Author
Layne Watson
Bio: Layne Watson is an academic researcher from Michigan State University. The author has contributed to research in topics: Nonlinear programming & Mathematical structure. The author has an hindex of 3, co-authored 3 publications receiving 213 citations.
Papers
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TL;DR: Chow, Mallet-Paret, and Yorke have recently proposed an algorithm for computing Brouwer fixed points of C^2 maps as discussed by the authors, and a numerical implementation of that algorithm is presented here.
158 citations
TL;DR: These digraphs show that such algorithms based on complementary pivoting for solving the linear complementarity problem can cycle even for symmetric, positive deFinite M, and provide some insight into the algorithms' behavior.
Abstract: For M ∈ En×n and q ∈ En, the linear complementarity problem is to find vectors w, z ∈ En such that w-Mz = q, w ≥ 0, z ≥ 0, wtz = 0. A family of algorithms based on complementary pivoting for solving this problem is modelled by digraphs. These digraphs show that such algorithms can cycle even for symmetric, positive deFinite M, and provide some insight into the algorithms' behavior. For a P-matrix M, it is proved that if the solution to the complementarity problem can be obtained by k principal pivots, then it can be obtained by k Bard-type pivots. Furthermore, the digraphs provide simple geometric proofs of some of Murty's algebraic results. The digraphs, apart from their use as models, also raise some interesting graph-theoretic questions.
57 citations
TL;DR: In this paper, a class of nonlinear optimization problems arising in various fields of continuum physics with a common mathematical structure is defined and several techniques for solving the structured nonlinear programming problem are presented and compared with computational results for a sample problem in plasticity.
4 citations
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Book•
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.
Abstract: From the Publisher:
Introduction to Numerical Continuation Methods continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra is adequate preparation for reading this book; some knowledge from a first course in numerical analysis may also be helpful.
889 citations
TL;DR: HOMPACK provides three qualitatively different algorithms for tracking the homotopy zero curve: ordinary differential equation-based, normal flow, and augmented Jacobian matrix.
Abstract: There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK provides three qualitatively different algorithms for tracking the homotopy zero curve: ordinary differential equation-based, normal flow, and augmented Jacobian matrix. Separate routines are also provided for dense and sparse Jacobian matrices. A high-level driver is included for the special case of polynomial systems.
393 citations
TL;DR: It is shown that the problem can be reduced to that of solving a system of eight second-degree equations in eight unknowns, it-is-fttrther-demrmstrate^-thafmis second- Degree system can be routinely solved using a continuation algorithm.
Abstract: This paper presents a unique approach to the kinematic analysis of the most general six-degree-of-freedom, six-revolute-joint manipulators (Previously, the problem of computing all possible configurations of a manipulator corresponding to a given hand position was approached by reducing the problem to that of solving a high degree polynomial equation in one variable. -In-this-paper Jt is shown that the problem can be reduced to that of solving a system of eight second-degree equations in eight unknowns, it-is-fttrther-demrmstrate^-thafmis second-degree system can be routinely solved using a continuation algorithm. To complete the general analysis, a second numerical method—a continuation heuristic —is shown to generate partial solution sets quickly. Finally, in some special cases, closed form solutions wereobtained for some commonly used industrial manipulators. The results can be applied to the~analysis~of--both-six-~and-~fme'degreeuf-freedmn manipulators composed of mixed revolute and prismatic joints. The numerical stability of continuation on small second-degree systems opens the way for routine use in offline robot programming applications. \\
268 citations
TL;DR: The design of a package of continuation procedures called PITCON to handle the following tasks is described: follow numerically any a priori specified curve on an equilibrium manifold determine the exact location of target points where a given variable has a specified value.
Abstract: The design of a package of continuation procedures called PITCON to handle the following tasks is described' (1) follow numerically any a priori specified curve on an equilibrium manifold; (2) on such a curve determine the exact location of target points where a given variable has a specified value; and (3) on such a curve identify and compute exactly the (simple) limit points where stability may be lost. The process is based on the local parametenzatmn wh]ch uses an estimate of the curvature to control the chome of parameter varmble. Categorms and Subject Descriptors. G 1.5 [Numerical Analysis]Roots of Nonlinear Equations-tteratwe methods, systems of equattons General Terms" Algorithms, Design Addltmnal
212 citations