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A textbook of Computer Based Numerical and Statiscal Techniques part 13

A textbook of Computer Based Numerical and Statiscal Techniques part 13. 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. | 106 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 1. Construct a forward difference table for the following values x 0 5 10 15 20 25 f x 7 11 14 18 24 32 Sol. Forward difference table for given data is x y Ay A2 y a3 y A4 y a5 y 0 5 10 15 20 25 7 11 14 18 24 32 4 3 4 6 8 -1 1 2 2 2 1 0 -1 -1 0 Example 2. If y x3 x2 - 2x 1 calculate values of y for x 0 1 2 3 4 5 and form the difference table. Alsofind the value of y at x 6 by extending the table and verify that the same value is obtained by substitution. Sol. For x 0 1 2 3 4 5 we get the values of y are 1 1 9 31 73 141. Therefore difference table for these data is as x y Ay A2 y a3 y 0 1 0 1 1 8 8 6 2 9 22 14 6 3 31 42 20 6 4 73 68 26 6 5 141 32 6 100 Because third differences are zero therefore A3y3 6 A2y4 - A2y3 6 A2y4-26 6 A2y4 32 Now A2y4 32 Ay5 - Ay4 32 CALCULUS OF FINITE DIFFERENCES 107 Ay5 - 68 32 Ay5 100 Further Ay5 100 y6 - y5 100 y6 - 141 100 y6 241 Verification For given function x3 x2 - 2x 1 at x 6 y 6 6 3 6 2 - 2 6 1 241 Hence Verified. Example 3. Given f 0 3 f 1 12 f 2 81 f 3 200 f 4 100 and f 5 8. From the difference table and find A5 f 0 . Sol. The difference table for given data is as follows x f x Af x A2 f x A3 f x A 4 f x A5 f x 0 3 9 1 12 69 60 -10 2 81 119 50 -269 -259 755 3 200 -100 -219 227 496 4 100 -92 8 5 8 Hence A5f 0 755. Example 4. Construct the forward difference table given that x 5 10 15 20 25 30 y 9962 9848 9659 9397 9063 8660 and point out the values of A2y10 A4y5. Sol. For the given data forward difference table is as x y Ay a2 y a3 y A4 y 5 9962 -114 10 9848 -189 -75 2 15 9659 -262 -73 1 -1 20 9397 -334 -72 3 2 25 9063 -403 -69 30 8660 From the table A2y10 A4y5 is as A2y10 -73 and A4y5 -1. 108 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 5. Find f 6 given that f 0 -3 f 1 6 f 2 8 f 3 12 the third differences being constant. Sol. For given data we construct the difference table x f x Af x A2f x A3 f x 0 -3 9 1 6 -7 2 9 2 8 2 4 3 12 We have f 6 f 0 6 E6