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Handbook of mathematics for engineers and scienteists part 121. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 808 Integral Equations 16.1.5. Method of Quadratures 16.1.5-1. Quadrature formulas. The method of quadratures is a method for constructing an approximate solution of an integral equation based on the replacement of integrals by finite sums according to some formula. Such formulas are called quadrature formulas and in general have the form x dx Ai xi e . 16.1.5.1 Ja i i where xi i 1 . n are the abscissas of the partition points of the integration interval a b or quadrature interpolation nodes Ai i 1 . n are numerical coefficients independent of the choice of the function x and en -0 is the remainder the truncation n error of formula 16.1.5.1 . As a rule Ai 0 and C Ai b - a. i 1 There are quite a few quadrature formulas of the form 16.1.5.1 . The following formulas are the simplest and most frequently used in practice. Rectangle rule A1 A-2 An-1 h An - 0 h -b a . - a h i - i n - 1 i - 1 2 . n . 16.1.5.2 Trapezoidal rule Ai - An - 2h A2 - A3 - - An-1 - h h _b a . - a h i - i n-1 i - 1 2 . n . 16.1.5.3 Simpson s rule orprismoidal formula A1 - A2m 1 - 3h A2 - - A2m - 3h A3 - - A2m-1 - 3h h -b a . - a h i - 1 n - 2m 1 n-1 i - 1 . n 16.1.5.4 where m is a positive integer. In formulas 16.1.5.2 - 16.1.5.4 h is a constant integration step. The quadrature formulas due to Chebyshev and Gauss with various numbers of interpolation nodes are also widely applied. Let us illustrate these formulas by an example. Example. For the interval -1 1 the parameters in formula 16.1.5.1 acquire the following values Chebyshev s formula n 6 A1 - A2 - - - n 3 X2 - -X5 - -0.4225186538 X1 --X6 --0.8662468181 X3 - -X4 - -0.2666354015. 16.1.5.5 Gauss formula n - 7 A1 - A7 - 0.1294849662 A2 - A6 - 0.2797053915 A3 - A5 - 0.3818300505 A4 - 0.4179591837 X1 - -X7 - -0.9491079123 X2 --X6 --0.7415311856 16.1.5.6 X3 - -X5 - -0.4058451514 X4 - 0. Note that a vast literature is devoted to quadrature formulas and the reader can find books of interest e.g. see Bakhvalov 1973 Korn and Korn 2000 . 16.1. Linear .
