ELA, Volume 15, pp. 159-177, May 2006, abstract.
Structured Condition Numbers and Backward Errors in
Scalar Product Spaces
Francoise Tisseur and Stef Graillat
The effect of structure-preserving perturbations on
the solution to a linear system, matrix inversion,
and distance to singularity is investigated.
Particular attention is paid to linear and nonlinear
structures that form Lie algebras, Jordan algebras and
automorphism groups of a scalar product. These include
complex symmetric, pseudo-symmetric, persymmetric,
skew-symmetric, Hamiltonian, unitary, complex orthogonal
and symplectic matrices. Under reasonable assumptions on
the scalar product, it is shown that there is little or
no difference between structured and unstructured
condition numbers and distance to singularity for matrices
in Lie and Jordan algebras. Hence, for these classes of
matrices, the usual unstructured perturbation analysis is
sufficient. It is shown that this is not true in general
for structures in automorphism groups. Bounds and
computable expressions for the structured condition numbers
for a linear system and matrix inversion are derived for
these nonlinear structures.
Structured backward errors for the approximate solution of
linear systems are also considered. Conditions are given
for the structured backward error to be finite. For Lie and
Jordan algebras it is proved that, whenever the structured
backward error is finite, it is within a small factor of or
equal to the unstructured one. The same conclusion holds for
orthogonal and unitary structures but cannot easily be
extended to other matrix groups.
This work extends and unifies earlier analyses.