STRUCTURED EIGENVALUE CONDITION NUMBER ANDBACKWARD ERROR OF A CLASS OF POLYNOMIALEIGENVALUE PROBLEMS

摘要

Necessary and sufficient conditions are obtained for simple eigenvalues of complexmatrix polynomials with *-even/odd and *-palindromic/antipalindromic structures to have the samenormwise condition number with respect to structure preserving and arbitrary perturbations. Here,* denotes either the transpose T or the conjugate transpose *. Exact expressions for the normwisestructured condition number of simple eigenvalues of *-palindromic/antipalindromic and T-even/oddpolynomials and tight bounds that localize the structured condition number of simple eigenvaluesof T-palindromic/antipalindromic and *-even/odd polynomials are also obtained. The backwarderror of complex numbers z as approximate eigenvalues of these polynomials is also considered, andnecessary and sufficient conditions are obtained for a given complex number z to have the samestructured and unstructured backward errors.