Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces II: numerical stability

Authors:
R. Barrio;S. Serrano
Affiliations:
GME, Dpt. Matemática Aplicada, Facultad de Ciencias, Edificio de Matemáticas, University of Zaragoza, E-50009 Zaragoza, Spain;GME, Dpt. Matemática Aplicada, CPS, University of Zaragoza, E-50015 Zaragoza, Spain
Venue:
Journal of Computational and Applied Mathematics
Year:
2005

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Abstract

In this paper, we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. In a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence. Here, we study the numerical stability of these algorithms for the generation and evaluation of a finite series of Sobolev orthogonal polynomials. Besides, we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence.