Research On The Pseudospectra Of Matrix And The Numerical Methods

The pseudospectrum of a matrix is a powerful tool to explain the behavior of non-normal matrices and linear operators,which have been widely used in many fields.When the matrix is non-normal,however,especially for large matrix,the computation of the pseudospectra becomes a very expensive computational task.Therefore,the research on the pseudospectra of matrix and the numerical methods for computing the pseudospectra is interesting and challenging.In this thesis,we focus on the pseudospectra of matrix and the numerical methods for computing the pseudospectra.Firstly,using the Hessenberg decomposition on Krylov subspace,we propose an Induced Dimension Reduction(IDR)method for computing the pseudospectra of matrix.This methods is more efficient with less computational cost.The numerical experiments and comparisons are carried out to show the IDR method is efficient.Then,the weighted pseudospectra are considered.We propose an Induced Dimension Reduction(IDR)method for computing the weighted pseudospectra of matrix.The numerical experiments and comparisons are carried out to show the IDR method for computing weighted pseudospectra is effective.Finally,the pseudospectra of polynomial eigenvalue problem are investigated.Based on the linearization of polynomial eigenvalue problem,an implicit restart IDR method for computing the pseudospectra of polynomial eigenvalue problem is proposed.The advantage of this method is that it can compute the pseudospectra in the interested fields,especially for the large linearization structured matrix.The given numerical experiments and comparisons show the implicit restart IDR method for computing the pseudospectra of polynomial eigenvalue is efficient.