tailieunhanh - A textbook of Computer Based Numerical and Statiscal Techniques part 21

A textbook of Computer Based Numerical and Statiscal Techniques part 21. 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. | 186 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Take the mean of equation 1 and 2 rz A_ 0 f IL -1 UU 0w U U-1 K f 0 A2 f I 1 j U - 2 1 2 IAf 0 2 s 2 u -1 r u - 2 u 1 1 3 u 1 u u - 1 u - 2 f A4 f -1 A4 f -2 Ï --------S---------------r A f -1 ----------------------------------------- 3 I 2 I 4 2 u 1 u u - 1 u - 2 5 u - 3 A5 f -1 u 2 A5 f -2 2 f u f 0 f 1 u - i Af 0 u u-1 ÎA2f 0 A2f -1 I u u 1 fU 2L n Sr A t -1 2 I 2 I 3 447 u 1 u u - 1 u - 2 A f -1 A f -2 4 2 v u 1 u u -1 u - 2 u -12 -----------5---------- A5 f -2 . This formula is very useful when u - and gives best result when u . 2 4 4 Laplace-Everett s Formula Gausss forward formula is given by f u - f 0 uAf 0 u u A2 f -1 u 1 u u -1 A3 f -1 u 1 u u - 1 u - 2 A4 f -2 2 3 4 u 2 u 1 u u - 1 u 2 a 5 f -2 u 1 u u - 1 u - 2 u - 3 a6 f -2 1 5 5 We know Af 0 - f 1 - f 0 A3 f -1 -A2 f 0 -A2 f -1 A5 f -2 -A4 f -1 -A4 f -2 Therefore using this in equation 1 we get f u - f 0 u f 1 - f 0 u u-1 A2 f -1 u 1 u u -1 A2 f 0 - A2 f -1 2 3 u 1 u u -1 u - 2 A4 f -2 u 2 u 1 u u 1 u 2 A4 f _1 _ A4 f _2 4 5 - 1 - u f 0 uf 1 u u A2 f -1 - u 1 u u -1 A2 f -1 2 3 u 1 u u -1 a2f 0 u 2 u 1 u u -1 u - 2 a4f -1 u 1 u u- 1 u-2 a4 f -2 - u 2 u 1 u u-1 u- 2 a4 f -2 4 44 5 44 . INTERPOLATION WITH EQUAL INTERVAL 187 1 - u f 0 uf 1 U U - 1 2 - U A2 f -1 U 1 u u -1 A2 f 0 3 3 u 2 u 1 u u - 1 u - 2 a 4 f -1 u 1 u u - 1 u - 2 3 - u a 4 f -2 5 t 5 J f u uf 1 U U 1 U -1 A2 f 0 U 2 U 1 U U - 1 U - 2 A4 f -1 . j 1 - U f 0 U U -1 2 - U A2f -1 U 1 U U - 15 U - 2 3 - U A4f -2 . . 2 Substitute 1 - u w in second part of equation 2 f u uf 1 U 1 u u -1 A2 f 0 U 2 U 1 u u - 1 U - 2 A4 f -1 j J I J 3 5 J J wf 0 w - 1 w w 1 A 2 f -1 w 2 w 1 w w - 1 w - 2 A 4 f -2 3 5 1 This is called Laplace-Everett s formula. It gives better estimate value when u . Example 1. From the following table find the value of e117 using Gauss forward formula x 1 ex f x Sol. The difference table is