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，该公式以高斯求积的精神近似同一函数$f$关于$r$个测度$\mu_1,\ldots,\mu_r$的$r$个积分。这一思想最初由Borges于1994年提出，尽管他并未提及多重正交性。我们给出一种计算求积节点和求积权重的方法，该方法通过使用带状Hessenberg矩阵的特征值以及左、右特征向量，推广了Golub-Welsch方法。上述方法曾由Coussement和Van Assche于2005年描述，但似乎未被广泛关注。我们详细描述了$r=2$时的结果，并给出若干实例。