General Rayleigh Quotient Iteration

Abstract

The Rayleigh quotient is generalized to the form $\rho _C = {{\langle {Ax,Bx}\rangle _C } /{\| {Bx} \|_C^2 }}$, where C is a Hermitian, positive definite matrix. General Rayleigh quotient iteration is applied to the problem $Ax = \lambda Bx$, and under certain conditions quadratic local convergence is proved. An example is given in which it is shown that Rayleigh quotient iteration diverges in a neighborhood of an eigenvector, while Blum–Rodrigue quotient iteration converges.