ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

15. Numerical Analysis and Scientific Computing

1991 MS Classification: 65B15, 65D32, 41 Iričanin, Bratislav, University of Belgrade, Yugoslavia On the modification of Euler-Maclaurin summation formula connected with the Chebyshev quadrature formulae Starting from the Euler-Maclaurin formula for calculating the finite sums

S_{n} − \frac{1}{h} ∫_{x_{0}}^{x_{n}}f(x)dx = −\frac{1}{2}(f_{n} −f_{0})+ \frac{h}{12}(fn^{′} −f0^{′})− \frac{h^{3}}{720}(fn^{′′′} −f0^{′′′})+…this paper

proposes the modification of the summation formula S_{n} = \frac{1}{m} ∑_{ν=0}^{n−1} ∑_{i=1}^{m} f_{a_{i}+ν} y

means of the maximal degree of algebraic accuracy. Using some properties of Bernoulli numbers, it is proved that the values a_{i} (which are equal to zero in original Euler-Maclaurin formula) can be acquired from the fol-

lowing system of equations: ∑_{i=1}^{m} a_{i}^{k} = \frac{m}{k+1}, k = 1, 2, 3, …, m, which

is solved. The obtained modified summation formula has some features which lead to the Chebyshev formulae for the numerical integration. Finally, the error term estimation of the obtained summation formula gives:

R ≈ h^{m} ∑_{j=0}^{m} \frac{B_{j}}{(m+2−j)!j!} (f_{n}^{(m)} − f_{0}^{(m)})which is additionally elucidated.

This work was supported by The Mathematical Institute of SASA, Belgrade, Project No.04M03.