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1

On some quadrature rules with Gregory end corrections

Bożek B., Solak W., Szydełko Z.

Opuscula Mathematica

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2009

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Vol. 29, no. 2

117-129

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].

2

A note on a family of quadrature formulas and some applications

Bożek B., Solak W., Szydełko Z.

Opuscula Mathematica

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2008

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Vol. 28, no. 2

109-121

EN

In this paper a construction of a one-parameter family of quadrature formulas is presented. This family contains the classical quadrature formulas: trapezoidal rule, mid-point rule and two-point Gauss rule. One can prove that for any continuous function there exists a parameter for which the value of quadrature formula is equal to the integral. Some applications of this family to the construction of cubature formulas, numerical solution of ordinary differential equations and integral equations are presented.

3

A remark on power series estimation via boundary corrections with parameter

Solak W.

Opuscula Mathematica

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1999

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Vol. 19

75-80

EN

We propose bilateral estimation of the sum of by means of the integral and the values of the function f (for which f(n) = an) at the appropriately chosen points. The parameters included in the inequality guarantee the stability of the results and its lower limits give the best estimation.

PL

W pracy podano dwustronne oszacowanie sumy szeregu ∑ ∞1 an przy pomocy całki ∫ ∞1 f(x)dx oraz wartości funkcji f(n) = an w stosownie wybranych punktach. Parametry podane w nierówności gwarantują stabilność wyników, ich kresy dolne dają najlepsze oszacowania.