On some quadrature rules with Gregory end corrections

• Autorzy: Bożek B. , Solak W. , Szydełko Z.
• Języki publikacji: EN

Abstrakty

EN

How can one compute the sum of an infinite series s := a1 + a2 + ź ź ź ? 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 a1 +a2 +ź ź ź+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 ƒ(x) to the value n: an = ƒ(n). In such situations, we can get more accurate estimates if instead of using the upper bounds on the remainder infinite sum R = ƒ(n + 1) + ƒ(n + 2) + . . ., we approximate this remainder by the corresponding integral I of ƒ(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 [3] and [2].

Słowa kluczowe

EN

numerical integration; quadrature formulas; summation of series

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2009
• Tom: Vol. 29, no. 2
• Strony: 117--129
• Opis fizyczny: Bibliogr. 3 poz., wykr., tab.

Twórcy

autor

Bożek B.

autor

Solak W.

autor

Szydełko Z.

• AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30, 30-059 Krakow, Poland,

Bibliografia

• [1] D. Kincaid, W. Cheney, Numerical Analysis, Mathematics of Scientific Computing, 3rd ed., The University of Texas at Austin, Brooks/Cole–Thomson Learning, 2002.
• [2] W. Solak, A remark on power series estimation via boundary corections with parameter, Opuscula Mathematica 19 (1999), 75–80.
• [3] W. Solak, Z. Szydełko, Quadrature rules with Gregory–Laplace end corrections, Journal of Computational and Applied Mathematics 36 (1991), 251–253.