Showing papers by "Liqing Zhang published in 1993"
TL;DR: In this article, two structure-variable homotopy algorithms called local straightenup method and global straighten-up method are developed to approximate the optimal homotope, and an adaptive step-size control strategy for these algorithms is proposed respectively.
Abstract: This paper deals with the monotone homotopy methods for solving the system of nonlinear equations. In Sect. 2 the homotopy with a distancemonotone homotopy path is discussed and it is proved that a homotopy with a given structure has a distance monotone homotopy path under some regular conditions. Then two structure-variable homotopy algorithms called local straighten-up method and global straighten-up method are developed to approximate the optimal homotopy, and an adaptive step-size control strategy for these algorithms is proposed respectively. A new method for analyzing the convergence is presented, which is based on the geometrical properties of the algorithm. Then the convergence of the algorithm is proved under certain regular conditions. Finally two numerical examples are given to illustrate the effectiveness of the above two algorithms.
5 citations
TL;DR: In this article, a two-sided approximation to the periodic orbit of an autonomous ODE is considered, where the explicit difference solution and the implicit one from the inner part are compared.
Abstract: A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.
2 citations
TL;DR: It is proved that a modified structure variable homotopy algorithm, called the descent structure variable Homotopy, or DSVH, algorithm, converges globally and quadratically in the neighborhood of the solution under the hypothesis of the nonsingularity of the nonlinear systems.
1 citations