$M$-QR decomposition and hyperpower iterative methods for computing outer inverses of tensors

TL;DR

This paper introduces tensor QR decomposition and hyperpower iterative methods under the M-product structure to efficiently compute outer inverses of tensors, with theoretical convergence analysis and numerical validation.

Contribution

It presents a novel approach combining M-QR decomposition and hyperpower iterative methods for tensor outer inverses, including convergence analysis and algorithm design.

Findings

Nineteen-order convergence of the proposed iterative method.

Effective tensor algorithms validated by numerical examples.

Applicability across different tensor modes with M-product structure.

Abstract

The outer inverse of tensors plays increasingly significant roles in computational mathematics, numerical analysis, and other generalized inverses of tensors. In this paper, we compute outer inverses with prescribed ranges and kernels of a given tensor through tensor QR decomposition and hyperpower iterative method under the M-product structure, which is a family of tensor-tensor products, generalization of the t-product and c-product, allows us to suit the physical interpretations across those different modes. We discuss a theoretical analysis of the nineteen-order convergence of the proposed tensor-based iterative method. Further, we design effective tensor-based algorithms for computing outer inverses using M-QR decomposition and hyperpower iterative method. The theoretical results are validated with numerical examples demonstrating the appropriateness of the proposed methods.