On Componentwise Condition Numbers For Generalized Eigenvalue Problems With Structured Matrices

In this paper, we consider the componentwise condition numbers for generalized eigenvalue problems (GEP) with structured matrices. Based on the previous work by Higham et al.[D.J. Higham and N.J. Higham, Structured backward error and condition of generalized eigenvalue problems.SIAM Journal on Matrix Analysis and Applications [J],1998,20:493-512.], with the part-componentwise and structured perturbation analysis, we define the structured componentwise condition numbers for GEP. The explicit expres-sions are derived, which can be applied to the following significant structured matrices: Toeplitz and Hankel. Numerical examples show that our results are smaller than the pre-vious ones, and can reveal the true conditioing of structured GEP. Our work also can be generalized to non-linear GEP with Cauchy and Vandermonde matrices.