Numerical Methods For Computing Large Matrix Pseudospectra

The computation of matrix pseudospectra, which have been widely used in many fields, is an interesting and important problem for discussion with special theoretic sense and engineering value. The pseudospectra of matrix is a powerful concept that broadens our understanding of the bahaviour of various matrix process and phenomena based on matrix computation. For non-normal matrices and operator, matrix pseudospectra had been proved to be more useful than eigenvalues. However, the compution of pseudospectra is a very expensive computational task. Thus, the use of high performace computing recources becomes key to obtaining useful answers in acceptable amouts of time.This paper, based on a brief summary of pseudospectra theory and algorithms, proposed an advanced kind of projection methods for computing the pseudospectra of large scale matrices, including orthogonalization projection method and oblique projection method respectively, which had been generalized to the computing of matrix polynomials pseudospectra. Different form the methods that have been developed, this paper develop a method called optimal rank-k method for the computing of matrix pseudospectra. This method has good performance for certain matrices. Also, this paper give a new definition of pseudospectra of matrix polynomials by using QR decomposition, which is generalized form the the QR decomposition definition of matrix pseudospectra. Nnumerical experiments illustrate the efficiency of different methods.