Istituto di Matematica Applicata e Tecnologie Informatiche     
Simoncini V., Gallopoulos E. Transfer function and resolvent norm approximations of large matrices. Preprint ercim.cnr.ian//1997-1069, 1997.
Recent applications have shown that the eigenvalue inspection of a nonnormal matrix $A$ may be misleading in predicting the behavior of numerical methods that rely on spectral information for their convergence. To this end, a better accordance between theory and practice may be achieved by inspecting the resolvent function $R(z) = (A-zI)^{-1}$, where $zin {cal C} subsetCC$ for which $R(z)$ is defined. Studying the variation of the resolvent norm corresponds to analyzing how the eigenvalues change under small perturbations of $A$. The locus of $CC$ on which $|R(z)|ge varepsilon^{-1}$ for some $varepsilon>0$ corresponds to the set of eigenvalues that solve the problem [ (A+{cal E})x = lambda x, qquad |{cal E}|le varepsilon, ] for some matrix norm cite{Trefethen.90}. A lot of effort has been put in computing approximations to $|R(z)|$ without explicitly addressing the possibly large matrix $(A-zI)^{-1}$. The usual approach consists of finding approximations to $|(A-zI)^{-1}|$ by means of singular value or eigenvalue solvers. Projection type methods tend to be preferred, since they only reference $A$ with matrix-vector products. In this note we show that methods that rely on reduction can be written in terms of a unifying function. Given two tall rectangular full rank matrices $D^*$ and $E$ of suitable dimension (Here `*' denotes conjugate transposition), we consider the projected resolvent function onto the subspaces spanned by the two bases, that is $G_z(A,E,D):= D(A-zI)^{-1}E$. The norm $|G_z(A,E,D)|$ measures the size of the perturbation for $zin {cal C}$ when $A$ is restricted to have structured perturbations of type $EDelta D$ for small norm disturbance matrices $Delta$. Most usual projection methods are shown to be related to transfer functions corresponding to certain selections of $E$ and $D$. In order to approximate $|(A-zI)^{-1}|$, these methods in fact compute $|G_z(A,E,D)|$ for given $E, D$. This setting also allows us to justify the poor approximation obtained by means of some reduction approaches.
Subject Eigenvalues, nonnormal matrices

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