Sensitivity and Perturbation analysis of Polynomial Eigenvalue problem

The conference dinner will take place in Café Campus behind the Mathematics Building on the campus of TU Berlin. We discuss a general framework for sensitivity and perturbation analysis of polynomial eigen-value problem. More specifically, we discuss first order variations of simple eigenvalues of matrix polynomials, determine the gradients and present a general definition of condition number of a simple eigenvalue. We show that our treatment unifies various condition numbers of simple eigenvalues of matrix pencils and matrix polynomials proposed in the literature. We present an alternative expression of condition number of a simple eigenvalue that does not involve left and right eigenvectors associated with the eigenvalue. We show that the sensitivity of a simple eigenvalue is inversely proportional to the absolute value of the derivative of the characteristic polynomial at the eigenvalue. We also construct fast perturbations for moving simple eigenval-ues of matrix polynomials and discuss perturbation bounds for approximate eigenvalues. The Golub–Kahan–Lanczos bidiagonal reduction generates a factorization of a matrix X ∈ R m×n , m ≥ n, such that X = U BV T where U ∈ R m×n is left orthogonal, V ∈ R n×n is orthogonal, and B ∈ R n×n is bidiagonal. Since, in the Lanczos recurrence, the columns of U and V tend to lose orthogonality, a reorthog-onalization strategy is necessary to preserve convergence of the singular values of the leading k × k submatrix B k = B(1: k, 1: k) to those of B. the computation of matrix functions, it is essential