For EpsilonEpsilonEpsilonEpsilonepsilonepsilon = 0, the inverse is computed. The type of the
Matrix 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.
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".
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 Matrix 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 Matrix 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 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
Multithreading type: reentrant (runs in parallel with non-exclusive operators).
Multithreading scope: global (may be called from any thread).
If the parameters are valid, the operator invert_matrixinvert_matrixInvertMatrixInvertMatrixInvertMatrixinvert_matrix
returns the value TRUE. If necessary, an exception is raised.
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.