Opuscula Math. 29, no. 2 (2009), 117-129
http://dx.doi.org/10.7494/OpMath.2009.29.2.117

Opuscula Mathematica

On some quadrature rules with Gregory end corrections

Bogusław Bożek
Wiesław Solak
Zbigniew Szydełko

Abstract. How can one compute the sum of an infinite series $$s := a_1 + a_2 + \ldots$$? If the series converges fast, i.e., if the term $$a_n$$ tends to $$0$$ fast, then we can use the known bounds on this convergence to estimate the desired sum by a finite sum $$a_1 + a_2 + \ldots + a_n$$. However, the series often converges slowly. This is the case, e.g., for the series $$a_n = n^{-t}$$ that defines the Riemann zeta-function. In such cases, to compute $$s$$ with a reasonable accuracy, we need unrealistically large values $$n$$, and thus, a large amount of computation. Usually, the $$n$$-th term of the series can be obtained by applying a smooth function $$f(x)$$ to the value $$n$$: $$a_n = f(n)$$. In such situations, we can get more accurate estimates if instead of using the upper bounds on the remainder infinite sum $$R = f(n + 1) + f(n + 2) + \ldots$$, we approximate this remainder by the corresponding integral $$I$$ of $$f(x)$$ (from $$x = n + 1$$ to infinity), and find good bounds on the difference $$I - R$$. First, we derive sixth order quadrature formulas for functions whose 6th derivative is either always positive or always negative and then we use these quadrature formulas to get good bounds on $$I - R$$, and thus good approximations for the sum $$s$$ of the infinite series. Several examples (including the Riemann zeta-function) show the efficiency of this new method. This paper continues the results from [W. Solak, Z. Szydełko, Quadrature rules with Gregory-Laplace end corrections, Journal of Computational and Applied Mathematics 36 (1991), 251–253] and [W. Solak, A remark on power series estimation via boundary corrections with parameter, Opuscula Mathematica 19 (1999), 75–80].

Keywords: numerical integration, quadrature formulas, summation of series.

Mathematics Subject Classification: 65D30, 65D32, 65G99, 65B10.

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• Bogusław Bożek
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
• Wiesław Solak
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
• Zbigniew Szydełko
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
• Revised: 2009-04-08.
• Accepted: 2009-04-08.