Gradient based iterative methods for solving symmetric tensor equations

Abstract

The steepest descent and conjugate gradient methods are classical gradient based iterative methods for solving symmetric positive definite linear system Ax=b. In this article, we are concerned with the numerical solution of the tensor equation 𝒜xm−1=b by those well‐known iterations in which 𝒜 is an mth order n‐dimensional symmetric tensor. Then we prove that the developed iterative methods converge locally linearly under some appropriate conditions. Finally, some numerical experiments are provided to illustrate the feasibility and effectiveness of the proposed methods.

Preconditioned tensor format conjugate gradient squared and biconjugate gradient stabilized methods for solving stein tensor equations

“…Recently, many scholars have found that many problems in control theory, partial differential equations, mechanics, and many other problems can be attributed to solving various tensor equations. [8][9][10][11][12][13][14][15][16] Therefore, iterative methods in their tensor forms have become one of the hot topics in numerical algebra. In this article, we consider the following high order Stein tensor equation…”

“…It is well known that the Stein matrix equation $$$$ X+{A}^{(1)}X{\left({A}^{(2)}\right)}^T=B $$$$ is widely used in the fields of optimal or robust stabilization of stochastic linear control systems, filtering theory for discrete‐time large‐scale dynamical systems, and the stability analysis of linear time invariant systems, see, for example, Reference 1‐7. Recently, many scholars have found that many problems in control theory, partial differential equations, mechanics, and many other problems can be attributed to solving various tensor equations 8‐16 . Therefore, iterative methods in their tensor forms have become one of the hot topics in numerical algebra.…”

On RGI Algorithms for Solving Sylvester Tensor Equations

“…It was not merely applied to control theory [12,17,18,34,49], also extensively penetrated into model reduction [4], image processing [8], quantum information [43], disturbance decoupling problem [11] and system identification [15,33,44]. Existing methods for solving (1.2) are classified into two categories: direct methods and iterative methods.…”

A relaxed gradient based iterative algorithm of the generalized Sylvester-conjugate matrix equation over centro-symmetric and centro-Hermitian matrices

“…A family of iterative methods for linear systems and a least-squares iterative solution to coupled matrix equations were presented by Ding and Chen (2005b). Li et al (2021) concerned with numerical solution of the tensor equation AXm−1=b, in which A is an m th order n -dimensional symmetric tensor. Zhang and Wang (2021) concerned with solving high order Sylvester tensor equation arising in Control theory, they also, developed preconditioned iterative algorithms based on the nearest Kronecker product for finding its solution.…”