Showing papers on "Homotopy analysis method published in 1986"
TL;DR: Probability one homotopy algorithms as mentioned in this paper are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one, and if constructed and implemented properly, are robust, numerically stable, accurate, and practical.
Abstract: Probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one These methods are theoretically powerful, and if constructed and implemented properly, are robust, numerically stable, accurate, and practical The concomitant numerical linear algebra problems deal with rectangular matrices, and good algorithms require a delicate balance (not always achieved) of accuracy, robustness, and efficiency in both space and time The author's experience with globally convergent homotopy algorithms is surveyed here, and some of the linear algebra difficulties for dense and sparse problems are discussed
159 citations
TL;DR: In this article, a continuation method for finding all solutions to polynomial systems is presented, which features a simpler homotopy than has been previously published, and is shown to be more efficient than the one presented in this paper.
78 citations
TL;DR: In this article, the authors describe two homotopies and their use in the solution of difficult equilibrium stage separation process problems, one based on Newton's method and the other based on the thermodynamic properties of the systems involved.
Abstract: In solving the nonlinear algebraic equations that are normally used to model multicomponent separation processes, one is not guaranteed that Newton's method or any of its relatives will converge to the solution. This paper describes two homotopies and their use in the solution of difficult equilibrium stage separation process problems. The first, the Newton homotopy, is able to solve more problems than a standard implementation of Newton's method but is not the most reliable homotopy. Since the equations being solved no longer model a separation process unit, the Newton homotopy sometimes suffers from intermediate solutions that are physically meaningless. The second, the thermodynamic homotopy, is strongly based on the thermodynamic properties of the systems involved. This new homotopy is able to handle not only the more traditional distillation problems (hydrocarbon systems and mildly nonideal systems), but is also extremely effective at solving azeotropic and extractive distillation problems. The implementation of these methods requires only minor modifications to existing software.
41 citations
TL;DR: This note considers the solution of a linear program, using suitably adapted homotopy techniques of nonlinear programming and equation solving that move through the interior of the polytope of feasible solutions, using quadratic regularizing term in an appropriate metric.
Abstract: In this note, we consider the solution of a linear program, using suitably adapted homotopy techniques of nonlinear programming and equation solving that move through the interior of the polytope of feasible solutions. The homotopy is defined by means of a quadratic regularizing term in an appropriate metric. We also briefly discuss algorithmic implications and connections with the affine variant of Karmarkar's method.
21 citations
TL;DR: In this article, the authors present a complete set of ordinary differential equations for generating solutions to parameterized systems of nonlinear equations over parameter intervals of interest, and a simple finite algorithm for the systematic exact evaluation of higher-order partial derivatives.
8 citations
3 citations
3 citations
TL;DR: The homotopy operators in high-dimensional parameter space are introduced and the high order Chern-Simons-like characteristic polynomials are precisely defined.
2 citations
01 Jan 1986
TL;DR: In this article, homotopy theoretic problems are intro-duced in nonlinear differential geometric control theory such as ho-motopic controls, e-controlled homotopic invariant distributions of vector fields and almoust decoupling of nonlinear systems.
Abstract: Homotopy theoretic problems are for the first time intro-duced in nonlinear differential geometric control theory such as: ho-motopic controls, e-controlled homotopic invariant distributions of vector fields and almoust decoupling of nonlinear systems; (feedback) homotopic equivalence of nonlinear control systems; nonlinear systems that generate control homotopies, etc.
2 citations