orthogonal_decompose_matrixOrthogonalDecomposeMatrixOrthogonalDecomposeMatrixorthogonal_decompose_matrixT_orthogonal_decompose_matrix
Short description
orthogonal_decompose_matrixOrthogonalDecomposeMatrixOrthogonalDecomposeMatrixorthogonal_decompose_matrixT_orthogonal_decompose_matrix — Perform an orthogonal decomposition of a matrix.
Signature
orthogonal_decompose_matrix( matrix MatrixID, string DecompositionType, string OutputMatricesType, string ComputeOrthogonal, out matrix MatrixOrthogonalID, out matrix MatrixTriangularID )void OrthogonalDecomposeMatrix( const HTuple& MatrixID, const HTuple& DecompositionType, const HTuple& OutputMatricesType, const HTuple& ComputeOrthogonal, HTuple* MatrixOrthogonalID, HTuple* MatrixTriangularID )static void HOperatorSet.OrthogonalDecomposeMatrix( HTuple matrixID, HTuple decompositionType, HTuple outputMatricesType, HTuple computeOrthogonal, out HTuple matrixOrthogonalID, out HTuple matrixTriangularID )def orthogonal_decompose_matrix( matrix_id: HHandle, decomposition_type: str, output_matrices_type: str, compute_orthogonal: str ) -> Tuple[HHandle, HHandle]
Herror T_orthogonal_decompose_matrix( const Htuple MatrixID, const Htuple DecompositionType, const Htuple OutputMatricesType, const Htuple ComputeOrthogonal, Htuple* MatrixOrthogonalID, Htuple* MatrixTriangularID )
HMatrix HMatrix::OrthogonalDecomposeMatrix( const HString& DecompositionType, const HString& OutputMatricesType, const HString& ComputeOrthogonal, HMatrix* MatrixTriangularID ) const
HMatrix HMatrix::OrthogonalDecomposeMatrix( const char* DecompositionType, const char* OutputMatricesType, const char* ComputeOrthogonal, HMatrix* MatrixTriangularID ) const
HMatrix HMatrix::OrthogonalDecomposeMatrix( const wchar_t* DecompositionType, const wchar_t* OutputMatricesType, const wchar_t* ComputeOrthogonal, HMatrix* MatrixTriangularID ) const (Windows only)
HMatrix HMatrix.OrthogonalDecomposeMatrix( string decompositionType, string outputMatricesType, string computeOrthogonal, out HMatrix matrixTriangularID )
Description
The operator orthogonal_decompose_matrixOrthogonalDecomposeMatrix decomposes the
Matrix defined by the matrix handle MatrixIDmatrixIDmatrix_id. The
results are stored in the two generated matrices
MatrixOrthogonal and MatrixTriangular. The
operator returns the matrix handles MatrixOrthogonalIDmatrixOrthogonalIDmatrix_orthogonal_id and
MatrixTriangularIDmatrixTriangularIDmatrix_triangular_id of these two matrices. Access
to the elements of the matrices is possible e.g., with the operator
get_full_matrixGetFullMatrix.
For OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'full'"full" all results of
the decomposition are stored in the matrices
MatrixOrthogonal and MatrixTriangular. For
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'reduced'"reduced" only a part of
result elements of the matrices MatrixOrthogonal and
MatrixTriangular are stored. Therefore the sizes of these
matrices are smaller than for OutputMatricesTypeoutputMatricesTypeoutput_matrices_type =
'full'"full".
For the parameter ComputeOrthogonalcomputeOrthogonalcompute_orthogonal = 'true'"true" both
output matrices are computed. For the ComputeOrthogonalcomputeOrthogonalcompute_orthogonal =
'false'"false" only the matrix MatrixTriangular is
computed. Thus, the runtime of the operation takes fewer time.
The type of the Matrix can be selected via the parameter
DecompositionTypedecompositionTypedecomposition_type. For DecompositionTypedecompositionTypedecomposition_type =
'qr'"qr" a QR decomposition (Quadratic/Right) or for
DecompositionTypedecompositionTypedecomposition_type = 'ql'"ql" a QL decomposition
(Quadratic/Left) is computed. The decomposition is
written as
\[\begin{eqnarray*}
\texttt{Matrix} \quad = \quad \texttt{MatrixOrthogonal}
\quad \cdot \quad \texttt{MatrixTriangular}.
