Deterministic complexity analysis of Hermitian eigenproblems

TL;DR

This paper develops deterministic algorithms for Hermitian eigenproblems, improving complexity bounds and derandomizing previous randomized methods, with detailed analysis of divide-and-conquer eigensolvers and stability in finite precision.

Contribution

It provides the first deterministic complexity bounds for Hermitian eigenproblems, matching or improving upon randomized algorithms, and analyzes the stability of classical reduction algorithms in floating point arithmetic.

Findings

Deterministic diagonalization in $O(n^{ ext{omega}} ext{log}(n)+n^2 ext{polylog}(n/ extepsilon))$ operations.

Stable reduction of Hermitian matrices to tridiagonal form in floating point model.

Improved deterministic eigenvalue computation complexity to nearly $O(n^{ ext{omega}} ext{log}(n/ extepsilon))$ bit operations.

Abstract

In this work we revisit the arithmetic and bit complexity of Hermitian eigenproblems. Recently, [BGVKS, FOCS 2020] proved that a (non-Hermitian) matrix can be diagonalized with a randomized algorithm in $O(n^{\omega}\log^2(n/\epsilon))$ arithmetic operations, where $\omega\lesssim 2.371$ is the square matrix multiplication exponent, and [Shah, SODA 2025] significantly improved the bit complexity for the Hermitian case. Our main goal is to obtain similar deterministic complexity bounds for various Hermitian eigenproblems. In the Real RAM model, we show that a Hermitian matrix can be diagonalized deterministically in $O(n^{\omega}\log(n)+n^2\mathrm{polylog}(n/\epsilon))$ arithmetic operations, improving the classic deterministic $\widetilde O(n^3)$ algorithms, and derandomizing the aforementioned state-of-the-art. The main technical step is a complete, detailed analysis of a well-known divide-and-conquer tridiagonal eigensolver of Gu and Eisenstat [GE95], when accelerated with the Fast Multipole Method, asserting that it can accurately diagonalize a symmetric tridiagonal matrix in nearly-$O(n^2)$ operations. In finite precision, we show that an algorithm by Sch\"onhage [Sch72] to reduce a Hermitian matrix to tridiagonal form is stable in the floating point model, using $O(\log(n/\epsilon))$ bits of precision. This leads to a deterministic algorithm to compute all the eigenvalues of a Hermitian matrix in $O\left(n^{\omega}\mathcal{F}\left(\log(n/\epsilon)\right)+n^2\mathrm{polylog}(n/\epsilon)\right)$ bit operations, where $\mathcal{F}(b)\in\widetilde O(b)$ is the bit complexity of a single floating point operation on $b$ bits. This improves the best known $\widetilde{O}(n^3)$ deterministic and $O\left(n^{\omega}\log^2(n/\epsilon)\mathcal{F}\left(\log(n/\epsilon)\right)\right)$ randomized complexities. We conclude with some other useful subroutines and with open problems.