Inverse, Shifted Inverse, and Rayleigh Quotient Iteration as Newton's Method

Two-norm normalized inverse, shifted inverse, and Rayleigh quotient iteration are well-known algorithms for approximating an eigenvector of a symmetric matrix. In this work we establish rigorously that each one of these three algorithms can be viewed as a standard form of Newton’s method from the nonlinear programming literature, followed by the normalization. This equivalence adds considerable understanding to the formal structure of inverse, shifted inverse, and Rayleigh quotient iteration and provides an explanation for their good behavior despite the possible need to solve systems with nearly singular coe cient matrices; the algorithms have what can be viewed as removable singularities. A thorough historical development of these eigenvalue algorithms is presented. Utilizing our equivalences we construct traditional Newton’s method theory analysis in order to gain understanding as to why, as normalized Newton’s method, inverse iteration and shifted inverse iteration are only linearly convergent and not quadratically convergent, and why a new linear system need not be solved at each iteration. We also investigate why Rayleigh quotient iteration is cubically convergent and not just quadratically convergent.