Abstract
A symbolic/variable-precision procedure is described (and implemented in Matlab) that generates an arbitrary number N of recurrence coefficients for orthogonal polynomials to any given precision nofdig. The only requirement is the availability of a variable-precision routine for computing the first 2 N moments of the underlying weight function to any precision dig > nofdig. The procedure is applied to Freud, Bose–Einstein, and Fermi–Dirac orthogonal polynomials.
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Cvetković, A., Milovanović, G.V.: The mathematica package “OrthogonalPolynomials”. Facta Univ. (Niš), Math. Inform. 19, 17–36 (2004)
Gautschi, W.: Orthogonal polynomials: computation and approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2004)
Gautschi, W.: Orthogonal polynomials, quadrature, and approximation: computational methods and software (in Matlab). In: Marcellán, F., Van Assche, W. (eds.) Orthogonal Polynomials and Special Functions: Computation and Applications. Lecture Notes in Mathematics, vol. 1883. Springer, Berlin (2006)
Gautschi, W., Milovanović, G.V.: Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series. Math. Comput. 44(169), 177–190 (1985)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 7th edn. Elsevier/Academic, Amsterdam (2007)
Kriecherbauer, T., McLaughlin, K.T.-R.: Strong asymptotics of polynomials orthogonal with respect to Freud weights. Int. Math. Res. Not. 6, 299–333 (1999)
Noschese, S., Pasquini, L.: On the nonnegative solution of a Freud three-term recurrence. J. Approx. Theory 99(1), 54–67 (1999)
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Gautschi, W. Variable-precision recurrence coefficients for nonstandard orthogonal polynomials. Numer Algor 52, 409–418 (2009). https://doi.org/10.1007/s11075-009-9283-2
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DOI: https://doi.org/10.1007/s11075-009-9283-2
Keywords
- Variable-precision recurrence coefficients
- Symbolic Chebyshev algorithm
- Freud orthogonal polynomials
- Bose–Einstein orthogonal polynomials
- Fermi–Dirac oerthogonal polynomials