Insertion of Parametric Anchors to Facilitate Newton-Raphson Iterations Near Pseudo Roots

Xiaojian Lin, Tienmo Shih

Abstract: In a nonlinear system, when both function values $f_i(x_1, x_2, \dots, x_i, \dots, x_n)$ and corresponding Jacobian determinants approach zeros, their ratios may remain finite. Under this circumstance, iterations of the traditional Newton-Raphson method (NRM) tend to wander around local extrema, resulting in non-convergence (not necessarily divergence). Herein, we propose to parametrically modify the given nonlinear system primarily based on the concept of matrix diagonal dominance. In addition to faithfully following the linearization formula of first-order Taylor Series Expansion adopted by NRM, we manage to guide iterations to travel along diminishing-parameter paths that are established by roots of these modified systems. When the parametrized system eventually reverts to the original one, iterated solutions have already passed extrema and approached the desired root. Using four examples governed by scientific and engineering laws, we illustrate the strategy of the proposed algorithm and, in passing, introduce the benefit of finding complex roots. Hopefully, the proposed study will serve as a reference for the community that is interested in using NRM to solve scientific and engineering nonlinear systems.