\end{eqnarray*}\]
Example:
DecompositionTypedecompositionTypedecomposition_type = 'qr'"qr",
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'full'"full"
\[\begin{eqnarray*}
\texttt{Matrix} =
\left[ \begin{array}{rrr}
6.0 & -3.0 & 4.0 \\
5.0 & 5.0 & -2.0 \\
-3.0 & 12.0 & 5.0 \\
3.0 & 4.0 & 7.0 \\
7.0 & 5.0 & 3.0
\end{array} \right]
\end{eqnarray*}\]
\[\begin{eqnarray*}
\to \qquad
\texttt{MatrixOrthogonal} &=&
\left[ \begin{array}{rrrrr}
-0.5303 & 0.2613 & -0.4277 & -0.5566 & -0.3972 \\
-0.4419 & -0.2920 & -0.6383 & 0.1727 & -0.5312 \\
0.2652 & -0.8443 & -0.2059 & -0.3961 & -0.1326 \\
-0.2652 & -0.2432 & -0.6008 & 0.7000 & -0.1400 \\
-0.6187 & -0.2729 & 0.0799 & -0.1160 & 0.7231
\end{array} \right]\\
\texttt{MatrixTriangular} &=&
\left[ \begin{array}{rrr}
-11.3137 & -1.5910 & -3.6239 \\
0 & -14.7129 & -5.1135 \\
0 & 0 & -7.9824 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array} \right]
\end{eqnarray*}\]
Example:
DecompositionTypedecompositionTypedecomposition_type = 'qr'"qr",
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'reduced'"reduced"
\[\begin{eqnarray*}
\texttt{Matrix} =
\left[ \begin{array}{rrr}
6.0 & -3.0 & 4.0 \\
5.0 & 5.0 & -2.0 \\
-3.0 & 12.0 & 5.0 \\
3.0 & 4.0 & 7.0 \\
7.0 & 5.0 & 3.0
\end{array} \right]
\end{eqnarray*}\]
\[\begin{eqnarray*}
\to \qquad
\texttt{MatrixOrthogonal} &=&
\left[ \begin{array}{rrr}
-0.5303 & 0.2613 & -0.4277 \\
-0.4419 & -0.2920 & -0.6383 \\
0.2652 & -0.8443 & -0.2059 \\
-0.2652 & -0.2432 & -0.6008 \\
-0.6187 & -0.2729 & 0.0799
\end{array} \right]\\
\texttt{MatrixTriangular} &=&
\left[ \begin{array}{rrr}
-11.3137 & -1.5910 & -3.6239 \\
0 & -14.7129 & -5.1135 \\
0 & 0 & -7.9824
\end{array} \right]
\end{eqnarray*}\]
Example:
DecompositionTypedecompositionTypedecomposition_type = 'ql'"ql",
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'full'"full"
\[\begin{eqnarray*}
\texttt{Matrix} =
\left[ \begin{array}{rrr}
6.0 & -3.0 & 4.0 \\
5.0 & 5.0 & -2.0 \\
-3.0 & 12.0 & 5.0 \\
3.0 & 4.0 & 7.0 \\
7.0 & 5.0 & 3.0
\end{array} \right]
\end{eqnarray*}\]
\[\begin{eqnarray*}
\to \qquad
\texttt{MatrixOrthogonal} &=&
\left[ \begin{array}{rrrrr}
0.6806 & 0.0657 & 0.3659 & -0.4932 & -0.3941 \\
0.1161 & 0.5464 & 0.6091 & 0.5274 & 0.1971 \\
0.4093 & -0.0832 & -0.4046 & 0.6474 & -0.4927 \\
-0.5362 & 0.4713 & 0.0072 & -0.1208 & -0.6897 \\
-0.2611 & -0.6842 & 0.5757 & 0.2119 & -0.2956
\end{array} \right]\\
\texttt{MatrixTriangular} &=&
\left[ \begin{array}{rrr}
0 & 0 & 0 \\
0 & 0 & 0 \\
10.5059 & 0 & 0 \\
-1.1429 & 12.4620 & 0 \\
-4.0399 & -7.9812 & -10.1489
\end{array} \right]
\end{eqnarray*}\]
Example:
DecompositionTypedecompositionTypedecomposition_type = 'ql'"ql",
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'reduced'"reduced"
\[\begin{eqnarray*}
\texttt{Matrix} =
\left[ \begin{array}{rrr}
6.0 & -3.0 & 4.0 \\
5.0 & 5.0 & -2.0 \\
-3.0 & 12.0 & 5.0 \\
3.0 & 4.0 & 7.0 \\
7.0 & 5.0 & 3.0
\end{array} \right]
\end{eqnarray*}\]
\[\begin{eqnarray*}
\to \qquad
\texttt{MatrixOrthogonal} &=&
\left[ \begin{array}{rrrrr}
0.3659 & -0.4932 & -0.3941 \\
0.6091 & 0.5274 & 0.1971 \\
-0.4046 & 0.6474 & -0.4927 \\
0.0072 & -0.1208 & -0.6897 \\
0.5757 & 0.2119 & -0.2956
\end{array} \right]\\
\texttt{MatrixTriangular} &=&
\left[ \begin{array}{rrr}
10.5059 & 0 & 0 \\
-1.1429 & 12.4620 & 0 \\
-4.0399 & -7.9812 & -10.1489
\end{array} \right]
\end{eqnarray*}\]
For DecompositionTypedecompositionTypedecomposition_type = 'rq'"rq" a RQ decomposition
(Right/Quadratic) or for DecompositionTypedecompositionTypedecomposition_type = 'lq'"lq"
a LQ decomposition (Left/Quadratic) is computed. The decomposition
is written as
\[\begin{eqnarray*}
\texttt{Matrix} \quad = \quad \texttt{MatrixTriangular}
\quad \cdot \quad \texttt{MatrixOrthogonal}.
