Abstract
Procedures and corresponding Matlab software are presented for generating Gauss–Turán quadrature rules for the Laguerre and Hermite weight functions to arbitrarily high accuracy. The focus is on the solution of certain systems of nonlinear equations for implicitly defined recurrence coefficients. This is accomplished by the Newton–Kantorovich method, using initial approximations that are sufficiently accurate to be capable of producing n-point quadrature formulae for n as large as 42 in the case of the Laguerre weight function, and 90 in the case of the Hermite weight function.
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References
Gautschi, W.: Orthogonal polynomials: computation and approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2004)
Milovanović, G.V.: Construction of s-orthogonal polynomials and Turán quadrature formulae. In: Milovanović, G.V. (ed.) Numerical methods and approximation III (Niš, 1987), pp 311–328. Univ. Niš, Niš (1988)
Milovanović, G.V., Spalević, M.M.: Quadrature formulae connected to σ-orthogonal polynomials. J. Comput. Appl. Math 240, 619–637 (2002)
Milovanović, G.V., Spalević, M.M., Cvetković, A.S.: Calculation of Gaussian-type quadratures with multiple nodes. Math. Comput. Model. 39, 325–347 (2004)
Shi, Y.G., Xu, G.: Construction of σ-orthogonal polynomials and gaussian quadrature formulas. Adv. Comput. Math. 27, 79–94 (2007)
Stroud, A.H., Stancu, D.D.: Quadrature formulas with multiple Gaussian nodes. J. SIAM Numer. Math. Ser. B 2, 129–143 (1965)
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Gautschi, W. High-precision Gauss–Turán quadrature rules for Laguerre and Hermite weight functions. Numer Algor 67, 59–72 (2014). https://doi.org/10.1007/s11075-013-9774-z
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DOI: https://doi.org/10.1007/s11075-013-9774-z