AbstractLet L, T, S, and R be closed densely defined linear operators from a Hubert space X into X where L can be factored as L = TS + R. The equation Lu = f is equivalent to the linear system Tv + Ru = f and Su = v. If Lu = f is a two-point boundary value problem, numerical solution of the split system admits cruder approximations than the unsplit equations. This paper develops the theory of such splittings together with the theory of the Methods of Least Squares and of Collocation for the split system. Error estimates in both L2 and L∞ norms are obtained for both methods.
Abstract
This paper presents and compares two optimal synthesis techniques for direct application in creating a robomech-II, the second manipulator presented in a new class of linkage arms called Robomcchs. The first optimal synthesis approach will solve the problem as a nonlinear optimization, with a subset of the device parameters described in a linear system and solved directly in a least squares sense. The second approach will employ a least squares optimization using Lagrange Multipliers to contend with nonlinear constraints. In this paper, each optimal synthesis procedure is developed for the general case and then applied to robomcch-II through an example.
Three kinds of preconditioners are proposed to accelerate the generalized AOR (GAOR) method for the linear system from the generalized least squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned generalized AOR (PGAOR) methods is better than that of the original GAOR methods. Finally, some numerical results are reported to confirm the validity of the proposed methods.
Splitting least squares and collocation procedures for two-point boundary value problems
