The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However, the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

计算相对于离散内积正交的有理函数序列的递归系数的问题被公式化为Hessenberg矩阵铅笔的反特征值问题。通过有理的Arnoldi迭代和通过使用unit相似变换的更新过程，提出了两种方法来解决该反特征值问题。后者被证明是数值稳定的。通过考虑双线性形式的双正交有理函数，可以概括该问题和两个过程。这导致三对角矩阵铅笔的特征值反问题。三对角铅笔暗示了双正交有理函数的短递归关系，这比正交情况更有效。然而，