1. A new algorithm for computing the Geronimus transformation with large shifts
- Author
-
Bueno Cachadina, Maria Isabel, Deaño, Alfredo, and Tavernetti, Edward
- Subjects
Computer Science ,Theory of Computation ,Algebra ,Mathematics, general ,Algorithms ,Numeric Computing ,Geronimus transformation ,Accuracy ,Roundoff error analysis ,Orthogonal polynomials ,Three-term recurrence relations - 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.
- Published
- 2010