| TR#: | CS0877 |
| Class: | CS |
| Title: | KRYLOV SUBSPACE METHODS FOR EIGENVALUES WITH SPECIAL PROPERTIES AND THEIR ANALYSIS FOR NORMAL MATRICES |
| Authors: | Avram Sidi |
| Not Available | |
| Abstract: | In this paper we propose a general approach by which eigenvalues with a special property of a given matrix $A$ can be obtained. In this approach we first determine a scalar function $\psi:\C \rightarrow \C$ whose modulus ia maximized by the eigenvalues that have the special property. Next, we compute the generalized power iterations $u_{j+1} = \psi(A) u_j, j=0,1,...,$ where $u_0$ is an arbitrary initial vector. Finally, we apply known Krylov subspace methods, such as the methods of Arnoldi and Lanczos, to the vector $u_n$ for some sufficiently large $n$. We can also apply the simultaneous iteration method to the subspace span $\{x^{(n)}_1,..., x^{(n)}_k\}$ with some sufficiently large $n$, where $x^{(j+1)}_m = \psi(A) x^{(j)}_m,\ j = 0,1,..., m=1,...,k$. In all cases the resulting Ritz pairs are approximations to the eigenpairs of $A$ with the special property. We provide a pretty thorough convergence analysis of the approach involving all three methods as $n \rightarrow \infty$ for the case in which $A$ is a normal matrix. We also discuss the connections and similarities of our approach with the existing methods and approaches in the literature. |
| Copyright | The above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information |
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