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

Generation of orthogonal rational functions by procedures for structured matrices