svd_matrixSvdMatrixSvdMatrixsvd_matrixT_svd_matrix🔗

Short description🔗

svd_matrixSvdMatrixSvdMatrixsvd_matrixT_svd_matrix — Compute the singular value decomposition of a matrix.

Signature🔗

svd_matrix( matrix MatrixID, string SVDType, string ComputeSingularVectors, out matrix MatrixUID, out matrix MatrixSID, out matrix MatrixVID )

Description🔗

The operator svd_matrixSvdMatrix computes a full or reduced singular value decomposition (SVD) of the Matrix defined by the matrix handle MatrixIDmatrixIDmatrix_id. The operator returns the matrix handle MatrixSIDmatrixSIDmatrix_sid of the matrix MatrixS with singular values in descending order. Optionally, the matrices MatrixU with the left and MatrixV with the right singular vectors are computed and the matrix handles MatrixUIDmatrixUIDmatrix_uid and MatrixVIDmatrixVIDmatrix_vid are returned. Access to the elements of the matrices is possible e.g., with the operator get_full_matrixGetFullMatrix. The SVD is written

\[\begin{eqnarray*} \texttt{Matrix} \quad = \quad \texttt{MatrixU} \quad \cdot \quad \texttt{MatrixS} \quad \cdot \quad \texttt{MatrixV}^T. \end{eqnarray*}\]

For SVDTypeSVDTypesvdtype = 'full'"full", a full SVD is computed.

Example:

SVDTypeSVDTypesvdtype = 'full'"full", ComputeSingularVectorscomputeSingularVectorscompute_singular_vectors = 'both'"both"

\[\begin{eqnarray*} \texttt{Matrix} = \left[ \begin{array}{rr} 6.0 & -5.0 \\ 10.0 & 4.0 \\ -3.0 & 5.0 \end{array} \right] \end{eqnarray*}\]

\[\begin{eqnarray*} \to \qquad \texttt{MatrixU} &=& \left[ \begin{array}{rrr} -0.5228 & 0.5691 & 0.6346 \\ -0.8070 & -0.5702 & -0.1535 \\ 0.2745 & -0.5924 & 0.7574 \end{array} \right]\\ \texttt{MatrixS} &=& \left[ \begin{array}{rr} 12.0547 & 0 \\ 0 & 8.1046 \\ 0 & 0 \end{array} \right]\\ \texttt{MatrixV} &=& \left[ \begin{array}{rrr} -0.9980 & -0.0629 \\ 0.0629 & -0.9980 \end{array} \right] \end{eqnarray*}\]

For SVDTypeSVDTypesvdtype = 'reduced'"reduced", a reduced SVD is computed.

Example:

SVDTypeSVDTypesvdtype = 'reduced'"reduced", ComputeSingularVectorscomputeSingularVectorscompute_singular_vectors = 'both'"both"

\[\begin{eqnarray*} \texttt{Matrix} = \left[ \begin{array}{rr} 6.0 & -5.0 \\ 10.0 & 4.0 \\ -3.0 & 5.0 \end{array} \right] \end{eqnarray*}\]

\[\begin{eqnarray*} \to \qquad \texttt{MatrixU} &=& \left[ \begin{array}{rrr} -0.5228 & 0.5691 \\ -0.8070 & -0.5702 \\ 0.2745 & -0.5924 \end{array} \right]\\ \texttt{MatrixS} &=& \left[ \begin{array}{rr} 12.0547 & 0 \\ 0 & 8.1046 \end{array} \right]\\ \texttt{MatrixV} &=& \left[ \begin{array}{rrr} -0.9980 & -0.0629 \\ 0.0629 & -0.9980 \end{array} \right] \end{eqnarray*}\]

For ComputeSingularVectorscomputeSingularVectorscompute_singular_vectors = 'left'"left", the matrix MatrixU with the left singular vectors is computed. For ComputeSingularVectorscomputeSingularVectorscompute_singular_vectors = 'right'"right", the matrix MatrixV with the right singular vectors is computed. For ComputeSingularVectorscomputeSingularVectorscompute_singular_vectors = 'both'"both", the matrices MatrixU and MatrixV with the left and right singular vectors are computed.

For ComputeSingularVectorscomputeSingularVectorscompute_singular_vectors = 'none'"none", no matrices with the singular vectors are computed. The matrix MatrixS is a matrix with \(n\) rows and one column, where the number \(n\) = min(number of rows of the input Matrix, number of columns of the input Matrix).

Example:

SVDTypeSVDTypesvdtype = 'reduced'"reduced" or 'full'"full", ComputeSingularVectorscomputeSingularVectorscompute_singular_vectors = 'none'"none"

\[\begin{eqnarray*} \texttt{Matrix} = \left[ \begin{array}{rr} 6.0 & -5.0 \\ 10.0 & 4.0 \\ -3.0 & 5.0 \end{array} \right] \end{eqnarray*}\]

\[\begin{eqnarray*} \to \qquad \texttt{MatrixS} = \left[ \begin{array}{r} 12.0547 \\ 8.1046 \end{array} \right] \end{eqnarray*}\]

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.

Execution information🔗

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.

SVDTypeSVDTypesvdtype (input_control) string → (string)HTuple (HString)HTuple (string)strHtuple (char*)

Type of computation.

Default: 'full'"full"
List of values: 'full', 'reduced'"full", "reduced"

ComputeSingularVectorscomputeSingularVectorscompute_singular_vectors (input_control) string → (string)HTuple (HString)HTuple (string)strHtuple (char*)

Computation of singular values.

Default: 'both'"both"
List of values: 'both', 'left', 'none', 'right'"both", "left", "none", "right"

MatrixUIDmatrixUIDmatrix_uid (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)

Matrix handle with the left singular vectors.

MatrixSIDmatrixSIDmatrix_sid (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)

Matrix handle with singular values.

MatrixVIDmatrixVIDmatrix_vid (output_control) matrix → (handle)HTuple (HHandle)HMatrix, HTuple (IntPtr)HHandleHtuple (handle)

Matrix handle with the right singular vectors.

Result🔗

If the parameters are valid, the operator svd_matrixSvdMatrix 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

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