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

Let Lˆ be a positive definite bilinear functional on the unit circle defined on PnPn, the space of polynomials of degree at most nn. Then its Geronimus transformation LL is defined by Lˆ(p,q)=L((z−α)p(z),(z−α)q(z)) for all p,q∈Pnp,q∈Pn, α∈Cα∈C. Given Lˆ, there are infinitely many such LL 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 LL and Lˆ, respectively, are related by a QR step where all the matrices involved are of order n+1n+1. 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 algorithms 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)O(n) flops. We present several numerical experiments to analyse the accuracy and to explain the behaviour of the algorithms.