โ Scribed by Abdellatif El Ghazi; Said El Hajji; Luc Giraud; Serge Gratton
No coin nor oath required. For personal study only.
In El Ghazi et al. [Backward error for the common eigenvector problem, CERFACS Report TR/PA/06/16, Toulouse, France, 2006], we have proved the sensitivity of computing the common eigenvector of two matrices A and B, and we have designed a new approach to solve this problem based on the notion of the backward error.
If one of the two matrices (saying A) has n eigenvectors then to find the common eigenvector we have just to write the matrix B in the basis formed by the eigenvectors of A. But if there is eigenvectors with multiplicity > 1, the common vector belong to vector space of dimension > 1 and such strategy would not help compute it.
In this paper we use Newton's method to compute a common eigenvector for two matrices, taking the backward error as a stopping criteria.
We mention that no assumptions are made on the matrices A and B.
๐ SIMILAR VOLUMES
A kind of generalized inverse eigenvalue problem is proposed which includes the additive, multiplicative and classical inverse eigenvalue problems as special cases. Newton's method is applied, and a local convergence analysis is given for both the distinct and the multiple eigenvalue cases. When the
formulations of Newton's method abound in the literature (see, e.g., [2], [6], [8], [12], [13]). Our aim here is twofold: to present in a unified setting a number of old and some new results that may be of particular interest in applications, and to discuss as illustrations two numerical techniques