tailieunhanh - A textbook of Computer Based Numerical and Statiscal Techniques part 16

A textbook of Computer Based Numerical and Statiscal Techniques part 16. 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. | 136 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Sol. The difference table for the given table is x 104 y 104 Ay 104 A3 y 104 A3 y 104 A4 y 104 A5 y 1 10000 5191 2 15191 5545 354 -24 3 20736 5875 330 0 24 0 4 26611 6205 330 24 24 27 5 32816 6559 354 75 51 -135 6 39375 6988 429 -9 -84 270 7 46363 7408 420 177 186 -270 8 53771 8005 597 93 -84 135 9 61776 8695 690 144 51 10 70471 9529 834 11 80000 Here sum of fifth differences is small which may be neglected and -270 are the adjacent values which are equal in magnitude and opposite in sign. Horizontal lines between these values point out the incorrect functional value 46363 coefficient of first middle term in 1 - p 5 is 10. Error is given by 10e 270 e 27 Hence correct functional value 46363 - 27 10000 . TECHNIQUE TO DETERMINE THE MISSING TERM Let given a set of equidistant values of arguments and its corresponding value of f x . Suppose for n 1 equidistant argument values x a a h a 2h . a nh are given. CALCULUS OF FINITE DIFFERENCES 137 y f x f Xo f Xi f x2 f a nh . . f xn . Let one of the value of f x is missing. Say it f i . To determine this missing value of f x assume that f x can be represented by a polynomial of degree n - 1 since n values of f x are known. Hence An-1 f x constant and Anf x 0 Therefore E - I n f x 0 because A E - 1 En - nC1En-11 nC2 En-21 -. -1 n En-n In f x 0 or Enf x - nC1En-1 f x nC2En-2 f x -. -1 n f x 0 For first tabulated value of x put x 0 Enf 0 - nEn-1 f 0 n n 1 En-2 f 0 -. -1 n f 0 0 or f n -nf n-1 n n-1 f n-2 -. -1 nf 0 0 . 1 In equation 1 except missing term each term is known and hence from this way missing term can be obtained. If two values of f x are mssing then in that case only n - 1 values of f x can be given by a polynomial of degree n -2 . . An-1 f x 0 or E -1 n-1 f x 0. This gives for x 0 the first tabulated value and for x 1 second tabulated value and by solving these two we get the two missing values for given function f x . Similarly method .