Some Research On The Pseudospectra Of Matrices

The computation of matrix pseudospectra, which have been widely used in many fields, is aninteresting 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 bahaviourof various matrix process and phenomena based on matrix computation.For non-normal matrices andoperator, matrix pseudospectra had been proved to be more useful than eigenvalues. However, thecompution of pseudospectra is a very expensive computational task. Thus, the use of high performacecomputing recources becomes key to obtaining useful answers in acceptable amouts of time.This paper,based on a brief summary of pseudospectra theory and algorithms,we studied thefollowing problems.First,we researched pseudospctra problem of companion matrix.With the sparsestructural characteristics of companion matrix,we givend a promotion pseudospectral defines based onGivens QR decomposition,it is different from the existing the pseudospectral definition based onmatrix QR decomposition,but also expressed the pseudospectral implication.Moreover,it is calculatedthe pseudospectra of companion matrix based on this definition more economical. Furthermore,discuddion the problem of determining the pseudospectral region,we givend a Gerschgorin diskmethod to determine the pseudospectral region by used of sparse structure of companion matrix.Andcompared with FOV method of classic pseudospectral region determining,calculated quantity isgreatly reduced,and pseudospectral region issmaller,so making calculate the companion matrixpseudospectral more economical.Secondly,we studied the projection methods of polynomialpseudospectral, we gived a inear projection method of calculating matrix polynomial pseudospectraland comparison with direct projection algorithm of calculation large matrix polynomialpseudospectral,and studied the good and bad.Further combined with many examples,givend numeric.To all sorts of algorithm, this paper gives the numerical experiments to verify their effectiveness.