# Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors

## TL;DR

This paper introduces a novel numerical integrator for symmetric and anti-symmetric low-rank matrices and Tucker tensors that preserves symmetry properties and exhibits robust error behavior, improving upon existing methods.

## Contribution

It presents a new integrator that maintains symmetry or anti-symmetry in low-rank matrix and tensor approximations, unlike previous projector-splitting methods.

## Key findings

- Exact reproduction of low-rank matrices and tensors.
- Robust error behavior in the presence of small singular values.
- Numerical experiments demonstrating the integrator's effectiveness.

## Abstract

A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. A related algorithm is given for the approximation of symmetric or anti-symmetric time-dependent tensors by symmetric or anti-symmetric Tucker tensors of low multilinear rank. The proposed symmetric or anti-symmetric low-rank integrator is different from recently proposed projector-splitting integrators for dynamical low-rank approximation, which do not preserve symmetry or anti-symmetry. However, it is shown that the (anti-)symmetric low-rank integrators retain favourable properties of the projector-splitting integrators: low-rank time-dependent matrices and tensors are reproduced exactly, and the error behaviour is robust to the presence of small singular values, in contrast to standard integration methods applied to the differential equations of dynamical low-rank approximation. Numerical experiments illustrate the behaviour of the proposed integrators.