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Showing papers on "Homotopy analysis method published in 1994"


Journal ArticleDOI
J. K. Park1, Byung Man Kwak1
TL;DR: In this paper, a three dimensional contact problem with the orthotropic Coulomb friction is formulated in the form of a system of nonlinear equations, and the nonlinear complementarity formulation derived naturally from the tree dimensional frictional contact phenomenon is used in the numerical analysis without such linearization as previously introduced.
Abstract: A three dimensional contact problem with the orthotropic Coulomb friction is formulated in the form of a system of nonlinear equations The nonlinear complementarity formulation derived naturally from the tree dimensional frictional contact phenomenon is used in the numerical analysis without such linearization as previously introduced The probability-one homotopy method known as a globally convergent zero finding algorithm is implemented as an exact method and applied to each incremental step

25 citations


Journal ArticleDOI
TL;DR: The construction of a symmetric homotopy is presented, where only the paths according to the generating solutions have to be traced during continuation.

21 citations



Journal ArticleDOI
TL;DR: Practical homotopy construction methods are described by presenting estimators to obtain bounds for polynomials over a bounded domain and applications illustrate the usefulness of the approach.

14 citations


Journal ArticleDOI
TL;DR: A homotopy algorithm based on the input normal form characterization of the reduced-order model is developed here and is compared with the Homotopy algorithms based on Hyland and Bernstein's optimal projection equations.
Abstract: In control system analysis and design, finding a reduced-order model, optimal in the L/sup 2/ sense, to a given system model is a fundamental problem. The problem is very difficult without the global convergence of homotopy methods, and a homotopy based approach has been proposed. The issues are the number of degrees of freedom, the well posedness of the finite dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. A homotopy algorithm based on the input normal form characterization of the reduced-order model is developed here and is compared with the homotopy algorithms based on Hyland and Bernstein's optimal projection equations. The main conclusions are that the input normal form algorithm can be very efficient, but can also be very ill conditioned or even fail. >

14 citations



01 Jan 1994
TL;DR: In this paper, a path-tracking method based on the technique of homotopy is proposed for real-time computation of line spectral pairs (LSPs) of incoming speech data.
Abstract: Line spectral pairs (LSPs) provide an alternate parametrization of the analysis and synthesis filters used in linear predictive coding (LPC) of speech. Effective use of LSPs in low bit-rate transmission of speech requires fast and accurate computation of these parameters. Special properties make LSFs highly amenable to computation by a path-tracking method based on the technique of homotopy, which is presented in this report. The system polynomial for the analysis filter is converted to two even-order symmetric polynomials whose roots give the line spectral pairs. The proposed method adaptively computes LSPs of incoming speech data by first defining continuous paths from known roots of the LSP polynomials of a speech frame to unknown roots of the next frame in the sequence. A gradient-search based numerical predictor-corrector procedure, having been initialized appropriately, is then used for tracking these paths in order to to compute the unknown roots. Conditions guaranteeing the existence of continuous paths are established by using a transformation between the s and z planes to translate the rules of root-locus construction in feedback control systems. Finally, simulation results are presented which verify the derived conditions and attest the applicability of the homotopy method for real-time computation of LSPs.

4 citations


Proceedings ArticleDOI
14 Dec 1994
TL;DR: In this article, a new homotopy approach was proposed to avoid the large dimensionality of the previous approaches by efficiently solving a pair of Lyapunov equations coupled by low rank linear operators.
Abstract: Homotopy approaches have previously been developed for synthesizing H/sub 2/ optimal reduced-order models. Some of the previous homotopy were based on directly solving the optimal projection equations, a set of two Lyapunov equations mutually coupled by a nonlinear term involving a projection matrix /spl tau/, that characterize the optimal reduced-order model. These algorithms are numerically robust but suffer from the curse of large dimensionality. Subsequently, gradient-based homotopy algorithms were developed. To make these algorithms efficient and to eliminate singularities along the homotopy path, the basis of the reduced-order model was constrained to minimal parameterization. However the resultant homotopy algorithms sometimes experienced numerical ill-conditioning or failure due to the minimal parameterization constraint. This paper presents a new homotopy approach the algorithm avoids the large dimensionality of the previous approaches by efficiently solving a pair of Lyapunov equations coupled by low rank linear operators. >

