The zeros of type II multiple orthogonal polynomials can be used for quadrature formulas that approximate $r$ integrals of the same function $f$ with respect to $r$ measures $\mu_1,\ldots,\mu_r$ in the spirit of Gaussian quadrature. This was first suggested by Borges in 1994. We give a method to compute the quadrature nodes and the quadrature weights which extends the Golub-Welsch approach using the eigenvalues and left and right eigenvectors of a banded Hessenberg matrix. This method was already described by Coussement and Van Assche in 2005 but it seems to have gone unnoticed. We describe the result in detail for $r=2$ and give some examples.

翻译：第二类多重正交多项式的零点可用于构造求积公式，以高斯求积的思想近似逼近同一函数$f$关于$r$个测度$\mu_1,\ldots,\mu_r$的$r$个积分。这一思想最早由Borges于1994年提出。我们给出一种计算求积节点和求积权重的算法，该算法将Golub-Welsch方法推广至带状Hessenberg矩阵的特征值及左右特征向量的应用场景。此方法虽已于2005年由Coussement与Van Assche提出，但似乎未被学界充分关注。本文针对$r=2$的情形详述该结果并给出若干算例。