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The paper presents a method for estimating the condition number of Chebyshev filtered vectors, which arise in the context of Chebyshev filtered subspace iteration for solving algebraic eigenproblems. They derive precise and inexpensive estimates to bound the condition number from above. These estimates are then used to dynamically select the QR-factorization algorithm within the ChASE library, improving its performance without sacrificing accuracy.
Dynamically selecting QR factorization based on condition number estimates dramatically improves the performance of the ChASE library for solving eigenproblems.
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.
