Calculation method for singular solutions of a class of nonlinear equations and its application

Abstract

To resolve the peculiar problem of the Jacobian matrix for a special class of nonlinear equations,an improved Newton mtheod was proposed based on the dual space.This paper proposed an explicit formula to compute the dual space of an ideal in a point through polynomial multiplication,and constructed augmented equations using the dual space.Meanwhile,the Jacobian matrix of augmented equations at initial point was full rank,and then the algorithm recovered quadratical convergence of Newton's iteration.The experimental results show that after three iterations,the accuracy of computation can achieve 10-15.The proposed method further enriches the theories of the dual space of ideal in algebra geometry and provides a new method for the numerical calculation in engineering applications.