A new dimension—reducing method for solving systems of nonlinear equations

Abstract

A new method for the numerical solution of systems of nonlinear algebraic and/or transcendental equations in R ⁿ is presented. Firstly, 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 and subsequently it perturbs the corresponding Jacobian by using proper perturbation parameters. The remaining component of the solution is evaluated separately using the final approximations of the other components. This reduced iterative formula generates a sequence of points in R ⁿ⁻¹ which converges quadratically to the 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:

• Newton's method
• dimension-reducing methods
• nonlinear SOR
• reduction to one-dimensional equations
• imprecise function values
• systems of nonlinear equations
• numerical solution
• zeros
• quadratic convergence

^∗Part of the work of the second author was done at the European Organization for Nuclear Research (CERN, Geneva) and the National Institute of Nuclear Physics (INFN, Bologna).

^∗Part of the work of the second author was done at the European Organization for Nuclear Research (CERN, Geneva) and the National Institute of Nuclear Physics (INFN, Bologna).

Notes

^∗Part of the work of the second author was done at the European Organization for Nuclear Research (CERN, Geneva) and the National Institute of Nuclear Physics (INFN, Bologna).