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.