Research On Algorithm Of A Class Of Multilinear Systems Based On Levenberg-Marquardt Method

Multilinear system is a special tensor equation,which has a wide application background in physics,numerical solution of differential equation,Markov process and control theory.Therefore,the method of solving this equation has gradually become the focus of scholars.In this paper,the algorithm of multilinear systems based on Levenberg-Marquardt(LM)method is studied.Firstly,the background,development history,application fields,research status at home and abroad and LM method of tensors and tensor equations are introduced.Then,the traditional LM method,the improved LM method and the trust region modified LM method are applied to solve the multilinear system.Then a new preprocessing LM method is proposed to solve multilinear systems with strong ?-tensor coefficients.The convergence of the preprocessed LM method is analyzed,and it is shown that the method is globally convergent and locally quadratic convergent under the condition of local error bounds.Thirdly,The convergence analysis is carried out for the LM method,and the results show that the method is globally and locally quadratic under the condition of local error bounds.Finally,Numerical experiments are carried out.The improved LM method and the trust region modified LM method are applied to solve multilinear systems,and the effectiveness of these methods is illustrated by experimental comparison.For the preprocessed LM method,the effectiveness of the algorithm is illustrated by exploring the classical high-order Markov chain.