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A textbook of Computer Based Numerical and Statiscal Techniques part 25

A textbook of Computer Based Numerical and Statiscal Techniques part 25. 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. | 226 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 1. Using Lagrange s formula find the value of i yx if yi 4 y3 120 y4 340 y5 2544 ii yoif y-30 30 y-12 34 y3 38 yis 42 Sol. i Here xo 1 x1 3 x2 4 x3 5 f x0 4 f xi 120 f x2 340 f xf 2544 Now using Lagrange s interpolation formula we have x - xi x - x2 x x3 x x xo x x2 x x3 f x f x xo - x1 xo - x1 xo - x3 x1 - xo x1 - x2 x1 - x3 x x0 x - x1 x - x3 x2 - x0 x2 - x1 x2 - x3 f x2 x - x0 x - xi x - x2 f x3 - x0 x3 - x1 x3 - x2 x -3 x -4 x -5 x - 1 x -4 x -5 f x - ----77----T7-----T x 4 ---77---V x 120 f x - 1 - 3 1 - 4 1 - 5 3-1 3- 4 3- 5 x -1 x - 3 x - 5 x - 1 x - 3 x - 4 7---TT----77------7 x 340 19--------------2 x 2544 4 -1 4 - 3 4 - 5 5 -1 5 - 3 5 - 4 1 yx -f x - -6 x - 3 x - 4 x - 5 30 x - 1 x - 4 x - 5 - 340 x - 1 x - 3 x - 5 318 x - 1 x - 3 x - 4 ii Here xo - 3o x1 - 12 x2 3 x3 18 yo 3o y1 34 y2 38 y3 42 Now from Lagrange s interpolation formula we have x - x1 x - x2 x - x3 x - x0 x - x2 x - x3 f x - 7------w----------77-------Ä f x0 7-------77------w---------Ä f x1 x0 - x1 x0 - x2 x0 - x3 x1 - x0 x1 - x2 x1 - x3 x - x0 x - x1 x - x3 x2 - x0 x2 - x1 x2 - x3 f x2 x - x0 x - x1 x - x2 x x3 - x0 x3 -x1 x3 - x2 - x 12 x - 3 x - 18 x 30 x 30 x - 3 x -18 x 34 yx -30 12 -30 - 3 -30 -18 -12 30 -12 -3 -12 -18 x 30 x 12 x -18 x 30 x 12 x - 3 3 30 3 12 3 -18 x 38 18 30 18 12 18- 3 x 42 yx - - 0.001052188 x 12 x - 3 x - 18 0.00419753 x 30 x - 3 x - 18 - 0.005117845 x 30 x 12 x - 18 0.001944444 x 30 x 12 x - 3 for x - 0 y0 - - 0.001052188 12 -3 -18 0.00419753 30 -3 -18 - 0.005117895 30 12 -18 0.001944444 30 12 -3 INTERPOLATION WITH UNEQUAL INTERVAL 227 y0 - 0.681817824 6.7999986 33.1636356 - 2.09999952 y0 39.9636546 - 2.781817344 y0 37.1818. Ans. 1 Example 2. If y0 y1 y 2 y3. y9 are consecutive terms of a senes. Prove that y5 56 y4 y6 - 28 y3 y7 8 y2 y8 - y1 y9 Sol. Here the arguments are 1 2 3 . 9 so for these values Lagrange s formula is given by y __ x -1 x - 2 x -