Babolian, E.; Masjed-Jamei, M.; Eslahchi, M. R. On numerical improvement of Gauss–Legendre quadrature rules. (English) Zbl 1062.65028 Appl. Math. Comput. 160, No. 3, 779-789 (2005). Summary: Among all integration rules with \(n\) points, it is well-known that \(n\)-point Gauss-Legendre quadrature rule \[ \int^1_{-1}f(x) dx\simeq\sum^n_{i=1} w_i f(x_i) \] has the highest possible precision degree and is analytically exact for polynomials of degree at most \(2n-1\), where nodes \(x_i\) are zeros of Legendre polynomial \(P_n(x)\), and \(w_i\)’s are corresponding weights. In this paper we are going to estimate numerical values of nodes \(x_i\) and weights \(w_i\) so that the absolute error of introduced quadrature rule is less than a preassigned tolerance \(\varepsilon_0\), say \(\varepsilon_0=10^{-8}\), for monomial functions \[ f(x)=x^j,\quad j=0,1,\dots,2n+1. \] (Two monomials more than precision degree of Gauss-Legendre quadrature rules.) We also consider some conditions under which the new rules act numerically, more accurate than the corresponding Gauss-Legendre rules. Some examples are given to show the numerical superiority of presented rules. Cited in 1 ReviewCited in 17 Documents MSC: 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures Keywords:numerical examples; error bound; numerical integration; method of undetermined coefficients; method of solving nonlinear systems; Gauss-Legendre quadrature rule PDFBibTeX XMLCite \textit{E. Babolian} et al., Appl. Math. Comput. 160, No. 3, 779--789 (2005; Zbl 1062.65028) Full Text: DOI References: [1] M.R. Eslahchi, M. MasjedJamei, E. Babolian, On numerical improvement of Gauss-Lobatto quadrature rules, submitted for publication; M.R. Eslahchi, M. MasjedJamei, E. Babolian, On numerical improvement of Gauss-Lobatto quadrature rules, submitted for publication [2] M. Dehghan, M. MasjedJamei, M.R. Eslahchi, On numerical improvement of closed Newton-Cotes quadrature rules, submitted for publication; M. Dehghan, M. MasjedJamei, M.R. Eslahchi, On numerical improvement of closed Newton-Cotes quadrature rules, submitted for publication [3] Burden, R. L.; Douglas Faires, J., Numerical Analysis (2001), Thomson Learning [4] Davis, P.; Rabinowitz, P., Methods of Numerical Integration (1984), Academic Press: Academic Press New York [5] Golub, G. H.; Welsh, J. H., Calculation of Gauss quadrature rules, Math. Comput, 23, 221-230 (1969) · Zbl 0179.21901 [6] Krylov, V. I., Approximate Calculation of Integrals (1962), Macmillan: Macmillan NY · Zbl 0111.31801 [7] Ralston, A.; Rabinowitz, P., A First Course in Numerical Analysis (1978), McGraw-Hill: McGraw-Hill New York · Zbl 0408.65001 [8] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1993), Springer Verlag: Springer Verlag New York · Zbl 0771.65002 [9] Szego, G., Orthogonal Polynomials (1975), American Mathematical Society · JFM 65.0278.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.