Convergence factors of Newton methods for nonlinear eigenvalue problems
2012 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 436, no 10, 3943-3953 p.Article in journal (Refereed) Published
Consider a complex sequence convergent to λ∗∈C with order p∈N. The convergence factor is typically defined as the fraction ck:=(λk+1-λ∗)/(λk-λ∗)p in the limit k→∞. In this paper, we prove formulas characterizing ck in the limit k→∞ for two different Newton-type methods for nonlinear eigenvalue problems. The formulas are expressed in terms of the left and right eigenvectors.
The two treated methods are called the method of successive linear problems (MSLP) and augmented Newton and are widely used in the literature. We prove several explicit formulas for ck for both methods. Formulas for both methods are found for simple as well as double eigenvalues. In some cases, we observe in examples that the limit ck as k→∞ does not exist. For cases where this limit does not appear to exist, we prove other limiting expressions such that a characterization of ck in the limit is still possible.
Place, publisher, year, edition, pages
Elsevier, 2012. Vol. 436, no 10, 3943-3953 p.
Nonlinear eigenvalue problems, Newton’s method, Convergence factors
Computer and Information Science
IdentifiersURN: urn:nbn:se:kth:diva-53257DOI: 10.1016/j.laa.2010.08.045ISI: 000303095700010ScopusID: 2-s2.0-84858797047OAI: oai:DiVA.org:kth-53257DiVA: diva2:469726
FunderSwedish e‐Science Research Center
QC 201412162011-12-262011-12-262014-12-16Bibliographically approved