On the computation of orthogonal rational functions

Several techniques are known to compute a new orthogonal polynomial φk+1 of degree k + 1 from Lk := span{φ0, ..., φk} in case of (discrete) orthogonality on the real line. In the Arnoldi approach one chooses Φk ∈ Lk and makes xΦk orthogonal against φ0, ..., φk. By taking as Φk a linear combination of φk and the kernel (or GMRES) polynomial ψk(x) = ∑k j=0 φj(0)φj(x), one needs to orthogonalize only against φk−2, φk−1, φk, and obtains what in numerical linear algebra is called Orthores, Orthomin or SymLQ [1]. A construction of an orthogonal basis of rational Krylov subspaces for given prescribed poles zj can be done via orthogonal rational functions (ORF) [2], and is required for instance in the approximate computation of matrix functions. Here, following [4], the choice of the continuation vector Φk which is multiplied by x/(x− zk+1) becomes essential, for instance for preserving orthogonality in a numerical setting. By generalizing the techniques of [2, 3], we compare several approaches and find optimal ones.