Structured Eigenvalue Condition Number and Backward Error of a Class of Polynomial Eigenvalue Problems
Shreemayee Bora · SIAM Journal on Matrix Analysis and Applications · 2009
Necessary and sufficient conditions are obtained for simple eigenvalues of complex matrix polynomials with $\star$-even/odd and $\star$-palindromic/antipalindromic structures to have the same normwise condition number with respect to structure preserving and arbitrary perturbations. Here, $\star$ denotes either the transpose T or the conjugate transpose $*$. Exact expressions for the normwise structured condition number of simple eigenvalues of $*$-palindromic/antipalindromic and T-even/odd polynomials and tight bounds that localize the structured condition number of simple eigenvalues of T-palindromic/antipalindromic and $*$-even/odd polynomials are also obtained. The backward error of complex numbers z as approximate eigenvalues of these polynomials is also considered, and necessary and sufficient conditions are obtained for a given complex number z to have the same structured and unstructured backward errors.