Shifted Inverse Iteration and Rayleigh Quotient Iteration as Newton’s Method

Richard Tapia
Rice University

Shifted inverse and Rayleigh quotient are well-known algorithms for computing an eigenvector of a symmetric matrix. In this talk we demonstrate that each of these algorithms can be viewed as a standard form of Newton’s method from the nonlinear programming literature. 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 Rayleigh quotient iteration is cubically convergent with constant at worst 1.


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