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, even though he does not mention multiple orthogonality. 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。这一思路最初由Borges于1994年提出，尽管他并未提及多重正交性。我们给出了一种计算求积节点与求积权值的方法，该方法将Golub-Welsch算法推广至带状Hessenberg矩阵的特征值及左右特征向量的应用场景。该法曾由Coussement与Van Assche于2005年描述，但似乎未被学界充分注意。我们针对$r=2$的情形详细阐述了这一结果，并给出了若干数值实例。