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.
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The first author’s work was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant MTM2006-06671. The third author’s work was funded by EAF during the UCSB Summer Research Program for Undergraduates in 2007. A. Deaño acknowledges financial support from the Spanish Ministry of Education and Science, under the program of postdoctoral grants (Programa de becas postdoctorales) and project MTM2006-09050.
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Bueno Cachadina, M.I., Deaño, A. & Tavernetti, E. A new algorithm for computing the Geronimus transformation with large shifts. Numer Algor 54, 101–139 (2010). https://doi.org/10.1007/s11075-009-9325-9
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DOI: https://doi.org/10.1007/s11075-009-9325-9
Keywords
- Geronimus transformation
- Accuracy
- Roundoff error analysis
- Orthogonal polynomials
- Three-term recurrence relations