On The Pseudospectra Of Matrix And Computations

The pseudospectra is a useful tool to explain the behavior of the systems associated with non-normal matrices and the eigenvalue perturbation theory,which have great backgrounds in many fields such as atmospheric science,control theory,lasers,hydrodynamic stability,numerical solution of differential equations and matrix iterations.In this paper,we study the theory of matrix pseudospectra and its computational problems.Firstly,the problem on the pseudospectra of companion matrix is discussed.A definition and a fast algorithm for the pseudospectra of companion matrix are analyzed,and the balance technique is presented using the pseudospectra as a tool.Moreover,the structured pseudospectra of companion matrix is discussed in order to analyse the perturbation of the zeros of polynomial.Secondly,the pseudospectra of Kronecker product is considered.For the Kronecker product of two matrices,the definitions of pseudospectra are given and the equivalence is proved,including four different disturbance situations.At the same time,the corresponding pseudospectral calculation method is deduced.Similarly,the pseudospectral definitions of Kronecker product of three matrices under three different disturbances are proposed.Furthermore,the definitions and algorithms of unstructured pseudospectra based on the QR factorization and real structured pseudospectra for Kronecker product of two matrices are derived.Besides,the selection of pseudospectral region is explored.A method of selecting region and general recommendations on regional options are given.Regional exclusion for calculating pseudospectra is improved.Finally,the numerical experiments and comparisons of the algorithms proposed in this paper are presented.The numerical results show the correctness and effectiveness of the proposed theory and algorithms.