Geodesic
Quadrature Formulas with High Order Approximation
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 87-100

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In the article the method of creation the quadrature formulas with high order approximation for a wide class of the areas is given. This method is based on approach of smooth function on the plane by the semilocal smoothing spline or S-spline. Semilocal smoothing splines are initiated by D. A. Silaev. Earlier the splines of the third and fifth degree are considered and applied. This work is devoted to use of S-splines of higher degrees. Steady $S$-splines of a class of $C^0$ (only continuous), consisting of polynoms of high degree of $n$ ($n=9, 10$) makes it possible to receive quadrature formulas of the 10th and 11th orders of approximation. It is supposed that integrand function belongs to $C^p$ class (to $p=10, 11$) in a bigger area, than initial area on which integration is conducted. It is also supposed that the border of area is set parametrically that helps to consider area border with a fine precision. Similar approach is possible for the construction of cubature formulas.
Keywords: an approximation; a spline; integrals; quadrature formulas; numerical methods. 
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     author = {D. A. Silaev},
     title = {Quadrature {Formulas} with {High} {Order} {Approximation}},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {87--100},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2013_6_4_a8/}
}
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D. A. Silaev. Quadrature Formulas with High Order Approximation. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 87-100. http://geodesic.mathdoc.fr/item/VYURU_2013_6_4_a8/