Globally convergent homotopy methods: a tutorial
TLDR
Homotopy algorithms for solving nonlinear systems of (algebraic) equations, which are convergent for almost all choices of starting point, are globally convergent with probability one and exhibit a large amount of coarse grain parallelism.About:
This article is published in Applied Mathematics and Computation.The article was published on 1989-05-01 and is currently open access. It has received 123 citations till now. The article focuses on the topics: n-connected & Homotopy analysis method.read more
Citations
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A series solution of the nonlinear Volterra and Fredholm integro-differential equations
TL;DR: In this paper, the convergence-parameter is used to adjust and control the convergence region of the solution series of a nonlinear Volterra and Fredholm problem with power-law nonlinearity.
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Novel Homotopy Theory for Nonlinear Networks and Systems and Its Applications to Electrical Grids
Hsiao-Dong Chiang,Tao Wang +1 more
TL;DR: A novel homotopy theory and a convergence theorem are presented and it is shown that the derived analytical results are applicable to a class of large-scale integrated-circuit designs and protein interaction networks.
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On higher-order differentiation in nonlinear mechanics
TL;DR: This interdisciplinary paper reviews higher-order numerical methods for the solution of nonlinear problems, and proposes a synthesis of two different conceptual frameworks, namely automatic differentiation and the asymptotic numerical method.
Posted Content
Modernizing PHCpack through phcpy
TL;DR: This paper describes the development of phcpy, a Python interface to PHCpack that solves systems in the Python shell or via scripts instead of navigating through menus.
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Computing Zeros of Sections of Vector Bundles Using Homotopies and Relocalization
TL;DR: An algorithm is described for computing fixed points on a Grassmannian manifold and it is shown that this method can be applied in the more general setting of solving equations on abstract smooth manifolds into vector bundles.
References
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Book
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Book
Numerical methods for unconstrained optimization and nonlinear equations
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
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An incremental approach to the solution of snapping and buckling problems
TL;DR: In this paper, an incremental approach to the solution of buckling and snapping problems is explored, where the authors use the length of the equilibrium path as a control parameter, together with the second order iteration method of Newton.