On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type
نویسندگان
چکیده
Abstract In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of error these Gauss–Kronrod formulae analytic functions are obtained, using techniques contour integration that were introduced by Gautschi and Varga (cf. SIAM J. Numer. Anal. 20 , 1170–1186 1983). Some illustrative numerical examples which show both accuracy sharpness our estimations displayed. Though sake brevity restrict ourselves to first kind weight, similar analysis may be carried out other three type weights; part corresponding computations included in final appendix.
منابع مشابه
A historical note on Gauss-Kronrod quadrature
The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch. The idea of inserting n+1 nodes into an n-point Gaussian quadrature rule and choosing them and the weights of the resulting (2n+1)-point quadrature rule in such a manner as to maximize the polynomial degree of exactness is generally attributed to A.S. Kronrod [2], [3]. This is entirely justi...
متن کاملOn Stieltjes Polynomials and Gauss-kronrod Quadrature
Let D be a real function such that D(z) is analytic and D(z) ± 0 for \z\ < 1. Furthermore, put W(x) = \J\ x2\D(e'v)\2 , x = costp , tp e [0, 71 ], and denote by pn(', rV) the polynomial which is orthogonal on [-1, +1] to Pn_[ (P„_! denotes the set of polynomials of degree at most n 1 ) with respect to W . In this paper it is shown that for each sufficiently large n the polynomial En+X(-, W) (ca...
متن کاملA Family of Gauss - Kronrod Quadrature Formulae
We show, for each n > 1, that the (2ra + l)-point Kronrod extension of the n-point Gaussian quadrature formula for the measure do-^t) = (1 + 7)2(1 t2)^2dt/((l + -y)2 47t2), -K -y < 1, has the properties that its n + 1 Kronrod nodes interlace with the n Gauss nodes and all its 2ra + 1 weights are positive. We also produce explicit formulae for the weights.
متن کاملError Bounds for Gauss-kronrod Quadrature Formulae
The Gauss-Kronrod quadrature formula Qi//+X is used for a practical estimate of the error R^j of an approximate integration using the Gaussian quadrature formula Q% . Studying an often-used theoretical quality measure, for ߣ* , we prove best presently known bounds for the error constants cs(RTMx)= sup \RlK+x[f]\ ll/(l»lloo<l in the case s = "Sn + 2 + tc , k = L^J LfJ • A comparison with the Ga...
متن کاملComputation of Gauss-Kronrod quadrature rules
Recently Laurie presented a new algorithm for the computation of (2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n+ 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to b...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Numerical Algorithms
سال: 2022
ISSN: ['1017-1398', '1572-9265']
DOI: https://doi.org/10.1007/s11075-022-01325-8