decompose_matrixT_decompose_matrixDecomposeMatrixDecomposeMatrixdecompose_matrix (Operator)

Name

decompose_matrixT_decompose_matrixDecomposeMatrixDecomposeMatrixdecompose_matrix — Decompose a matrix.

Signature

decompose_matrix( : : MatrixID, MatrixType : Matrix1ID, Matrix2ID)

Herror T_decompose_matrix(const Htuple MatrixID, const Htuple MatrixType, Htuple* Matrix1ID, Htuple* Matrix2ID)

void DecomposeMatrix(const HTuple& MatrixID, const HTuple& MatrixType, HTuple* Matrix1ID, HTuple* Matrix2ID)

HMatrix HMatrix::DecomposeMatrix(const HString& MatrixType, HMatrix* Matrix2ID) const

HMatrix HMatrix::DecomposeMatrix(const char* MatrixType, HMatrix* Matrix2ID) const

HMatrix HMatrix::DecomposeMatrix(const wchar_t* MatrixType, HMatrix* Matrix2ID) const   (Windows only)

static void HOperatorSet.DecomposeMatrix(HTuple matrixID, HTuple matrixType, out HTuple matrix1ID, out HTuple matrix2ID)

HMatrix HMatrix.DecomposeMatrix(string matrixType, out HMatrix matrix2ID)

def decompose_matrix(matrix_id: HHandle, matrix_type: str) -> Tuple[HHandle, HHandle]

Description

The operator decompose_matrixdecompose_matrixDecomposeMatrixDecomposeMatrixDecomposeMatrixdecompose_matrix decomposes the square input Matrix given by the matrix handle MatrixIDMatrixIDMatrixIDMatrixIDmatrixIDmatrix_id. The results are stored in two generated matrices Matrix1 and Matrix2. The operator returns the matrix handles Matrix1IDMatrix1IDMatrix1IDMatrix1IDmatrix1IDmatrix_1id and Matrix2IDMatrix2IDMatrix2IDMatrix2IDmatrix2IDmatrix_2id. Access to the elements of the matrices is possible e.g., with the operator get_full_matrixget_full_matrixGetFullMatrixGetFullMatrixGetFullMatrixget_full_matrix.

The type of the input Matrix can be selected via the parameter MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type. The following values are supported: 'general'"general""general""general""general""general" for general, 'symmetric'"symmetric""symmetric""symmetric""symmetric""symmetric" for symmetric, 'positive_definite'"positive_definite""positive_definite""positive_definite""positive_definite""positive_definite" for symmetric positive definite, and 'tridiagonal'"tridiagonal""tridiagonal""tridiagonal""tridiagonal""tridiagonal" for tridiagonal matrices.

The decomposition MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type = 'general'"general""general""general""general""general" or 'tridiagonal'"tridiagonal""tridiagonal""tridiagonal""tridiagonal""tridiagonal" is a LU factorization (Lower/Upper) with the form \texttt{Matrix} = \texttt{Matrix1} * \texttt{Matrix2} The output Matrix1 is a lower triangular matrix with unit diagonal elements and interchanged rows. The output Matrix2 is an upper triangular matrix.

Example for a factorization of a general matrix:

Example for a factorization of a tridiagonal matrix:

For MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type = 'symmetric'"symmetric""symmetric""symmetric""symmetric""symmetric" the factorization is a UDU^T decomposition (Upper/Diagonal/Upper) with the form where the output Matrix1 is an upper triangular matrix with interchanged columns. The output matrix Matrix2 is a symmetric block diagonal matrix with 1 x 1 and 2 x 2 diagonal blocks.

Example for a factorization of a symmetric matrix:

For MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type = 'positive_definite'"positive_definite""positive_definite""positive_definite""positive_definite""positive_definite" a Cholesky factorization is computed with the form \texttt{Matrix} = \texttt{Matrix1} * \texttt{Matrix2} where the output Matrix1 is a lower triangular matrix and the output matrix Matrix2 is an upper triangular matrix. Furthermore, the Matrix2 is the transpose of the matrix Matrix1.

Example for a factorization of a positive definite matrix:

It should be noted that in the examples there are differences in the meaning of the values of the output matrices: If a value is shown as an integer number, e.g., 0 or 1, the value of this element is per definition this certain value. If the number is shown as a floating point number, e.g., 0.0 or 1.0, the value is computed by the operator.

Attention

For MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type = 'symmetric'"symmetric""symmetric""symmetric""symmetric""symmetric" or 'positive_definite'"positive_definite""positive_definite""positive_definite""positive_definite""positive_definite", the upper triangular part of the input Matrix must contain the relevant information of the matrix. The strictly lower triangular part of the matrix is not referenced. For MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type = 'tridiagonal'"tridiagonal""tridiagonal""tridiagonal""tridiagonal""tridiagonal", only the main diagonal, the superdiagonal, and the subdiagonal of the input Matrix are used. The other parts of the matrix are not referenced. If the referenced part of the input Matrix is not of the specified type, an exception is raised.

Execution Information

Parameters

MatrixIDMatrixIDMatrixIDMatrixIDmatrixIDmatrix_id (input_control)  matrix HMatrix, HTupleHHandleHTupleHtuple (handle) (IntPtr) (HHandle) (handle)

Matrix handle of the input matrix.

MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type (input_control)  string HTuplestrHTupleHtuple (string) (string) (HString) (char*)

Type of the input matrix.

Default value: 'general' "general" "general" "general" "general" "general"

List of values: 'general'"general""general""general""general""general", 'positive_definite'"positive_definite""positive_definite""positive_definite""positive_definite""positive_definite", 'symmetric'"symmetric""symmetric""symmetric""symmetric""symmetric", 'tridiagonal'"tridiagonal""tridiagonal""tridiagonal""tridiagonal""tridiagonal"

Matrix1IDMatrix1IDMatrix1IDMatrix1IDmatrix1IDmatrix_1id (output_control)  matrix HMatrix, HTupleHHandleHTupleHtuple (handle) (IntPtr) (HHandle) (handle)

Matrix handle with the output matrix 1.

Matrix2IDMatrix2IDMatrix2IDMatrix2IDmatrix2IDmatrix_2id (output_control)  matrix HMatrix, HTupleHHandleHTupleHtuple (handle) (IntPtr) (HHandle) (handle)

Matrix handle with the output matrix 2.

Result

If the parameters are valid, the operator decompose_matrixdecompose_matrixDecomposeMatrixDecomposeMatrixDecomposeMatrixdecompose_matrix returns the value TRUE. If necessary, an exception is raised.

Possible Predecessors

create_matrixcreate_matrixCreateMatrixCreateMatrixCreateMatrixcreate_matrix

Possible Successors

get_full_matrixget_full_matrixGetFullMatrixGetFullMatrixGetFullMatrixget_full_matrix, get_value_matrixget_value_matrixGetValueMatrixGetValueMatrixGetValueMatrixget_value_matrix

Alternatives

orthogonal_decompose_matrixorthogonal_decompose_matrixOrthogonalDecomposeMatrixOrthogonalDecomposeMatrixOrthogonalDecomposeMatrixorthogonal_decompose_matrix, solve_matrixsolve_matrixSolveMatrixSolveMatrixSolveMatrixsolve_matrix

References

David Poole: “Linear Algebra: A Modern Introduction”; Thomson; Belmont; 2006.
Gene H. Golub, Charles F. van Loan: “Matrix Computations”; The Johns Hopkins University Press; Baltimore and London; 1996.

Module

Foundation