# Closed orbit correction at synchrotrons for symmetric and near-symmetric lattices

## TL;DR

This paper explores how lattice symmetry in synchrotrons simplifies closed orbit correction by using structured matrices, with practical benefits demonstrated through simulations and experiments at FAIR.

## Contribution

It introduces the use of structured Circulant matrices for orbit correction in symmetric and near-symmetric lattices, enhancing computational efficiency and interpretability.

## Key findings

- Symmetric lattices lead to structured orbit response matrices.
- Nearest-Circulant approximation effectively handles broken symmetries.
- Simulations and experiments validate the advantages of symmetry exploitation.

## Abstract

This contribution compiles the benefits of lattice symmetry in the context of closed orbit correction. A symmetric arrangement of BPMs and correctors results in structured orbit response matrices of Circulant or block Circulant type. These forms of matrices provide favorable properties in terms of computational complexity, information compression and interpretation of mathematical vector spaces of BPMs and correctors. For broken symmetries, a nearest-Circulant approximation is introduced and the practical advantages of symmetry exploitation are demonstrated with the help of simulations and experiments in the context of FAIR synchrotrons.