TriangularMatrix¶

class
TriangularMatrix
(*args)¶ Hermitian Matrix.
 Available constructors:
TriangularMatrix(dim)
TriangularMatrix(dim, isLower)
 Parameters
 diminteger
The dimension of the triangular matrix (square matrix with dim rows and dim columns).
 isLowerbool
Flag telling if the matrix is triangular lower (True) or upper (False). Default is True.
See also
Notes
The triangular matrix is filled with 0.
Methods
clean
(self, threshold)Set elements smaller than a threshold to zero.
computeDeterminant
(self[, keepIntact])Compute the determinant.
computeEV
(self[, keepIntact])Compute the eigenvalues decomposition (EVD).
computeEigenValues
(self[, keepIntact])Compute eigenvalues.
computeGram
(self[, transpose])Compute the associated Gram matrix.
computeLogAbsoluteDeterminant
(self[, keepIntact])Compute the logarithm of the absolute value of the determinant.
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.
computeTrace
(self)Compute the trace of the matrix.
getClassName
(self)Accessor to the object’s name.
getDimension
(self)Accessor to the dimension (the number of rows).
getId
(self)Accessor to the object’s id.
getImplementation
(self)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.
isDiagonal
(self)Test whether the matrix is diagonal or not.
isEmpty
(self)Tell if the matrix is empty.
isLowerTriangular
(self)Test whether the matrix is lower triangular or upper triangular.
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 square linear system whose the present matrix is the operator.
transpose
(self)Transpose the matrix.
checkTriangularity

__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

computeDeterminant
(self, keepIntact=True)¶ Compute the determinant.
 Parameters
 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
 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

computeEV
(self, keepIntact=True)¶ Compute the eigenvalues decomposition (EVD).
The eigenvalues decomposition of a square matrix with size reads:
where is an diagonal matrix and is an orthogonal matrix.
 Parameters
 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
 eigen_values
ComplexCollection
The vector of eigenvalues with size that form the diagonal of the matrix of the EVD.
 Phi
SquareComplexMatrix
The left matrix of the EVD.
 eigen_values
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)

computeEigenValues
(self, keepIntact=True)¶ Compute eigenvalues.
 Parameters
 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
 eigenvalues
ComplexCollection
Eigenvalues.
 eigenvalues
See also
Examples
>>> import openturns as ot >>> M = ot.SquareMatrix([[1.0, 2.0], [3.0, 4.0]]) >>> M.computeEigenValues() [(0.372281,0),(5.37228,0)]

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 ]]

computeLogAbsoluteDeterminant
(self, keepIntact=True)¶ Compute the logarithm of the absolute value of the determinant.
 Parameters
 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
 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]

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]

computeTrace
(self)¶ 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

getClassName
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getDimension
(self)¶ Accessor to the dimension (the number of rows).
 Returns
 dimensionint

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

getImplementation
(self)¶ 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

isDiagonal
(self)¶ Test whether the matrix is diagonal or not.
 Returns
 testbool
Answer.

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

isLowerTriangular
(self)¶ 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
(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 square linear system whose the present matrix is the operator.
 Parameters
 rhssequence of float or
Matrix
with values or rows, respectively The right hand side member of the linear system.
 keep_intactbool, optional
A flag telling whether the present matrix can be overwritten or not. Default is True and leaves the present matrix unchanged.
 rhssequence of float or
 Returns
Notes
This will handle both matrices and vectors. Note that you’d better type explicitely 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 = ot.Point([1.0] * 2) >>> x = M.solveLinearSystem(b) >>> np.testing.assert_array_almost_equal(M * x, b)

transpose
(self)¶ Transpose the matrix.
 Returns
 MT
SquareMatrix
The transposed matrix.
 MT
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 ]]