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.

翻译：计算一个离散内部产品理性函数序列的复发系数的问题,是作为 Hessenberg 矩阵铅笔的反二次值问题拟订的。建议采用两种程序,通过理性的 Arnoldi 迭代和采用单一相似性转换的更新程序来解决这一反二次值问题。后者在数字上是稳定的。这个问题和两种程序都通过考虑双线形式的双向合理函数而普遍化。这导致三对角矩阵铅笔的反二次值问题。三对角铅笔意味着双向合理函数的短暂重复关系,比正向情况更有效。然而,解决这一问题的程序必须依靠非统一操作,而且可能不是数字稳定的。