Lagrange Multipliers and Rayleigh Quotient Iteration in Constrained Type Equations
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We generalize the Rayleigh quotient iteration to a class of functions called vector Lagrangians. The Rayleigh quotient is an expression used in literature as an estimate of the Lagrange multiplier in constrained optimization. We discuss two methods of solving the updating equation associated with the iteration. One method leads to a generalization of Riemannian Newton method for embedded manifolds in a Euclidean space while the other leads to a generalization of the classical Rayleigh quotient formula and its invariant subspace extension. We also show how to apply second order iteration in this context to obtain cubic convergence. We discuss applications of this result to linear and nonlinear eigenvalue and subspace problems as well as potential applications in optimization.
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