LOW-MEMORY KRYLOV SUBSPACE METHODS FOR OPTIMAL RATIONAL MATRIX FUNCTION APPROXIMATION

Tyler Chen, Anne Greenbaum, Cameron Musco, Christopher Musco

    Research output: Contribution to journalArticlepeer-review

    Abstract

    We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function's denominator, and it can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature-based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead.

    Original languageEnglish (US)
    Pages (from-to)670-692
    Number of pages23
    JournalSIAM Journal on Matrix Analysis and Applications
    Volume44
    Issue number2
    DOIs
    StatePublished - Jun 2023

    Keywords

    • Krylov subspace method
    • Lanczos
    • low-memory
    • matrix function approximation
    • optimal approximation

    ASJC Scopus subject areas

    • Analysis

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