Numerical algorithms that compute the eigenvalues of a nonsymmetric
matrix typically make roundoff errors of size roughly
, where
is the machine precision.
Therefore, applying a simple and accurate similarity transform
to reduce the norm of the matrix
, or to
reduce the condition numbers of some subset of
's eigenvalues,
can make the computed eigenvalues of
more accurate.
For example, consider the matrix
Osborne [346] first noted that the norm of a matrix can often be reduced
with a similarity transform of the form
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(120) |
Although balancing in the -norm is equivalent to minimizing
the Frobenius norm, balancing a matrix in an arbitrary norm may
not have such a simple effect on a matrix norm. Other work
discusses the mathematical properties of using diagonal scaling
to balance matrices and to minimize matrix norms [81,82].
Focusing on practice instead of theory, we present here two styles of
algorithms for balancing sparse matrices. The algorithms are
analyzed more thoroughly in [81] and [82];
software can be accessed through the book's homepage,
ETHOME.