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High-precision Gauss-Turán quadrature rules for Laguerre and Hermite weight functions. (English) Zbl 1298.65041

Summary: 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.

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
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[1] Gautschi, W.: Orthogonal polynomials: computation and approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2004) · Zbl 1130.42300
[2] Milovanović, GV; Milovanović, GV (ed.), Construction of s-orthogonal polynomials and Turán quadrature formulae, 311-328 (1988), Niš
[3] Milovanović, G.V., Spalević, M.M.: Quadrature formulae connected to σ-orthogonal polynomials. J. Comput. Appl. Math 240, 619-637 (2002) · Zbl 0997.65045 · doi:10.1016/S0377-0427(01)00476-9
[4] Milovanović, G.V., Spalević, M.M., Cvetković, A.S.: Calculation of Gaussian-type quadratures with multiple nodes. Math. Comput. Model. 39, 325-347 (2004) · Zbl 1049.65019 · doi:10.1016/S0895-7177(04)90014-3
[5] Shi, Y.G., Xu, G.: Construction of σ-orthogonal polynomials and gaussian quadrature formulas. Adv. Comput. Math. 27, 79-94 (2007) · Zbl 1122.65026 · doi:10.1007/s10444-007-9033-8
[6] Stroud, A.H., Stancu, D.D.: Quadrature formulas with multiple Gaussian nodes. J. SIAM Numer. Math. Ser. B 2, 129-143 (1965) · Zbl 0141.13803
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