# The Optimal Error Bound for the Method of Simultaneous Projections

## TL;DR

This paper derives the best possible linear convergence error bound for the simultaneous projection method on linear subspaces in Hilbert spaces, enhancing understanding of its efficiency and relation to other projection methods.

## Contribution

It provides the first optimal error bound for the simultaneous projection method, expressed via the Friedrichs number, and compares it with existing bounds for the alternating projection method.

## Key findings

- Established the smallest possible convergence rate estimate.
- Expressed the error operator norm in terms of the Friedrichs number.
- Connected the results to the alternating projection method and recent dichotomy theorems.

## Abstract

In this paper we find the optimal error bound (smallest possible estimate, independent of the starting point) for the linear convergence rate of the simultaneous projection method applied to closed linear subspaces in a real Hilbert space. We achieve this by computing the norm of an error operator which we also express in terms of the Friedrichs number. We compare our estimate with the optimal one provided for the alternating projection method by Kayalar and Weinert (1988). Moreover, we relate our result to the alternating projection formalization of Pierra (1984) in a product space. Finally, we adjust our results to closed affine subspaces and put them in context with recent dichotomy theorems.