A new algorithm for computing the Geronimus transformation with large shifts

A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. The basic Geronimus transformation with shift alpha transforms the monic Jacobi matrix associated with a measure d mu into the monic Jacobi matrix associated with d mu/(x -aEuro parts per thousand alpha) + C delta(x -aEuro parts per thousand alpha), for some constant C. In this paper we examine the algorithms available to compute this transformation and we propose a more accurate algorithm, estimate its forward errors, and prove that it is forward stable. In particular, we show that for C = 0 the problem is very ill-conditioned, and we present a new algorithm that uses extended precision.