Numerical Theories And Methods For Several Classes Of Tensor Equations

Tensor equations as higher-order generalizations of matrix equations have been inten-sively applied in the field of scientific and engineering computing.Especially,the study of solving tensor equations has become a hot topic in numerical algebra due to it can be used as a powerful tool in control theory,data mining,information retrieval,partial differential equations,Markov process,ect.By the Krylov subspace methods and the splitting iterative techniques for matrix equations,many efficient iterative methods were proposed for solving various tensor equations.From the viewpoint of numerical calculation,this dissertation is devoted to proposing some iterative methods in their tensor forms for solving several classes of linear and nonlinear tensor equations,the convergence analysis of which is also presented.We provide some numerical examples to illustrate the feasibility and validity of the algorithms proposed in here.We organize the rest of this dissertation as follows:In Chapter 1,we mainly introduce the background of several classes of tensor equations and some fundamental properties and notations associated with tensors.In Chapter 2,for the discrete Lyapunov tensor equation with respect to n-mode prod-uct,we firstly propose a simple iterative method based on its own structure for obtaining an approximate solution.The original equation is transformed into two unconstrained op-timization subproblems,then a gradient based iterative and its modified version to solve these problems are presented by the hierarchical identification principal.In order to improve their performance,we further develop a residual norm conjugate gradient method based on exact linear search technique for solving the tensor equation.The convergence analysis of all algorithms is also established.Under different conditions,the provided numerical results illustrate that the simple iterative method and the residual norm conjugate gradient method are more efficient.In Chapter 3,for the generalized coupled Sylvester tensor equations with respect to n-mode product,a modified conjugate residual method in its tensor form is derived,and the convergence analysis is then given.Secondly,a tensor form of bi-conjugate gradient stabilized method is devoted to finding an iterative solution of the tensor equations.Several numerical examples are tested to illustrate that our algorithms require less CPU time than some existing methods.In Chapter 4,for the tensor equation A*_NX=B with respect to Einstein product,we firstly propose the Hermitian and skew-Hermitian splitting method in its tensor form for obtaining an iterative solution.Then a new Hermitian method is derived for solving the tensor equation.Convergence analysis shows that the above algorithms are convergent to an exact solution for any initial value.Combining with the Smith technique,two modified algorithms to solve the tensor equation are established by choosing a suitable initial value.Numerical results are given to illustrate that our methods are feasible and effective.In Chapter 5,for the nonlinear symmetric tensor equation Ax^m-1=b,it is transformed into a unconstrained optimization problem.By the Taylor formula and exact linear search technique,we propose the steepest descent and conjugate gradient methods in their tensor forms for solving the transformed problem,and also provide the convergence analysis of the algorithms mentioned above.Compared with some existing methods,the obtained numerical results illustrate that the performance of our methods is much better.In Chapter 6,we conclude this dissertation by giving some remarks and questions.