Some Studies On Iterative Algorithms For Several Kinds Of Tensor Equations

Tensor equations are widely used in the fields of finite element,finite difference,spectral method,discretization of high-dimensional linear partial differential equa-tions,tensor complementarity problem,data mining,numerical partial differential equations and other fields.The iterative algorithm based on tensor format,which can overcome the shortcoming of rapidly increasing dimensionality when tensor e-quations are transformed into linear equations,has become one of the hot topics in numerical algebra.Tensor equation is the extension of linear equation and matrix equation.The algorithms for solving linear equations and matrix equations have been widely s-tudied,including the splitting iterative methods and the subspace methods.These algorithms are also extended to solve tensor equation.From the view of numerical calculation,this dissertation studies the algorithms of some linear tensor equations and multi-linear tensor equations by combining numerical algebra methods and opti-mization methods.It mainly focuses on establishing and designing the algorithms of the general solutions and constrained solutions of those linear and multi-linear tensor equations based on tensor format,and proving the convergence of these algorithms.At the same time,numerical experiments verify the effectiveness and validity of these algorithms.In chapter 2,firstly,a modified conjugate gradient(CG)method is designed for solving a class of third-order generalized coupled tensor equation.The convergence of the algorithm is proved and it is extended to the general order generalized coupled tensor equation.Secondly,for a class of generalized coupled Sylvester tensor equa-tion,a BCR algorithm based on tensor form is given,and the convergence of the algorithm is proved.Finally,the feasibility of the algorithm is verified by numerical experiments.In chapter 3,firstly,the definitions of the centrosymmetric tensor and anti-centrosymmetric tensor are given,which are the generalizations of centrosym-metric matrix and anti-centrosymmetric matrix.Secondly,an iterative algorithm for solving a class of generalized Sylvester tensor equations over centrosymmetric and anti-centrosymmetric tensors is proposed and the finite termination is proved in the absence of round off errors.In the case of infinite multiple solutions,the uniqueness of the minimum norm solution for a given special initial solution is proved.Finally,the effectiveness of the CG_BTF algorithm is validated by numerical experimentsIn chapter 4,the conjugate direction algorithm is extended to solve a class of generalized coupled Sylvester tensor equations.Firstly,the transpose of third-order tensor on mode is given and the relationship between the symmetry of third-order tensor and its transposition on mode is proposed.Secondly,a conjugate direction algorithm for solving the symmetric solution of third-order generalized coupled tensor equation is proposed,and the convergence of the algorithm is proved by the properties of transposition on mode.Thirdly,for the minimum norm solution,it is proved that the uniqueness of the solution for the compatible equation under the special initial solution.Finally,numerical experiments are presented to verify the effectiveness of the CD_BTF algorithmIn chapter 5,the iterative algorithms for solving the reflexive and anti-reflexive solutions of a class of generalized coupled tensor equations are studied.Firstly,the general reflexive and anti-reflexive tensors are defined by using quasi-reflexive ma-trices and product on mode between tensors and matrices.Secondly,a modified conjugate direction method is designed to solve the reflexive and anti-reflexive so-lutions of the generalized coupled tensor equations.Thirdly,the convergence of the algorithm is proved.For consistent equations,the minimum norm solution is unique when given a special initial solution.Finally,to conform the effectiveness of the algorithm,we proposed some numerical experiments.In chapter 6,firstly,it is transformed into unconstrained optimization problem for the semi-symmetric multi-linear tensor equation.Secondly,an LM method based on Armijo criterion is presented to solve the unconstrained problem.Thirdly,the global convergence of the algorithm and quadratic convergence under the local error bound condition are proved.Fourthly,as an application,an LM algorithm for solving the H-eigenpairs of a real semi-symmetric tensor is given.At last,numerical results show the effectiveness of the LM algorithmIn chapter 7,For the nonhomogeneous multi-linear tensor equations,a new BFGS algorithm is proposed by improving the correction matrix and the search tech-nique.Secondly,the global convergence of the algorithm is proved on the non-convex conditions for value functions.Finally,the effectiveness of the BFGS algorithm we have proposed is validated through numerical experiments.