Naslov
Euler Integral Identity, Quadrature Formulas and Related Inequalities

Autori
Franjić, Iva ; Pečarić, Josip ; Perić, Ivan

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
LUMS 2^nd International Conference on Mathematics and its Applications in Information Technology, Book of Abstracts / - , 2008

Skup
LUMS 2^nd International Conference on Mathematics and its Applications in Information Technology

Mjesto i datum
Pakistan, 09.03.2008. - 12.03.2008

Vrsta sudjelovanja
Plenarno

Vrsta recenzije
Nije recenziran

Ključne riječi
Euler Integral Identity; quadrature formulas; corrected quadrature formulas; Gauss-type quadrature formulas; Lobatto-type quadrature formulas

Sažetak
The aim of this talk was to develop a unified treatment for deriving generalizations of classical quadrature formulae and preserve the best error estimate at the same time. The main tool used are the extended Euler formulae. General Euler-Ostrowski approximate the integral with the values of the function in m equidistant points so they are in fact a generalization of the extended Euler formulae. Their remainder is expressed in terms of periodic functions $B^{; ; ; \ast}; ; ; _n(x-mt)$. Ostrowski's inequality and some of its generalizations are produced as a special case of the error estimate for these formulae. Further, families of general 3-point, 4-point and 5-point quadrature formulae, are considered. The idea is to adjoin every classical quadrature formula with a corresponding "corrected" quadrature formula. Corrected quadrature formulae have a degree of exactness higher than the "regular" quadrature formulae and beside the values of the function in the same points (but with different coefficients) include in the quadrature values of the first derivative at the end points of the interval of integration. This is not an obstacle when those values are easy to calculate. In fact, if the function obtains the same value in both ends, a quadrature formula with an even higher degree of exactness is produced. The key results which secure the preservation of the best error estimate for the studied formulae are the lemmas that show that functions in terms of which the remainders are expressed, have constant sign. As special cases the following formulae are obtained: Simpson's, dual Simpson's, Maclaurin's, Simpson 3/8, Boole's, the 2 and 3-point Gauss, the 3, 4 and 5-point Lobatto formula and further, for each of these formulae, a corresponding corrected one. The 4-point Gauss formula and the corrected 5-point Lobatto formula are also derived. They could not be obtained as special cases since they both have a degree of exactness higher than the families of quadrature formulae that were studied here. Finally, let us mention a somewhat different approach to Boole type formulae in the last section of the last chapter. Quadratures that include the values of up to $(2n-5)$-th derivative at the end points of the interval and reach the maximum degree of exactness are derived. Based on the so far known pairs of dual quadrature formulae, an identity is obtained which is then taken as a definition of a dual formula. Using this, general dual Boole's formulae are derived.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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