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# Communications in Mathematical Sciences

## Volume 19 (2021)

### Number 8

### Gradient-based iterative algorithms for the tensor nearness problems associated with Sylvester tensor equations

Pages: 2275 – 2290

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n8.a9

#### Authors

#### Abstract

This paper is concerned with the solution of the tensor nearness problem associated with the Sylvester tensor equation represented by the Einstein product. We first proposed a gradientbased iterative algorithm for the Sylvester tensor equation mentioned above, and then the solution to the tensor nearness problem under consideration can be obtained by finding the least F‑norm solution of another Sylvester tensor equation with special initial iteration tensors. It is shown that the solution to the above tensor nearness problem can be derived within finite iteration steps for any initial iteration tensors in the absence of roundoff errors. The performed numerical experiments show that the algorithm we propose here is efficient.

#### Keywords

Sylvester tensor equation, least F-norm solution, tensor nearness problem

#### 2010 Mathematics Subject Classification

15A69, 65F10

The first author was supported by the National Natural Science Foundation of China (Grant No. 11961057), and the Science Foundation of Tianshui Normal University (Nos. CXT2019-36, CXJ2020-11) as well as the Fuxi Scientific Research Innovation Team Foundation of Tianshui Normal University (No. FXD2020-03).

The second author was supported by National Natural Science Foundation of China (Grant No. 12071196).

Received 13 May 2020

Accepted 23 May 2021

Published 7 October 2021