Estimating the condition number of Chebyshev filtered vectors with application to the ChASE library

TL;DR

This paper introduces a method to estimate the condition number of Chebyshev filtered vectors efficiently, enabling adaptive QR-factorization choices in the ChASE library to improve performance without losing accuracy.

Contribution

It provides a novel, inexpensive approach to bound the condition number of Chebyshev filtered vectors and integrates this into the ChASE library for optimized QR-factorization.

Findings

The condition number can be accurately bounded with inexpensive estimates.

Adaptive QR-factorization improves performance of the ChASE library.

The method maintains the accuracy of eigenproblem solutions.

Abstract

Chebyshev filtered subspace iteration is a well-known algorithm for the solution of (symmetric/Hermitian) algebraic eigenproblems which has been implemented in several application codes~\cite{Kronik:2006ff, abinit:2020} or in stand alone libraries~\cite{ChASE}. An essential part of the algorithm is the QR-factorization of the array of vectors spanning the active subspace that have been filtered by the Chebyshev filter. Typically such an array has an a-priori unknown high condition number that directly influences the choice of QR-factorization algorithm. In this work we show how such condition number can be bound from above with precise and inexpensive estimates. We then proceed to use these estimates to implement a mechanism for the choice of QR-factorization in the ChASE library. We show how such mechanism enhance the performance of the library without compromising on its accuracy.