In this paper, we generate the recursion coefficients for rational functions with prescribed poles that are orthonormal with respect to a continuous Sobolev inner product. Using a rational Gauss quadrature rule, the inner product can be discretized, thus allowing a linear algebraic approach. The presented approach involves reformulating the problem as an inverse eigenvalue problem involving a Hessenberg pencil, where the pencil will contain the recursion coefficients that generate the sequence of Sobolev orthogonal rational functions. This reformulation is based on the connection between Sobolev orthonormal rational functions and the orthonormal bases for rational Krylov subspaces generated by a Jordan-like matrix. An updating procedure, introducing the nodes of the inner product one after the other, is proposed and the performance is examined through some numerical examples.

翻译：本文针对具有指定极点的有理函数，生成了其在连续Sobolev内积下正交归一化所需的递推系数。通过采用有理Gauss求积法则，可将该内积离散化，从而允许采用线性代数方法进行处理。所提出的方法将问题重新表述为涉及一个Hessenberg束的反特征值问题，该束将包含生成Sobolev正交有理函数序列的递推系数。此重构基于Sobolev正交有理函数与由类Jordan矩阵生成的有理Krylov子空间的正交基之间的关联。本文提出了一种逐步引入内积节点的更新过程，并通过若干数值算例检验了其性能。