# Strictly Balancing Matrices in Polynomial Time Using Osborne's Iteration

## TL;DR

This paper proves that Osborne's iteration can strictly balance matrices in polynomial time across any finite Lp norm, providing the first such guarantee and improving upon previous weak balancing results.

## Contribution

The paper introduces a polynomial-time implementation of Osborne's iteration for strict matrix balancing in any Lp norm, a novel theoretical guarantee.

## Key findings

- Converges to a strictly epsilon-balanced matrix in polynomial time.
- First proof of polynomial-time strict balancing in Lp norms.
- Improves upon previous weak balancing results.

## Abstract

Osborne's iteration is a method for balancing $n\times n$ matrices which is widely used in linear algebra packages, as balancing preserves eigenvalues and stabilizes their numeral computation. The iteration can be implemented in any norm over $\mathbb{R}^n$, but it is normally used in the $L_2$ norm. The choice of norm not only affects the desired balance condition, but also defines the iterated balancing step itself. In this paper we focus on Osborne's iteration in any $L_p$ norm, where $p < \infty$. We design a specific implementation of Osborne's iteration in any $L_p$ norm that converges to a strictly $\epsilon$-balanced matrix in $\tilde{O}(\epsilon^{-2}n^{9} K)$ iterations, where $K$ measures, roughly, the {\em number of bits} required to represent the entries of the input matrix. This is the first result that proves that Osborne's iteration in the $L_2$ norm (or any $L_p$ norm, $p < \infty$) strictly balances matrices in polynomial time. This is a substantial improvement over our recent result (in SODA 2017) that showed weak balancing in $L_p$ norms. Previously, Schulman and Sinclair (STOC 2015) showed strong balancing of Osborne's iteration in the $L_\infty$ norm. Their result does not imply any bounds on strict balancing in other norms.