A MATLAB package computing simultaneous Gaussian quadrature rules for Multiple Orthogonal Polynomials
<|reference_start|>A MATLAB package computing simultaneous Gaussian quadrature rules for Multiple Orthogonal Polynomials: The aim of this paper is to describe a Matlab package for computing the simultaneous Gaussian quadrature rules associated with a variety of multiple orthogonal polynomials. Multiple orthogonal polynomials can be considered as a generalization of classical orthogonal polynomials, satisfying orthogonality constraints with respect to $r$ different measures, with $r \ge 1$. Moreover, they satisfy $(r+2)$--term recurrence relations. In this manuscript, without loss of generality, $r$ is considered equal to $2$. The so-called simultaneous Gaussian quadrature rules associated with multiple orthogonal polynomials can be computed by solving a banded lower Hessenberg eigenvalue problem. Unfortunately, computing the eigendecomposition of such a matrix turns out to be strongly ill-conditioned and the \texttt{Matlab} function \texttt{balance.m} does not improve the condition of the eigenvalue problem. Therefore, most procedures for computing simultaneous Gaussian quadrature rules are implemented with variable precision arithmetic. Here, we propose a \texttt{Matlab} package that allows to reliably compute the simultaneous Gaussian quadrature rules in floating point arithmetic. It makes use of a variant of a new balancing procedure, recently developed by the authors of the present manuscript, that drastically reduces the condition of the Hessenberg eigenvalue problem.<|reference_end|>
@article{laudadio2024a,
    title={A MATLAB package computing simultaneous Gaussian quadrature rules for
  Multiple Orthogonal Polynomials},
    author={Teresa Laudadio, Nicola Mastronardi, Walter Van Assche, Paul Van
  Dooren},
    journal={J. Comput. Appl. Math. 451 (2024), 116109},
    year={2024},
    doi={10.1016/j.cam.2024.116109},
    archivePrefix={arXiv},
    eprint={2406.11269},
    primaryClass={math.NA cs.NA}
}