A textbook of Computer Based Numerical and Statiscal Techniques part 27. 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. | 246 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES The second divided difference for x0 x1 x2 is given by xiz x2 - o xi f x0z xi x2 xo xi x2 x2 - X0 The third divided difference for x0 x1 x2 x3 is given by f x0 x1z x2 x3 x0 x1z x2 x3 xi x2z x3 - xoz xiz x2 ---------------------- and so on. x3 - x0 f x0z xiz x2. xn xoz xiz x2. xn x1 xn - x0z xn-i Properties of Divided Differences 1. Divided differences are symmetric with respect to the arguments . independent of the order of arguments. . x0 x1 x1 x0 Also x0 x1 x2 x2 x0 x1 or x1 x2 x0 2. The nth divided differences of a polynomial of nth degree are constant. Let f x A0xn A1xn-1 . An1 x An by a polynomial of degree n provided A0 0 and arguments be equally spaced so that x1 - x0 x2 - x1 .xn - xn1 h Then first divided difference x0z xi yi - y0 xi - x0 Ay0 h Second divided difference x0z xiz x2 i 2 h2 2 A y0 x0z xiz x2 .xn-iz xn Any0 n h Since function is a nth degree polynomial Therefore Any0 constant nth divided difference will also be constant. 3. The nth divided difference can be expressed as the quotient of two determinants of order n 1 . . x0z xi yi xi - y0 - x0 yi y0 x i- x0 xi - x0 yi y0 xi x0 x0z xi i i i i y0 yi y2 x02 xi x2 Similarly x0z xiz x2 x0 xi x2 x0 xi x2 . and so on. i i i i i i The nth divided difference can be expressed as the product of multiple integral. 4. INTERPOLATION WITH UNEQUAL INTERVAL 247 . Relation Between Divided Differences and Ordinary Differences Let the arguments x0 x1 x2 . xn be equally spaced such that x1 - x0 x2 - x1 . xn - xn-1 h x1 x0 h x2 x0 2h xn x0 nh Now first divided difference for arguments x0 x1 be given by A f xo f x1 f x0 f x0 h f x0 Af x0 Similarly second divided difference be given as . 1 1 A2 f x0 ----- f x1 x2 f x0 x1 1 2 x x0 1 rf x2 f x1 f x1 f x0 _ V V V V V 2 x0 x2 x1 x1 x0 f x0 2h f x0 h f x0 h f x0 h h Z7T f x0 2h 2 f x0 h f x0 jL0 2h 2 h 1 A f x0 ------- f xV x3 xV x2 x1x2x3 x2 x0 . 2 1 3h A2f x1 A2f x0 2h2 2h2 A2 f x1 A2 f
