Algorithms for the Geronimus transformation for orthogonal polynomials on the unit circle

Let L̂ be a positive definite bilinear functional on the unit circle defined on Pn, the space of polynomials of degree at most n. Then its Geronimus transformation L is defined by L̂(p, q) = L ( (z − α)p(z), (z − α)q(z) ) for all p, q ∈ Pn, α ∈ C. Given L̂, there are infinitely many such L which can be described by a complex free parameter. The Hessenberg matrix that appears in the recurrence relations for orthogonal polynomials on the unit circle is unitary, and can be factorized using its associated Schur parameters. Recent results show that the unitary Hessenberg matrices associated with L and L̂, respectively, are related by a QR step. For the analogue on the real line of this so-called spectral transformation, the tridiagonal Jacobi matrices associated with the respective functionals are related by an LR step. In this paper we derive algorithms that compute the new Schur parameters after applying a Geronimus transformation. We present two forward and one backward algorithm. The QR step between unitary Hessenberg matrices plays a central role in the derivation of each of the algorithms, where the main idea is to do the inverse of a QR step. Making use of the special structure of unitary Hessenberg matrices, all the algorithms are efficient and need only O(n) flops. We present several numerical experiments to analyze the accuracy and to explain the behaviour of the algorithms.