Generalized Tensor Methods For Solving Nonlinear Systems

In recent years,because of the prosperity of artificial intelligence and neural net-work,solving nonlinear systems has become one of the hot topics in the field of science and engineering calculation again,and has attracted more attention.In this paper,we establish a new tensor method(the generalized tensor method)to large-scale nonlinear systems,the key point of which is to solve a tensor equation in each step.Firstly,for weakly nonlinear tensor equations,we analyze two cases whose co-efficient matrix is zero matrix or not,and propose some methods based on the HSS iterations(such as Newton-HSS,Picard-HSS and nonlinear HSS-like methods).Theo-retical analysis indicates that these methods are locally convergent under reasonable as-sumptions.The numerical results show that: when the coefficient matrix is zero matrix,the Newton-HSS method outperforms the Newton-GMRES method? when the coeffi-cient matrix is non-zero matrix,these methods based on HSS iteration is more efficient and stable,especially the nonlinear HSS-like method.Secondly,for the special case of multi-linear systems,symmetric M-systems,by using the idea of Picard and nonlin-ear iteration,we give two methods named HSS-D and HSS-F respectively,and analyze their local convergent properties.Numerical experiments show that the HSS-D method is superior to some existing iteration methods,while HSS-F method is only better than a few methods,and has some limitations.In addition,we prove the root to the nonhomogeneous M-equation is existent and unique,and give an approximate convergence order of the generalized tensor method.Finally,we propose three inexact forms of the generalized tensor method: the Tensor-Picard CG(T-P CG)?Tensor-Picard GMRES(T-P GMRES)and Tensor-nonlinear HSS(T-N HSS)methods.The numerical results show that the T-P CG method is the best,which is basically superior to other methods and the Newton-HSS,Newton-GMRES methods.