\end{eqnarray*}\]
Example:
DecompositionTypedecompositionTypedecomposition_type = 'rq'"rq",
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'full'"full"
\[\begin{eqnarray*}
\texttt{Matrix} =
\left[ \begin{array}{rrrrr}
6.0 & 5.0 & -3.0 & 3.0 & 7.0 \\
-3.0 & 5.0 & 12.0 & 4.0 & 5.0 \\
4.0 & -2.0 & 5.0 & 7.0 & 3.0
\end{array} \right]
\end{eqnarray*}\]
\[\begin{eqnarray*}
\to \qquad
\texttt{MatrixOrthogonal} &=&
\left[ \begin{array}{rrrrr}
0.6806 & 0.1161 & 0.4093 & -0.5362 & -0.2611 \\
0.0657 & 0.5464 & -0.0832 & 0.4713 & -0.6842 \\
0.3659 & 0.6091 & -0.4046 & 0.0072 & 0.5757 \\
-0.4932 & 0.5274 & 0.6474 & -0.1208 & 0.2119 \\
-0.3941 & 0.1971 & -0.4927 & -0.6897 & -0.2956
\end{array} \right]\\
\texttt{MatrixTriangular} &=&
\left[ \begin{array}{rrrrr}
0 \qquad & \qquad 0 \qquad & \qquad 10.5059 & -1.1429 & -4.0399 \\
0 \qquad & \qquad 0 \qquad & \qquad 0 & 12.4620 & -7.9812 \\
0 \qquad & \qquad 0 \qquad & \qquad 0 & 0 & -10.1489
\end{array} \right]
\end{eqnarray*}\]
Example:
DecompositionTypedecompositionTypedecomposition_type = 'rq'"rq",
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'reduced'"reduced"
\[\begin{eqnarray*}
\texttt{Matrix} =
\left[ \begin{array}{rrrrr}
6.0 & 5.0 & -3.0 & 3.0 & 7.0 \\
-3.0 & 5.0 & 12.0 & 4.0 & 5.0 \\
4.0 & -2.0 & 5.0 & 7.0 & 3.0
\end{array} \right]
\end{eqnarray*}\]
\[\begin{eqnarray*}
\to \qquad
\texttt{MatrixOrthogonal} &=&
\left[ \begin{array}{rrrrr}
0.3659 & 0.6091 & -0.4046 & 0.0072 & 0.5757 \\
-0.4932 & 0.5274 & 0.6474 & -0.1208 & 0.2119 \\
-0.3941 & 0.1971 & -0.4927 & -0.6897 & -0.2956
\end{array} \right]\\
\texttt{MatrixTriangular} &=&
\left[ \begin{array}{rrrrr}
10.5059 & -1.1429 & -4.0399 \\
0 & 12.4620 & -7.9812 \\
0 & 0 & -10.1489
\end{array} \right]
\end{eqnarray*}\]
Example:
DecompositionTypedecompositionTypedecomposition_type = 'lq'"lq",
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'full'"full"
\[\begin{eqnarray*}
\texttt{Matrix} =
\left[ \begin{array}{rrrrr}
6.0 & 5.0 & -3.0 & 3.0 & 7.0 \\
-3.0 & 5.0 & 12.0 & 4.0 & 5.0 \\
4.0 & -2.0 & 5.0 & 7.0 & 3.0
\end{array} \right]
\end{eqnarray*}\]
\[\begin{eqnarray*}
\to \qquad
\texttt{MatrixOrthogonal} &=&
\left[ \begin{array}{rrrrr}
-0.5303 & -0.4419 & 0.2652 & -0.2652 & -0.6187 \\
0.2613 & -0.2920 & -0.8443 & -0.2432 & -0.2729 \\
-0.4277 & 0.6383 & -0.2059 & -0.6008 & 0.0799 \\
-0.5566 & 0.1727 & -0.3961 & 0.7000 & -0.1160 \\
-0.3972 & -0.5312 & -0.1326 & -0.1400 & 0.7231
\end{array} \right]\\
\texttt{MatrixTriangular} &=&
\left[ \begin{array}{rrrrr}
-11.3137 & 0 & 0 & \qquad 0 \qquad & \qquad 0 \\
-1.5910 & -14.7129 & 0 & \qquad 0 \qquad & \qquad 0 \\
-3.6239 & -5.1135 & -7.9824 & \qquad 0 \qquad & \qquad 0
\end{array} \right]
\end{eqnarray*}\]
Example:
DecompositionTypedecompositionTypedecomposition_type = 'lq'"lq",
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type = 'reduced'"reduced"
