatrixImplementationDefaultSmallPivot 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)
