A textbook of Computer Based Numerical and Statiscal Techniques part 17. 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. | 146 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES _ 1 A2E-1 xux f-E -1 ux _ E E -1 2 E ux 1 2. . _ 2 x E2 - E 1 a ----E----- I ux 1 E E - 1 xE-xux _ 1 AE x u0 _ 1 xC1AE xC2A2E2 . u0 2 n _ uo xC1AEuo A E2uo . _ u0 xC1Au1 xC2A2 u2 . _ . f 1 ux- 1 2 A ux- 3 2 - I I A ux- 5 2 1 A6 Ux - 7 2 _ 1 E-1 2 2 1 1 A2p-3 2 1 2 3 2 f 1 A2 a4e-5 2 uv------A E uv -----------1 A E uY x 2 4 x 2 I 4 x 1 2 3 2 5 2 f 1 T A6E-7 2 . 3 I 4 J x 1E-1 2 _ 2 1X1Y a2e-1 1 -1 2 -3 2 i 1 I 2 JI4 2 4 2 -1 2 -3 2 -5 2 f 1 A2E_1 A3 A E 3 14 j ux _ 1 E 1 2 1 1 A2E-1 -1 2 ux 2 4 x 1E-1 2 2 4E A2 -1 2 4E ux 1 E 1 22E1 2 4 1 A A2 -1 2 ux _ 2 A 2 -1 2 ux 2 A -1 ux 1 E -1 ux _ 1 -E E2 -E3 E4 -E5 . ux _ ux - ux 1 ux 2 - ux 3 ux 4 - ux 5 _ . 3. . _ 1 x n u0 nC1 1 x n-1 x Au0 nC2 1 x n-2 x2 A2u0 1 x xA n u0 1 x 1 A n u0 1 xE n u0 1 nC1 xE nC2x2E2 nC3x3E3 . u0 u0 nC1u1 x nC2u2x2 nC3 u3x3 . . CALCULUS OF FINITE DIFFERENCES 147 1 3 . a m Example 10. Prove that Ax ta x 7 x A iz A x 2 1 1 2 Sol. . A 3 2 A 2 . 2 A2 . xn 1 -1A 2 A 1 A 1 2 xn AE 1 2xn Afx -1 x 1 -1T-i x -1T f x 1T-f x -1 T . 2 I 2 II 2 II 2 FACTORIAL NOTATIONS The product of n consecutive factors each at a constant difference and the first factor being x is called a factorial function or a factorial polynomial of degree n and is defined by x n x x - h x - 2h x - 3h . x - n - 1 n 0 If interval of differencing being unity then x n x x - 1 x - 2 x - 3 . x- n - 1 n 0 Because of their properties this function play an important role in the theory of finite differences and also it helps in finding the various order differences of a polynomial directly by simple rule of differentiation. Example 11. Obtain the function whose first difference is 9x2 11x 5. Sol. Let f x be the required function so that Af x 9x2 11x 5 Let 9x2 11x 5 9 x 2 A x B 9x x - 1 Ax B On substitution x 0 we get B 5 and for x 1 we get A 20. Therefore we have Af x 9 x 2 20 x 5 On integrating we have f x 9 y-