\[\begin{eqnarray*}
\texttt{Matrix} =
\left[ \begin{array}{rrrrr}
6.0 & 5.0 & -3.0 & 3.0 & 7.0 \\
-3.0 & 5.0 & 12.0 & 4.0 & 5.0 \\
4.0 & -2.0 & 5.0 & 7.0 & 3.0
\end{array} \right]
\end{eqnarray*}\]
\[\begin{eqnarray*}
\to \qquad
\texttt{MatrixOrthogonal} &=&
\left[ \begin{array}{rrrrr}
-0.5303 & -0.4419 & 0.2652 & -0.2652 & -0.6187 \\
0.2613 & -0.2920 & -0.8443 & -0.2432 & -0.2729 \\
-0.4277 & 0.6383 & -0.2059 & -0.6008 & 0.0799
\end{array} \right]\\
\texttt{MatrixTriangular} &=&
\left[ \begin{array}{rrr}
-11.3137 & 0 & 0 \\
-1.5910 & -14.7129 & 0 \\
-3.6239 & -5.1135 & -7.9824
\end{array} \right]
\end{eqnarray*}\]
It should be noted that in the examples there are differences in the
meaning of the numbers of the output matrices: The results of the
elements are per definition a certain value if the number of this
value is shown as an integer number, e.g., 0 or 1. If the number is
shown as a floating point number, e.g., 0.0 or 1.0, the value is
computed.
Execution information
-
Multithreading type: reentrant (runs in parallel with non-exclusive operators).
-
Multithreading scope: global (may be called from any thread).
-
Processed without parallelization.
Parameters
MatrixIDmatrixIDmatrix_id (input_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)
Matrix handle of the input matrix.
DecompositionTypedecompositionTypedecomposition_type (input_control) string → (string)HTuple (HString)HTuple (string)strHtuple (char*)
Method of decomposition.
Default: 'qr'"qr"
List of values: 'lq', 'ql', 'qr', 'rq'"lq", "ql", "qr", "rq"
OutputMatricesTypeoutputMatricesTypeoutput_matrices_type (input_control) string → (string)HTuple (HString)HTuple (string)strHtuple (char*)
Type of output matrices.
Default: 'full'"full"
List of values: 'full', 'reduced'"full", "reduced"
ComputeOrthogonalcomputeOrthogonalcompute_orthogonal (input_control) string → (string)HTuple (HString)HTuple (string)strHtuple (char*)
Computation of the orthogonal matrix.
Default: 'true'"true"
List of values: 'false', 'true'"false", "true"
MatrixOrthogonalIDmatrixOrthogonalIDmatrix_orthogonal_id (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)
Matrix handle with the orthogonal part of the
decomposed input matrix.
MatrixTriangularIDmatrixTriangularIDmatrix_triangular_id (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)
Matrix handle with the triangular part of the
decomposed input matrix.
Result
If the parameters are valid, the operator
orthogonal_decompose_matrixOrthogonalDecomposeMatrix returns the value 2 (H_MSG_TRUE). If
necessary, an exception is raised.
Combinations with other operators
Combinations
Possible predecessors
create_matrixCreateMatrix
Possible successors
get_full_matrixGetFullMatrix, get_value_matrixGetValueMatrix
See also
decompose_matrixDecomposeMatrix, solve_matrixSolveMatrix
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