Dynamical Bifurcation Theory Research Articles

Rayleigh quotient iteration for nonsymmetric matrices

Rayleigh quotient iteration is an iterative algorithm for the calculation of approximate eigenvectors of a matrix. Given a matrix, the algorithm supplies a function whose iteration of an initial vector,v0{v_0}, produces a sequence of vectors,vn{v_n}. If the matrix is symmetric, then for almost any choice ofv0{v_0}the sequence will converge to an eigenvector at an eventually cubic rate. In this paper we show that there exist open sets of real matrices, each of which possesses an open set of initial vectors for which the algorithm will not converge to an eigenspace. The proof employs techniques from dynamical systems and bifurcation theory.

Dynamical systems and bifurcation theory

Dynamical systems and bifurcation theory