Abstract
In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described.
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References
A.C. Berti and A. Sri Ranga, Companion orthogonal polynomials: Some applications, Appl. Numer. Math., to appear.
T.S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications Series (Gordon and Breach, London, 1978).
W. Gautschi, On generating Gaussian quadrature rules, in: Numerische Integration, ed. G. Hämmerlin, International Series of Numerical Mathematics, Vol. 45 (Birkhäuser, Basel, 1979) pp. 147–154.
W. Gautschi, A survey of Gauss–Christoffel quadrature formulae, in: E.B. Christoffel – The Influence of His Work in Mathematics and Physical Sciences, eds. P.L. Butzer and F. Fehér (Birkhäuser, Basel, 1981) pp. 72–147.
G.H. Golub and J.H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969) 221–230.
W.B. Jones, W.J. Thron and H. Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980) 503–528.
W.B. Jones and A. Magnus, Computation of poles of two-point Padé approximants and their limits, J. Comput. Appl. Math. 6 (1980) 105–119.
H. Rutishauser, Solution of eigenvalue problems with the LR transformation, in: Further Contributions to the Solution of Simultaneous Linear Equations and Determination of Eigenvalues, National Bureau of Standards Applied Mathematics Series, Vol. 49 (1958) pp. 47–81.
A. Sri Ranga, Another quadrature rule of highest algebraic degree of precision, Numer. Math. 68 (1994) 283–294.
A. Sri Ranga, Symmetric orthogonal polynomials and the associated orthogonal L-polynomials, Proc. Amer. Math. Soc. 123 (1995) 3135–3141.
A. Sri Ranga and E.X.L. de Andrade, Zeros of polynomials which satisfy a certain three term recurrence relation, Comm. Anal. Theory Contin. Fractions 1 (1992) 61–65.
A. Sri Ranga, E.X.L. de Andrade and G.M. Phillips, Associated symmetric quadrature rules, Appl. Numer. Math. 21 (1996) 175–183.
G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. 23 (Amer. Math. Soc., Providence, RI, 1975).
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de Andrade, E., Bracciali, C. & Sri Ranga, A. Gaussian Quadrature Rules with Simple Node-Weight Relations. Numerical Algorithms 27, 61–76 (2001). https://doi.org/10.1023/A:1016797317080
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DOI: https://doi.org/10.1023/A:1016797317080