Iterative Algorithms For Nonlinear Matrix Equations And Tensor Equations

Matrix and tensor equation problems have many applications in modern mathematics,applied physics,engineering technology,and biological sciences.Many of these problems are essentially transformed into matrix or tensor problems.Many scholars have studied this type of problem,and achieved a series of research results.This subject is the study of iterative algorithms for several types of matrix and tensor equations.These equations and algorithms have good theoretical significance and broad application prospects.The main research work in this paper is the iterative algorithm research of the following three types of equations:Problem I,Given matrices Ai,Q?Rp×p Rpxp,where Q is a symmetric positive definite matrix,find X?A +such thatProblem ?.Given tensors Ap,Bp,C?R1×…×IN×I1×…×IN,Ap,Bp are nonsingular tensors,find ??RI1×…×IN×I1×…×IN such thatProblem ?.Given tensors A ?R1×…×IN×I1×…×IN,B?RI1×…×IN×K1×…KP,C?RU1×…×US×I1×…IN,D?RI1×…×IN×V1×…×VT,??RU1×…×US×V1×…×VT,find X?RI1×…×IN×I1×…×IN such thatBased on the principle of fixed point iteration,an inversion-free iterative algorithm for solving nonlinear matrix equation problem ? is given,and the convergence of the algorithm is proved by the principle of positive definiteness of the matrix.Using the idea of Newton's iteration method combined with the LSQR method,an algorithm for solving tensor polynomial equation problem ? is given,and the property of local at least second-order convergence of the algorithm is proved.Based on the gradient of the tensor,an iterative algorithm for solving the tensor equation systems ? is proposed,and the characteristic of the finite step termination in the precise calculation of the algorithm is analyzed.There are corresponding numerical experiments to verify the effectiveness of the algorithm for the above problems.