3 citations


Proceedings ArticleDOI
30 May 1994
TL;DR: This paper shows using circuit examples and normal forms coupled with codimension arguments, that multi-parameter homotopy methods can avoid period-doubling and cyclic fold bifurcations along solution paths, and find all solutions along a period-Doubling path.
Abstract: This paper applies real and complex multi-parameter homotopy to finding periodic solutions of nonlinear circuits. We show using circuit examples and normal forms coupled with codimension arguments, that multi-parameter homotopy methods can avoid period-doubling and cyclic fold bifurcations along solution paths, and find all solutions along a period-doubling path. We distinguish between circuit-direct and formulation-indirect multiparameter homotopy, and show that the latter (with two real parameters) can avoid period-doubling bifurcations, while the former cannot. >

2 citations


Posted Content
TL;DR: In this paper, a homotopy method on the unit simplex is proposed to compute an economic equilibrium in a pure exchange economy with n−1 commodities, where the excess demand is a continuous function from the n −1 commodity to the (n t 1)-dimensional Euclidean space.
Abstract: In the model of a pure exchange economy with n.}-1 commodities, the excess demand is a continuous function from the n-dimensional unit simplex S" to the (n t 1)-dimensional Euclidean space R"}~. A zero point of this function is a price vector at which the demand is equal to the supply in the economy. Such a price vector yields an economic equilibrium. In this paper, we propose a homotopy method on the unit simplex to compute such an economic equilibrium. This method has a clear economic interpretation. Along the path of generated prices the excess demand of each commodity is a multiple of the difference between the current and initial prices of that commodity.

1 citations


Proceedings ArticleDOI
01 Jan 1994
TL;DR: The experimental results on both the parity checker and encoder/decoder problem show the excellent convergence behavior of homotopy continuation method in contrast with backpropagation algorithm.
Abstract: In this paper, the training of multilayer neural networks is expressed as the problem of solving a system of nonlinear equations. The weights in the network are considered as the variables of the nonlinear equations. Moreover, the nonlinear equations can be solved by using homotopy-based continuation methods after the entire training data are presented to the network. Unlike gradient-based algorithm, it can almost be constructed to be globally convergent. The experimental results on both the parity checker and encoder/decoder problem show the excellent convergence behavior of homotopy continuation method in contrast with backpropagation algorithm. >


17 Jan 1994
TL;DR: In this paper, homotopy algorithms for both full-and reduced-order LQG controller design problems with an H-to infinity constraint on disturbance attenuation are developed, which is enforced by replacing the covariance Lyapunov equation by a Riccati equation whose solution gives an upper boundary on H-squared performance.
Abstract: Homotopy algorithms for both full- and reduced-order LQG controller design problems with an H-to infinity constraint on disturbance attenuation are developed. The H-to infinity constraint is enforced by replacing the covariance Lyapunov equation by a Riccati equation whose solution gives an upper boundary on H-squared performance. The numerical algorithm, based on homotopy theory, solves the necessary conditions for a minimum of the upper bound on H-squared performance. The algorithms are based on two minimal parameter formulations: Ly, Bryson, and Cannon''s 2X2 block parametrization and the input normal Riccati form parametrization. An over-parametrization formulation is also proposed. Numerical experiments suggest that the combination of a globally convergent homotopy method and a minimal parameter formulation applied to the upper bound minimization gives excellent results for mixed-norm H-squared/H-to infinity synthesis. The nonmonocity of homotopy zero curves is demonstrated, proving that algorithms more sophisticated that standard continuation are necessary.

Proceedings ArticleDOI
07 Aug 1994
TL;DR: In this paper, the authors apply real and complex multi-parameter homotopy to finding periodic solutions of power electronic circuits, and distinguish between circuit-direct and formulation-indirect methods.
Abstract: Commonly used methods for calculating periodic steady state, such as forward integration and shooting, may fail for highly nonlinear circuits with multiple solutions and/or multiple time scales. Homotopy continuation methods, because of their potentially large or global regions of convergence, and suitability for finding multiple solutions, have been applied to the calculation of periodic steady state for such systems. This paper applies real and complex multi-parameter homotopy to finding periodic solutions of power electronic circuits. We show that multi-parameter homotopy methods can avoid period-doubling and cyclic fold bifurcations along solution paths, and find all stable and unstable periodic solutions along folding or period-doubling paths. We distinguish between circuit-direct and formulation-indirect homotopy, and show that the latter (with two real parameters) can avoid period-doubling bifurcations, while the former cannot. >