A dimension-reducing method for solving systems of nonlinear equations in

Abstract

A method for the numerical solution of systems of nonlinear algebraic and/or transcendental equations in is presented. This method reduces the dimensionality of the system in such a way that it can lead to an iterative approximate formula for the computation of n−1 components of the solution, while the remaining component of the solution is evaluated separately using the final approximations of the other components. This (n−1)-dimensional iterative formula generates a sequence of points in which converges quadratically to n−1 components of the solution. Moreover, it does not require a good initial guess for one component of the solution and it does not directly perform function evaluations, thus it can be applied to problems with imprecise function values. A proof of convergence is given and numerical applications are presented.

Keywords:

• Implicit function theorem
• Newton's method
• reduction to one-dimensional equations
• nonlinear SOR
• m-step SOR-Newton
• imprecise function values
• bisection method
• systems of nonlinear equations
• numerical solution
• zeros
• quadratic convergence

The work of the second author was done at the Department of Mathematics of Cornell University

The work of the second author was done at the Department of Mathematics of Cornell University

Notes

The work of the second author was done at the Department of Mathematics of Cornell University