tailieunhanh - A textbook of Computer Based Numerical and Statiscal Techniques part 34

A textbook of Computer Based Numerical and Statiscal Techniques part 34. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 316 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES TRAPEZOIDAL RULE Putting n 1 in equation 2 and taking the curve y f x through x0 y0 and x0 y0 as a polynomial of degree one so that differences of order higher than one vanish we get X0 h C 1 . h h J f x dx hI y0 -Ay0 y 2y Vi _ y0 t y0 Vi 2 2 2 x0 v y Similarly for the next sub interval X0 h X0 2h we get X0 2h h X0 nh h j f x dx - yi y2 j f x dx - yn_i Vn X0 h X0 n-1 h Adding the above integrals we get X0 nh j f x dx - Vn V0 2 yi y . yn-1 2 x0 which is known as Trapezoidal rule. J f x dx 2h SIMPSON S ONE-THIRD RULE Putting n 2 in equation 2 and taking the curve through x0 y0 x1 y1 and x2 y2 as a polynomial of degree two so that differences of order higher than two vanish we get 1 a2 y0 Ay0 7 A y0 6 _ x0 6y0 6 y1 - y0 y2 - 2y1 - y0 - y0 4y1 y2 X0 4h h Similarly J f x dx 3 2 . X0 2 h X0 nh h J f x dx 3 yn-2 4yn-1 yn X0 n-2 h Adding the above integrals we get X0 nh J f x dx - y0 yn 4 yi y3 . yn-1 2 y2 y4 . yn-2 3 x0 which is known as Simpson s one-third rule. Note Using the formula the given interval of integration must be divided into an even number of subintervals. NUMERICAL DIFFERENTIATION AND INTEGRATION 317 SIMPSON S THREE-EIGHT RULE Putting n 3 in equation 2 and taking the curve through xo yo x1 y1 x2 y2 and x3 y3 as a polynomial of degree three so that differences of order higher than three vanish we get xo 3h f f x dx - 3h yo 3 Ayo 3 A2yo 1A3yo J _ 2 4 8 x0 - 3h Oo 12 y - yo 6 y2 - 2yi yo y - 3y2 3yi - yo 3h - yo 3y1 3y2 y3 8 xoT 3h Similarly J f x dx y y3 3y4 3y5 y6 . xo 3h xo r6h 3h J f x dx y yn-3 3yn-2 3yn-1 yn xo n-3 h Adding the above integrals we get xo nh 3h I f x dx yQ yn 3 yi y2 y4 y5. yn-2 yn-1 2 y3 y6 . yn-3 8 xo which is known as Simpson s three-eighth rule. Note Using this formula the given interval of integration must be divided into sub-intervals whose number n is a multiple of 3. BOOLE S RULE Putting n 4 in equation 2 and neglecting all differences of order higher than .