Summary: Noname manuscript No.
(will be inserted by the editor)
A new algorithm for computing the Geronimus
transformation with large shifts
M. I. Bueno · A. Dea~no · E. Tavernetti
Received: date / Accepted: date
Abstract A monic Jacobi matrix is a tridiagonal matrix which contains the parame-
ters 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
transforms the monic Jacobi matrix associated with a measure dµ into the monic Ja-
cobi matrix associated with dµ/(x-)+C(x-), 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.
Keywords Geronimus transformation · accuracy · roundoff error analysis · orthogonal
polynomials · three-term recurrence relations.
Mathematics Subject Classification (2000) 15A21 · 15A23 · 05A05 · 05B25
The first author's work was supported by Direcci“on General de Investigaci“on (Ministerio de
Ciencia y Tecnolog“ia) of Spain under grant MTM2006-06671. The third author's work was