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Estimating the Quadratic Form x T A −m x for Symmetric Matrices: Further Progress and Numerical Computations

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  • Marilena Mitrouli

    (Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, 15784 Athens, Greece
    These authors contributed equally to this work.)

  • Athanasios Polychronou

    (Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, 15784 Athens, Greece
    These authors contributed equally to this work.)

  • Paraskevi Roupa

    (Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, 15784 Athens, Greece
    These authors contributed equally to this work.)

  • Ondřej Turek

    (Department of Mathematics, University of Ostrava, 701 03 Ostrava, Czech Republic
    These authors contributed equally to this work.)

Abstract

In this paper, we study estimates for quadratic forms of the type x T A − m x , m ∈ N , for symmetric matrices. We derive a general approach for estimating this type of quadratic form and we present some upper bounds for the corresponding absolute error. Specifically, we consider three different approaches for estimating the quadratic form x T A − m x . The first approach is based on a projection method, the second is a minimization procedure, and the last approach is heuristic. Numerical examples showing the effectiveness of the estimates are presented. Furthermore, we compare the behavior of the proposed estimates with other methods that are derived in the literature.

Suggested Citation

  • Marilena Mitrouli & Athanasios Polychronou & Paraskevi Roupa & Ondřej Turek, 2021. "Estimating the Quadratic Form x T A −m x for Symmetric Matrices: Further Progress and Numerical Computations," Mathematics, MDPI, vol. 9(12), pages 1-13, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:12:p:1432-:d:577970
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    References listed on IDEAS

    as
    1. Jianqing Fan & Yuan Liao & Han Liu, 2016. "An overview of the estimation of large covariance and precision matrices," Econometrics Journal, Royal Economic Society, vol. 19(1), pages 1-32, February.
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