A Generalized Projection method for systems of nonlinear equations

A Generalized Projection method for systems of nonlinear equations " (1971). Retrospective Theses and Dissertations. Paper 4896. 1971 Signature was redacted for privacy. In this work a Projection method for solving systems of linear equations is extended to cover the nonlinear case. Projection methods fall into the broad class of minimization methods, which includes the various types of gradient techniques for solving systems of equations. The general idea in minimization methods is to solve the system of equations Fx = 0 by reducing the norm (usually the Euclidean norm) of the residue vector to zero, or, sufficiently close to zero. The tech­ niques are iterative in nature and the approximation vector at the (k+1)®^ step is generally expressed in terms of the approximation vector by the relation + a. n.. where a, is a scalar value and • ' K*K' K. p^ is a vector. In the method presented in this study the p^'s used are the columns of the nxn identity matrix when an n^^ order system is being solved. The are determined at each step so as to reduce the Euclidean norm of the residue vector, R, a maximum amount. The method is a single step method since the choice of the p^'s allows only one compo­ nent of the approximation vector to change at any step. Under the current implementation of the method the Pj^'s are chosen in a definite order-the first component of the approximation vector is altered, then the second, third, etc., down to the n^^ component, for a system of 3 order n. After the component is changed the cycle is repeated starting at the first component again. The process continues until the convergence conditions are satisfied. The stepsize, is determined from an expression which involves inner products of the residue vector and columns of the Jacobian matrix of Fx. A proof of the convergence of the method is presented using basic theorems from functional analysis on continuous mappings. Convergence of the method requires the continuity of the mapping F and the non-singularity of the Jacobian matrix of F on the set of approximation vectors generated by the method. of the subroutines required are given in the Appendix. A set of examples, mainly from journal publications, is given in a set of tables in which computer CPU time and number of cycles are used as comparison norms with two other methods for solving systems …