Estimating the condition number of Chebyshev filtered vectors with application to the ChASE library
This work addresses a specific computational bottleneck in numerical linear algebra libraries for eigenproblems, offering an incremental improvement to existing methods.
The paper tackles the problem of unknown high condition numbers in Chebyshev filtered vectors during QR-factorization in eigenproblem algorithms, by providing inexpensive upper-bound estimates and implementing a mechanism in the ChASE library that enhances performance without compromising accuracy.
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.