Journal Article10.1287/MOOR.1.4.359
On Paths Generated by Fixed Point Algorithms
TL;DR: This paper considers algorithms that compute fixed points or more generally solve equations which are based on complementary pivoting, and treats some special applications of these algorithms from a global point of view, and looks at the behavior and properties of these paths sufficiently far from the solution.
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Abstract: This paper considers algorithms that compute fixed points or more generally solve equations which are based on complementary pivoting. These algorithms, starting with a map f0 and its fixed point x0, deform ft to f∞ = f as t goes from 0 to ∞, and follow the path xt of fixed points of ft. In this paper we study these paths. In particular, we treat some special applications of these algorithms from a global point of view, and thus look at the behavior and properties of these paths sufficiently far from the solution. Our methods are motivated by methods of global analysis.
We show that in many implementations of these algorithms the path can be specified. These include the cases when the mapping is linear and when it is smooth and monotone. The results of the linear analysis are used to study the local behavior of these paths i.e., properties and behavior sufficiently close to the solution for smooth mappings.
An important implication of this study is that the paths can be modified and thus the work done by these algorithms can be controlled. We also show, for smooth mappings, how our results can be implemented to increase the efficiency of some standard algorithms.
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Citations
•Book
Introduction to Numerical Continuation Methods
Eugene L. Allgower,Kurt Georg +1 more
- 01 Jan 1987
TL;DR: The Numerical Continuation Methods for Nonlinear Systems of Equations (NCME) as discussed by the authors is an excellent introduction to numerical continuuation methods for solving nonlinear systems of equations.
Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations
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TL;DR: In this paper, a digest of simplicial and continuation methods for approximating fixed-points or zero-points of nonlinear finite-dimensional mappings is presented, where the following curves are implicitly defined, as for example, in the case of homotopies.
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On the Convergence Rate of Algorithms for Solving Equations that are Based on Methods of Complementary Pivoting
TL;DR: This paper considers the problem of solving a system of n nonlinear equations in n variables, when the underlying functions are continuously differentiable and their derivative satisfies a Lipschitz condition, and shows that certain realizations of these methods achieve quadratic convergence when they reach sufficiently close to the solution.
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References
•Book
Iterative Solution of Nonlinear Equations in Several Variables
J.M. Ortega,Werner C. Rheinboldt +1 more
- 01 Jun 1970
TL;DR: In this article, the authors present a list of basic reference books for convergence of Minimization Methods in linear algebra and linear algebra with a focus on convergence under partial ordering.
7.9K
•Book
Theory of Ordinary Differential Equations
Earl A. Coddington,Norman Levinson +1 more
- 01 Jan 1955
TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.
7.2K
•Book
Differential Equations, Dynamical Systems, and Linear Algebra
Morris W. Hirsch,Steve Smale +1 more
- 12 May 1974
TL;DR: In this article, the structure theory of linear operators on finite-dimensional vector spaces has been studied and a self-contained treatment of that subject is given, along with a discussion of the relations between dynamical systems and certain fields outside pure mathematics.
Equilibrium Points of Bimatrix Games
C. E. Lemke,J. T. Howson +1 more
TL;DR: An algebraic proof of the existence of equilibrium points for two-person non-zero-sum games is given in this paper, leading to an efficient scheme for computing an equilibrium point, which is valid for any ordered field.
1.2K
Bimatrix Equilibrium Points and Mathematical Programming
TL;DR: In this paper, simple constructive proofs are given of solutions to the matric matric system Mz − ω = q; z ≧ 0; ω ≧ 1; zT = 0, for various kinds of data M, q, which embrace quadratic programming and the problem of finding equilibrium points of bimatrix games.