Approximating the weighted pseudospectra of large matrices

In different applications it is necessary to compute the pseudospectra of matrices in an energy or weighted norm. In this work we develop a practical method to convert an Arnoldi factorization based on the Euclidean inner product into another Arnoldi factorization in a weighted inner product. We also propose how to use this approach to find a portion of the weighted pseudospectra of large matrices. Numerical results on different test problems are presented.