Matrix¶

class
Matrix
(*args)¶ Real rectangular matrix.
 Parameters
 n_rowsint, , optional
Number of rows. Default is 1.
 n_columnsint, , optional
Number of columns. Default is 1.
 valuessequence of float with size , optional
Values. columnmajor ordering is used (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
(self, threshold)Set elements smaller than a threshold to zero.
computeGram
(self[, transpose])Compute the associated Gram matrix.
computeQR
(self[, fullQR, keepIntact])Compute the QR factorization.
computeSVD
(self[, fullSVD, keepIntact])Compute the singular values decomposition (SVD).
computeSingularValues
(self[, keepIntact])Compute the singular values.
getClassName
(self)Accessor to the object’s name.
getId
(self)Accessor to the object’s id.
getImplementation
(self, \*args)Accessor to the underlying implementation.
getName
(self)Accessor to the object’s name.
getNbColumns
(self)Accessor to the number of columns.
getNbRows
(self)Accessor to the number of rows.
isEmpty
(self)Tell if the matrix is empty.
reshape
(self, newRowDim, newColDim)Reshape the matrix.
reshapeInPlace
(self, newRowDim, newColDim)Reshape the matrix, in place.
setName
(self, name)Accessor to the object’s name.
solveLinearSystem
(self, \*args)Solve a rectangular linear system whose the present matrix is the operator.
transpose
(self)Transpose the matrix.

__init__
(self, *args)¶ Initialize self. See help(type(self)) for accurate signature.

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

computeGram
(self, transpose=True)¶ Compute the associated Gram matrix.
 Parameters
 transposedbool
Tells if matrix is to be transposed or not. Default value is True
 Returns
 MMT
Matrix
The Gram matrix.
 MMT
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
(self, 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_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
 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.
 Q1
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
(self, 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
 fullSVDbool, optional
Whether the null parts of the orthogonal factors are explicitely 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_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.
 singular_values
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
(self, keepIntact=True)¶ Compute the singular values.
 Parameters
 fullSVDbool, optional
Whether the null parts of the orthogonal factors are explicitely 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_values
Point
The vector of singular values with size that form the diagonal of the matrix of the SVD decomposition.
 singular_values
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
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getId
(self)¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getImplementation
(self, *args)¶ Accessor to the underlying implementation.
 Returns
 implImplementation
The implementation class.

getName
(self)¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getNbColumns
(self)¶ Accessor to the number of columns.
 Returns
 n_columnsint

getNbRows
(self)¶ Accessor to the number of rows.
 Returns
 n_rowsint

isEmpty
(self)¶ 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
(self, newRowDim, newColDim)¶ Reshape the matrix.
 Parameters
 newRowDimint
The row dimension of the reshaped matrix.
 newColDimint
The column dimension of the reshaped matrix.
 Returns
 MT
Matrix
The reshaped matrix.
 MT
Notes
If the size of the reshaped matrix is smaller than the size of the matrix to be reshaped, only the first elements are kept (in a columnmajor 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
(self, 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 first elements are kept (in a columnmajor 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
(self, name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

solveLinearSystem
(self, *args)¶ Solve a rectangular linear system whose the present matrix is the operator.
 Parameters
 Returns
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)