ISSN:
0945-3245
Keywords:
AMS(MOS): 65F15
;
CR: G1.3
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary For each λ in some domainD in the complex plane, letF(λ) be a linear, compact operator on a Banach spaceX and letF be holomorphic in λ. Assuming that there is a ξ so thatI−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 multiplicitym. For fixed λ leth(λ) denote the arithmetic mean of them eigenvalues of the pencil [I−F(λ),I] which are closest to 0. Thenh is holomorphic in a neighborhood of ξ andh(ξ)=0. Under suitable hypotheses the classical Muller's method applied toh converges locally with order approximately 1.84.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01462234
