On some quadrature rules with Gregory end corrections
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 seri...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2009-01-01
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| Series: | Opuscula Mathematica |
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| Online Access: | http://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2910.pdf |
| _version_ | 1811274345051324416 |
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| author | Bogusław Bożek Wiesław Solak Zbigniew Szydełko |
| author_facet | Bogusław Bożek Wiesław Solak Zbigniew Szydełko |
| author_sort | Bogusław Bożek |
| collection | DOAJ |
| description | 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]. |
| first_indexed | 2024-04-12T23:17:23Z |
| format | Article |
| id | doaj.art-7ed4fc0686244568be442ae494ccfd3c |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-04-12T23:17:23Z |
| publishDate | 2009-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-7ed4fc0686244568be442ae494ccfd3c2022-12-22T03:12:38ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742009-01-01292117129http://dx.doi.org/10.7494/OpMath.2009.29.2.1172910On some quadrature rules with Gregory end correctionsBogusław Bożek0Wiesław Solak1Zbigniew Szydełko2AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, PolandAGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, PolandAGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, PolandHow 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].http://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2910.pdfnumerical integrationquadrature formulassummation of series |
| spellingShingle | Bogusław Bożek Wiesław Solak Zbigniew Szydełko On some quadrature rules with Gregory end corrections Opuscula Mathematica numerical integration quadrature formulas summation of series |
| title | On some quadrature rules with Gregory end corrections |
| title_full | On some quadrature rules with Gregory end corrections |
| title_fullStr | On some quadrature rules with Gregory end corrections |
| title_full_unstemmed | On some quadrature rules with Gregory end corrections |
| title_short | On some quadrature rules with Gregory end corrections |
| title_sort | on some quadrature rules with gregory end corrections |
| topic | numerical integration quadrature formulas summation of series |
| url | http://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2910.pdf |
| work_keys_str_mv | AT bogusławbozek onsomequadratureruleswithgregoryendcorrections AT wiesławsolak onsomequadratureruleswithgregoryendcorrections AT zbigniewszydełko onsomequadratureruleswithgregoryendcorrections |