Complete Iterative Method for Computing Pseudospectra

Abstract

Efficient codes for computing pseudospectra of large sparse matrices usually use a Lanczos type method with the shift and invert technique and a shift equal to zero. Then, these codes are very efficient for computing pseudospectra on regions where the matrix is nonnormal (because k(A \Gamma zI) \Gamma1 k2 is large) but they lose their efficiency when they compute pseudospectra on regions where the spectrum of A is not sensitive (k(A \Gamma zI) \Gamma1 k2 is small). A way to overcome this loss of efficiency using only iterative methods associated with an adaptive shift is proposed. 1 Introduction The "-pseudoeigenvalue and "-pseudospectrum are defined as: ffl is an "-pseudoeigenvalue of A if is an eigenvalue of A+ E with kEk 2 "kAk 2 ffl The "-pseudospectrum of A is defined by " (A) = fz 2 l C ; z is an "\Gammapseudoeigenvalue of Ag For a fixed ", the contour of " (A) can be defined as fz 2 l C ; kAk 2 k(A \Gamma zI) \Gamma1 k 2 = " \Gamma1 g. The graphical representati..