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Lanczos Eigenvalue Algorithm With No Reorthogonalization for Nonsymmetric Matrices

IP.com Disclosure Number: IPCOM000065338D
Original Publication Date: 1985-Nov-01
Included in the Prior Art Database: 2005-Feb-19

Publishing Venue

IBM

Related People

Authors:
Cullum, JK Willoughby, RA [+details]

Abstract

The problem of computing eigenvalues and eigenvectors of large, sparse, but nonsymmetric, matrices has attracted much interest over the past few years, but the algorithms proposed to date are limited in power and range of applicability. This disclosure describes a new nonsymmetric Lanczos eigenvalue algorithm which can be used on very large matrices and has produced complete and accurate results for test problems. The general two-sided Lanczos recursion [1] is given in Equations (1) - (4). Let A be a general nxn matrix. Let v1 and w1 be two starting n-vectors with their Euclidean 'inner product' v1 T w1 = 1. For i = 1,2,...,M the following recursions define tridiagonal matrices Tm, m = 1, 2...with diagonal entries Tm(i,i) X ai and off-diagonal entries Tm(i,i + 1) X bi+1 and Tm(i + 1,i) X qi+1 .