Some Efficient Algorithms for Solving Systems of Nonlinear Equations

TLDR: A class of secant methods and a class of methods related to Brown’s methods, but using orthogonal rather than stabilized elementary transformations are introduced to avoid finding a new approximation to the Jacobian matrix of the system at each step, and thus increase the efficiency.

Abstract: We compare the Ostrowski efficiency of some methods for solving systems of nonlinear equations without explicitly using derivatives. The methods considered include the discrete Newton method, Shamanskii’s method, the two-point secant method, and Brown’s methods. We introduce a class of secant methods and a class of methods related to Brown’s methods, but using orthogonal rather than stabilized elementary transformations. The idea of these methods is to avoid finding a new approximation to the Jacobian matrix of the system at each step, and thus increase the efficiency. Local convergence theorems are proved, and the efficiencies of the methods are calculated. Numerical results are given, and some possible extensions are mentioned.