A textbook of Computer Based Numerical and Statiscal Techniques part 22

A textbook of Computer Based Numerical and Statiscal Techniques part 22. 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. | 196 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 5. If f x is a polynomial of degree four find the value of f using Gauss s backward formula from the following data f 4 270 f 5 648 Af 5 682 A3f 4 132. Sol. Given Af 5 682 f 6 - f 5 682 f 6 682 648 f 6 1330 Also A3f 4 132 E - 1 3 f 4 132 f 7 - 3f 6 3f 5 - f 4 132 f 7 3 x 1330 - 3 x 648 270 132 f 7 2448 Now form difference table as x f x Af x A2 f x A3 f x 4 270 378 5 648 682 304 132 6 1330 1118 436 7 2448 Take a 6 h 1 a hu u From Gauss backward formula f f 0 uAf -1 A2 f -1 U -1 A3 f -2 1330 x 682 x 436 -02 -1a x 132 2 6 1330 - - Hence f . INTERPOLATION WITH EQUAL INTERVAL 197 Example 6. Using Gauss backward interpolation formula find the population for the year 1936. Given that Year 1901 1911 1921 1931 1941 1951 Population in thousands 12 15 20 27 39 52 Sol. Here h 10. Take origin at 1941 to evaluate population in 1936 x - a 1936 -1941 -5 u ------- ------------ h 10 10 Difference table for given data is as u f u Af u A2f u A 3f u A 4 f u A5f u -4 12 3 -3 15 5 2 0 -2 20 7 2 3 3 -10 -1 27 12 5 -4 -7 0 39 13 1 1 52 Gauss backward formula is u 1 u 2 u 1 u u 1 3 u 2 u 1 u u 1 f u f 0 uAf -1 v 2 A2f -1 1---3-----iA3f -2 ----- L_4_ A4f -2 u 2 u 1 u u - 1 u - 2 a5 f -3 39 x 12 ------ -x 1 --------- -------x -4 2 6 39 - - - thousands Hence the population in 1936 is 32625 thousand. 198 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES PROBLEM SET 1. Given that V12500 V12510 V12520 V12530 . Using Gauss s backward formula show that V12516 2. Find the value of cos 51 421 by Gauss s backward formula from the following data x 50 51 52 53 54 cos x Ans. 3. The population of a town in the years are as follows Year 1931 1941 1951 1961 1971 Population in thousands 15 20 27 39 52 Find the .