A new algorithm for computing the Geronimus transformations for large shifts

Bueno, Maria Isabel, Deaño, Alfredo, Tavernetti, Edward (2009) A new algorithm for computing the Geronimus transformations for large shifts. Numerical Algorithms, 54 . pp. 101-139. ISSN 1017-1398. E-ISSN 1572-9265. (doi:10.1007/s11075-009-9325-9)

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http://dx.doi.org/10.1007/s11075-009-9325-9

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 ? transforms the monic Jacobi matrix associated with a measure d? into the monic Jacobi matrix associated with d?/(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.

Item Type: Article 10.1007/s11075-009-9325-9 Geronimus transformation Accuracy Roundoff error analysis Orthogonal polynomials Three-term recurrence relations Q Science > QA Mathematics (inc Computing science) > QA297 Numerical analysisQ Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics Alfredo Deano-Cabrera 21 Nov 2018 11:07 UTC 30 May 2019 08:21 UTC https://kar.kent.ac.uk/id/eprint/70238 (The current URI for this page, for reference purposes) https://orcid.org/0000-0003-1704-247X