Some Research On Matrix Pseudospectra Problems

The pseudospectra is a useful tool to explain the behavior of non-normal matrices and linear operators.In this paper,some matrix pseudospectra problems are studied on the basis of existing pseudospectral research.A new algorithm for calculating pseudospectra of large-scale matrix is proposed;The problem of pseudospectra of matrix pencil and its calculation method are discussed;The definition and calculation of matrix pseudospectra based on polar decomposition are investigated.Firstly,the block method is combined with the Krylov subspace projection method,and the pseudospectra of the large matrix is calculated by the Householder Arnoldi(ABHA)method.The numerical results show that this method is superior to the existing algorithm.Secondly,a new definition and complete equivalent proof for the pseudospectra of generalized eigenvalue problem is given.According to the definition,a new algorithm is proposed,and the definition of pseudospectra is extended to polynomial eigenvalue problem.Then,based on the polar decomposition,a new definition which is equivalent to the definition based on the classical decomposition — SVD decomposition is proposed,and a new algorithm is presented.Furthermore,the proposed algorithms are programmed and the numerical experiments and comparisons are carried out.The numerical results show that the proposed algorithms are effective and has advantages.Finally,a MATLAB Graphical User Interface(GUI)for calculating and visualizing pseudospectra is designed.In the interface,it is convenient to call the existing algorithm and all the algorithms proposed in this paper for matrix pseudospectra calculation and 2D and 3D pseudospectra Graphics.