Theories And Numerical Algorithms For Several Tensor Equations

The problem of solving tensor equations is an important subject in the field of numerical algebra research,which are widely applied in theoretical physics,numerical partial differential equations,data mining,and tensor complementation issues.In this paper,we study systematically the theories and numerical algorithms of several types of tensor equations.In Chapter 2,the tensor square root problem(?)is investigated.The tensor square root problem is a high-order generalization of the matrix square root problem.We design the Newton's method for solving this equation,and the algorithm contains only tensor computations without matricizations.Then the local quadratic convergence theorem of the algorithm is given.Finally,some numerical examples show that our algorithm is feasible.In Chapter 3,we focus on the generalized Sylvester tensor equation(?)We design a based-gradient algorithm to solve such tensor equations.By this iterative algorithm,the solvability of the tensor equation can be automatically determined.When the tensor equation is solvable,we show that the solutions of the tensor equation within the error range can be obtained in finite iteration steps,and by selecting a suitable initial tensor,the minimum F-norm solution of the equation can be obtained.Numerical experiments are performed to illustrate the feasibility of the proposed methods.In Chapter 4,we study the polynomial tensor equation(?)First,we construct a fixed point iterative method to solve the polynomial tensor equation,then we prove that the polynomial tensor equation has a unique positive solution when the right term b is a positive vector,the leading coefficient tensor is a H~+-tensor and B_j(j=2,...,m-1)are nonnegative tensors.Finally,some numerical examples show that our results are true.