Iterative Algorithms For Two Kinds Of Tensor Equations

In this paper,two kinds of tensor equations are studied.A gradient algorithm for Sylvester equation and a nested split conjugate gradient algorithm for tensor equations under Einstein product are proposed respectively.In chapter 2,we study a gradient-based iterative method for solving Sylvester tensor equations.First,we extend the iterative algorithm based on Jacobi gradient to solve Sylvester tensor equations.In order to improve the iterative algorithm based on Jacobi gradient,we further introduce N+1 relaxation factor to accelerate the convergence rate of the algorithm proposed in the previous step,thus obtaining an accelerated algorithm for solving Sylvester tensor equations based on Jacobi gradient.We analyze the convergence of the two algorithms,and the results show that the iterative solution of our algorithm always converges to the exact solution of any initial value under certain condition of step size.In chapter 3,based on the existing conjugate gradient algorithm for solving tensor equations under Einstein product,a nested split conjugate gradient algorithm for solving tensor equations under Einstein product is proposed.The convergence of the proposed algorithm is analyzed,and the convergence of the proposed algorithm is proved theoretically,and the effectiveness of the proposed algorithm is proved by numerical examples.