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Numerical Algorithms

, Volume 54, Issue 1, pp 101–139 | Cite as

A new algorithm for computing the Geronimus transformation with large shifts

  • Maria Isabel Bueno Cachadina
  • Alfredo Deaño
  • Edward Tavernetti
Open Access
Original Paper
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Abstract

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 α transforms the monic Jacobi matrix associated with a measure dμ into the monic Jacobi matrix associated with dμ/(x − α) + Cδ(x − α), 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 Classifications (2000)

15A21 15A23 05A05 05B25 
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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Maria Isabel Bueno Cachadina
    • 1
  • Alfredo Deaño
    • 2
  • Edward Tavernetti
    • 3
  1. 1.Department of Mathematics and College of Creative StudiesUniversity of CaliforniaSanta BarbaraUSA
  2. 2.DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  3. 3.Department of MathematicsUniversity of CaliforniaDavisUSA

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