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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:11:09 +0000Multiple limit point bifurcation
https://resolver.caltech.edu/CaltechAUTHORS:20170802-082136797
Authors: {'items': [{'id': 'Decker-D-W', 'name': {'family': 'Decker', 'given': 'Dwight W.'}}, {'id': 'Keller-H-B', 'name': {'family': 'Keller', 'given': 'Herbert B.'}}]}
Year: 1980
DOI: 10.1016/0022-247X(80)90090-6
In this paper we present a new bifurcation or branching phenomenon which
we call multiple limit point bifurcation. It is of course well known that bifurcation
points of some nonlinear functional equation G(u, λ) = 0 are solutions (u_0, λ_0) at which two distinct smooth branches of solutions, say [u(ε), λ(ε)] and [u^(ε), λ^(ε)], intersect nontangentially. The precise nature of limit points is less easy to specify but they are also singular points on a solution branch; that is, points (u_0, λ_0) = (u(0), λ(0)), say, at which the Frechet derivative G_u^0 ≡ G_u(u_0, λ_0) is singular.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ews4e-sxe88Convergence Rates for Newton's Method at Singular Points
https://resolver.caltech.edu/CaltechAUTHORS:20120712-112618470
Authors: {'items': [{'id': 'Decker-D-W', 'name': {'family': 'Decker', 'given': 'D. W.'}}, {'id': 'Keller-H-B', 'name': {'family': 'Keller', 'given': 'H. B.'}}, {'id': 'Kelley-C-T', 'name': {'family': 'Kelley', 'given': 'C. T.'}}]}
Year: 1983
DOI: 10.1137/0720020
If Newton's method is employed to find a root of a map from a Banach space into itself and the derivative is singular at that root, the convergence of the Newton iterates to the root is linear rather than quadratic. In this paper we give a detailed analysis of the linear convergence rates for several types of singular problems. For some of these problems we describe modifications of Newton's method which will restore quadratic convergence.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/zs4er-99k51