Computation Of Pseudospectra Of Large Matrix By Generalized Projectrion

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, presents some algorithms for calculation of pseudospectra of single matrix and matrix polynomial. For large matrix polynomial , we adopt the generalized Arnoldi projection algorithm to speed up calculation .So, we can compute pseudospectra by the projection matrix which dimension is lower instead, and the result is well. For a single matrix, this paper proposes an adaptive grid method, in order to accelerate pseudo-spectral calculation. Traditionaly,compared with the standard grid method, adaptive grid method can rapidly positioning prayer pseudospectral boundary,and it avoids calculating the smallest eigenvalue of the grid points that we do not interesteded.Where it speeds up caculation of pseudospectra. Of course, this method also can be applied to the matrix polynomial. To all sorts of algorithm, this paper gives the numerical experiments to verify their effectiveness.