Fast Hermitian Diagonalization with Nearly Optimal Precision

TL;DR

This paper introduces a nearly optimal precision Hermitian diagonalization algorithm that operates in near matrix multiplication time, significantly reducing the bits of precision needed compared to previous methods, with practical implications.

Contribution

It provides a new Hermitian diagonalization algorithm requiring substantially fewer bits of precision, with near optimal runtime, improving upon prior general algorithms for matrix diagonalization.

Findings

Requires only lg(1/ε)+O(log(n)+loglog(1/ε)) bits of precision.

Runs in near matrix multiplication time.

For n=4000, ε=10^{-15}, 92 bits suffice, compared to over 600 billion bits in previous work.

Abstract

Algorithms for numerical tasks in finite precision simultaneously seek to minimize the number of floating point operations performed, and also the number of bits of precision required by each floating point operation. This paper presents an algorithm for Hermitian diagonalization requiring only $\lg(1/\varepsilon)+O(\log(n)+\log\log(1/\varepsilon))$ bits of precision where $n$ is the size of the input matrix and $\varepsilon$ is the target error. Furthermore, it runs in near matrix multiplication time. In the general setting, the first complete analysis of the stability of a near matrix multiplication time algorithm for diagonalization is that of Banks et al. [BGVKS20]. They exhibit an algorithm for diagonalizing an arbitrary matrix up to $\varepsilon$ backward error using only $O(\log^4(n/\varepsilon)\log(n))$ bits of precision. This work focuses on the Hermitian setting, where we determine a dramatically improved bound on the number of bits needed. In particular, the result is close to providing a practical bound. The exact bit count depends on the specific implementation of matrix multiplication and QR decomposition one wishes to use, but if one uses suitable $O(n^3)$-time implementations, then for $\varepsilon=10^{-15},n=4000$, we show 92 bits of precision suffice (and 59 are necessary). By comparison, the same parameters in [BGVKS20] does not even show that 682,916,525,000 bits suffice.