A note on backward error and condition number of polynomial two-parameter eigenvalue problems

Study of eigenvalue problems associated with matrix polynomials with single or multiple parameters appears in many practical applications. Polynomial two-parameter eigenvalue problem (PTEP) is a special type of such problems, which consists of a system of two bivariate matrix polynomials with two spectral parameters. The numerical study of polynomial eigenvalue problem is an emerging area of research in recent times and sustainable progress has been made in the areas of numerical algorithms, sensitivity and backward error analysis. To know the error in a computed solution of eigenpairs, it is very much essential to analyse the backward errors and condition numbers of the problem. However, for PTEP of grade k, this analysis becomes more complex, than in the single-parameter case due to the interaction between the two parameters. In this paper, we consider the PTEP of grade k and provided new estimates of the normwise backward error and condition numbers.