A Lanczos-type procedure for tensors

TL;DR

This paper introduces a novel tensor-based Lanczos-type algorithm to efficiently compute bilinear forms involving the time-ordered exponential, addressing a key challenge in solving linear non-autonomous ODE systems.

Contribution

It extends the non-Hermitian Lanczos algorithm to 4-mode tensors, providing a new framework for the computation of the time-ordered exponential.

Findings

Theoretical properties of the tensor Lanczos extension are established.

Computational results demonstrate the method's effectiveness on real-world problems.

Abstract

The solution of linear non-autonomous ordinary differential equation systems (also known as the time-ordered exponential) is a computationally challenging problem arising in a variety of applications. In this work, we present and study a new framework for the computation of bilinear forms involving the time-ordered exponential. Such a framework is based on an extension of the non-Hermitian Lanczos algorithm to 4-mode tensors. Detailed results concerning its theoretical properties are presented. Moreover, computational results performed on real world problems confirm the effectiveness of our approach.