Globally convergent homotopy algorithms for nonlinear systems of equations
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TLDR
The theory of globally convergent homotopy algorithms was introduced in this article, which is a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely.Abstract:
Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely. These new globally convergent homotopy techniques have been successfully applied to solve Brouwer fixed point problems, polynomial systems of equations, constrained and unconstrained optimization problems, discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements, and finite difference, collocation, and Galerkin approximations to nonlinear partial differential equations. This paper introduces, in a tutorial fashion, the theory of globally convergent homotopy algorithms, deseribes some computer algorithms and mathematical software, and presents several nontrivial engineering applications.read more
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Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms
TL;DR: Changes to HOMPACK include numerous minor improvements, simpler and more elegant interfaces, use of modules, new end games, support for several sparse matrix data structures, and new iterative algorithms for large sparse Jacobian matrices.
Journal ArticleDOI
Axisymmetric flow due to a stretching sheet with partial slip
TL;DR: The steady, laminar, axisymmetric flow of a Newtonian fluid due to a stretching sheet when there is a partial slip of the fluid past the sheet is investigated and a solution based upon He's homotopy perturbation method has been developed.
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Theory of Globally Convergent Probability-One Homotopies for Nonlinear Programming
TL;DR: Some probability-one homotopy convergence theorems for unconstrained and inequality constrained optimization, for linear and nonlinear inequality constraints, and with and without convexity are derived.
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Globalization Techniques for Newton-Krylov Methods and Applications to the Fully Coupled Solution of the Navier-Stokes Equations
TL;DR: This paper reviews several representative globalizations of Newton-Krylov methods, discusses their properties, and reports on a numerical study aimed at evaluating their relative merits on large-scale two- and three-dimensional problems involving the steady-state Navier-Stokes equations.
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An efficient algorithm for finding multiple DC solutions based on the SPICE-oriented Newton homotopy method
TL;DR: This paper shows a very simple SFICE-oriented Newton homotopy method which can efficiently find out the multiple dc solutions in circuit simulations and proves an important theorem about how many variables should be chosen to implement this algorithm.
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