Experiments with Conjugate Gradient Algorithms for Homotopy Curve Tracking

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 is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. The HOMPACK algorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. Here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. The test problems used include actual large scale, sparse structural mechanics problems

Taylor-newton homotopy method for computing the depth of flow rate for a channel

Homotopy approximation methods (HAM) can be considered as one of the new methods belong to the general classification of the computational methods which can be used to find the numerical solution of many types of the problems in science and engineering. The general problem relates to the flow and the depth of water in open channels such as rivers and canals is a nonlinear algebraic equation which is known as continuity equation. The solution of this equation is the depth of the water. This paper represents attempt to solve the equation of depth and flow using Newton homotopy based on Taylor series. Numerical example is given to show the effectiveness of the purposed method using MATLAB language

A Domain Decomposition Preconditioner for Hermite Collocation Problems

We propose a preconditioning method for linear systems of equations arising from piecewise Hermite bicubic collocation applied to two dimensional elliptic PDEs with mixed boundary conditions. We construct an efficient, parallel preconditioner for the GMRES method. The main contribution of the paper is a novel interface preconditioner derived in the framework of substructuring and employing a local Hermite collocation discretization for the interface subproblems based on a hybrid fine-coarse mesh. Interface equations based on this mesh depend only weakly on unknowns associated with subdomains. The effectiveness of the proposed method is highlighted by numerical experiments that cover a variety of problems