invert_matrixT_invert_matrixInvertMatrixInvertMatrixinvert_matrix (Operator)

Name

invert_matrixT_invert_matrixInvertMatrixInvertMatrixinvert_matrix — Invert a matrix.

Signature

Description

The operator invert_matrixinvert_matrixInvertMatrixInvertMatrixInvertMatrixinvert_matrix computes the inverse of the MatrixMatrixMatrixMatrixmatrixmatrix defined by the matrix handle MatrixIDMatrixIDMatrixIDMatrixIDmatrixIDmatrix_id. A new matrix MatrixInvMatrixInvMatrixInvMatrixInvmatrixInvmatrix_inv is generated with the result and the matrix handle MatrixInvIDMatrixInvIDMatrixInvIDMatrixInvIDmatrixInvIDmatrix_inv_id of this matrix is returned. Access to the elements of the matrix is possible e.g. with the operator get_full_matrixget_full_matrixGetFullMatrixGetFullMatrixGetFullMatrixget_full_matrix.

For EpsilonEpsilonEpsilonEpsilonepsilonepsilon = 0, the inverse is computed. The type of the MatrixMatrixMatrixMatrixmatrixmatrix can be selected via 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, 'tridiagonal'"tridiagonal""tridiagonal""tridiagonal""tridiagonal""tridiagonal" for tridiagonal, 'upper_triangular'"upper_triangular""upper_triangular""upper_triangular""upper_triangular""upper_triangular" for upper triangular, 'permuted_upper_triangular'"permuted_upper_triangular""permuted_upper_triangular""permuted_upper_triangular""permuted_upper_triangular""permuted_upper_triangular" for permuted upper triangular, 'lower_triangular'"lower_triangular""lower_triangular""lower_triangular""lower_triangular""lower_triangular" for lower triangular, and 'permuted_lower_triangular'"permuted_lower_triangular""permuted_lower_triangular""permuted_lower_triangular""permuted_lower_triangular""permuted_lower_triangular" for permuted lower triangular matrices.

Example 1:

Example 2:

Example 3: For EpsilonEpsilonEpsilonEpsilonepsilonepsilon > 0, the pseudo inverse is computed using a singular value decomposition (SVD). During the computation, all singular values less than the value EpsilonEpsilonEpsilonEpsilonepsilonepsilon * the largest singular value are set to 0. For these values no internal division is done to prevent a division by zero. If a square matrix is computed with the SVD algorithm the computation takes more time. The type of the matrix must be set to MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type = 'general'"general""general""general""general""general".

Example:

Note: The relative accuracy of the floating point representation of the used data type (double) is EpsilonEpsilonEpsilonEpsilonepsilonepsilon = 2.2204e-16.

It should be also 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.

Attention

For MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type = 'symmetric'"symmetric""symmetric""symmetric""symmetric""symmetric", 'positive_definite'"positive_definite""positive_definite""positive_definite""positive_definite""positive_definite", or 'upper_triangular'"upper_triangular""upper_triangular""upper_triangular""upper_triangular""upper_triangular" the upper triangular part of the input MatrixMatrixMatrixMatrixmatrixmatrix must contain the relevant information of the matrix. The strictly lower triangular part of the matrix is not referenced. For MatrixTypeMatrixTypeMatrixTypeMatrixTypematrixTypematrix_type = 'lower_triangular'"lower_triangular""lower_triangular""lower_triangular""lower_triangular""lower_triangular" the lower triangular part of the input MatrixMatrixMatrixMatrixmatrixmatrix must contain the relevant information of the matrix. The strictly upper 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 MatrixMatrixMatrixMatrixmatrixmatrix are used. The other parts of the matrix are not referenced. If the referenced part of the input MatrixMatrixMatrixMatrixmatrixmatrix is not of the specified type, an exception is raised.

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

Result

If the parameters are valid, the operator invert_matrixinvert_matrixInvertMatrixInvertMatrixInvertMatrixinvert_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

invert_matrix_modinvert_matrix_modInvertMatrixModInvertMatrixModInvertMatrixModinvert_matrix_mod

See also

transpose_matrixtranspose_matrixTransposeMatrixTransposeMatrixTransposeMatrixtranspose_matrix, transpose_matrix_modtranspose_matrix_modTransposeMatrixModTransposeMatrixModTransposeMatrixModtranspose_matrix_mod

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