Interpreting the Classical Inverse, Shifted, Inverse, and Rayleigh Quotient Iteration Methods as a Standard Formulation of Newton’s Method from the Nonlinear Programming Literature
The inverse, shifted inverse, and Rayleigh quotient
iterations are well-known algorithms for computing an eigenvector of a
symmetric matrix. In this
talk we demonstrate that each one of these three algorithms can be viewed
as a standard form of Newton’s method from the nonlinear programming
literature, involving an
This provides an explanation for their good behavior despite the
need to solve systems with nearly singular coefficient matrices.
Our equivalence result also leads us naturally to a new proof that
the convergence of the Rayleigh quotient iteration is q-cubic with rate
constant at worst 1.