Nonlinear eigenvalue approximation
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For each λ in some domain D in the complex plane, let F(λ) be a linear, compact operator on a Banach space X and let F be holomorphic in λ. Assuming that there is a ξ so that I-F(ξ) is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue ξ. In the Newton method the smallest eigenvalue of the operator pencil [I-F(λ), F′(λ)] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I-F(ξ), I] with algebraic multiplicity m. For fixed λ let h(λ) denote the arithmetic mean of the m eigenvalues of the pencil [I-F(λ), I] which are closest to 0. Then h is holomorphic in a neighborhood of ξ and h(ξ)=0. Under suitable hypotheses the classical Muller's method applied to h converges locally with order approximately 1.84. © 1988 Springer-Verlag.
author list (cited authors)
Moss, W. F., Smith, P. W., & Ward, J. D.