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Quadrature rules from a RII type recurrence relation and associated quadrature rules on the unit circle

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Abstract

We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special RII recurrence relation. We also look into some methods for generating the nodes (which lie on the real line) and the positive weights of these quadrature rules. With a simple transformation, these quadrature rules on the real line also lead to certain positive quadrature rules of highest algebraic degree of precision on the unit circle. This way, we also introduce new approaches to evaluate the nodes and weights of these specific quadrature rules on the unit circle.

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Acknowledgments

The authors are grateful to the anonymous referees for the constructive comments and suggestions, which helped us to improve the manuscript.

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Correspondence to Cleonice F. Bracciali.

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This work is part of the PhD thesis of the author (JAP) at UNESP and supported by a grant from CAPES of Brazil. The research of the author (CFB) was partially supported by the research grants 305208/2015-2 and 402939/2016-6 from CNPq of Brazil. The research of the author (ASR) was partially supported by the research grant 305073/2014-1 from CNPq and the research grants 2016/09906-0 and 2017/12324-6 from FAPESP of Brazil.

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Bracciali, C.F., Pereira, J.A. & Ranga, A.S. Quadrature rules from a RII type recurrence relation and associated quadrature rules on the unit circle. Numer Algor 83, 1029–1061 (2020). https://doi.org/10.1007/s11075-019-00714-w

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Keywords

  • Orthogonal polynomials on the unit circle
  • Quadrature rules
  • RII type recurrence relation

Mathematics Subject Classification (2010)

  • 42C05
  • 58C40
  • 65D32