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A textbook of Computer Based Numerical and Statiscal Techniques part 23

A textbook of Computer Based Numerical and Statiscal Techniques part 23. 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. | 206 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 7. Apply Stirling s formula to find a polynomial of degree four which takes the values of y as given below x i 2 3 4 5 y i -1 1 -1 1 Ans. u4 - 8 u2 1 3 3 8. Apply Stirling s formula to interpolate the value of y at x 1.91 from the following data x 1.7 1.8 1.9 2.0 2.1 2.2 y 5.4739 6.0496 6.6859 7.3851 8.1662 9.0250 Ans. 6.7531 4.5 BESSEL S Example 1. Using Bessel s formula find the value of y at x 3.75 for the data given below x 2.5 3.0 3.5 4.0 4.5 5.0 y 24.145 22.043 20.225 18.644 17.262 16.047 Sol. Difference table for the given data is as x y A A2 A3 A4 A5 -2 2.5 24.145 -2.102 -1 3.0 22.043 -1.818 0.284 -0.047 0 3.5 20.225 -1.581 0.237 -0.038 0.009 -0.003 1 4.0 18.644 -1.382 0.199 -0.032 0.006 2 4.5 17.262 -1.215 0.167 3 5.0 16.047 Here h 0.5 x - a _ 3.75 - 3.5 u h 0.5 . INTERPOLATION WITH EQUAL INTERVAL 207 Now from Bessel s formula we have f u f 0 f 1 2 1 u - 2jAf 0 u u -1 a2f 0 A2f -1 2 2 L 1Ï u u 1 I u -------. A3 f -1 u 1 u u 1 u 2 A4 f 1 A4 f 2 4 2 u 1 u u 1 u 2 u 1 2 5 A5 f 2 . 1 18.644 20.225 0.5 0.5 1.581 0.5 2 05 x 1 1.99 2.37 0 0.5 1 0.5 0.5 2.5 x 16 9 2 0 19.407. Approx. Example 2. Following table gives the values of ex for certain equidistant values of x. Find the value of ex at x 0.644 using Bessel s formula x 0.61 0.62 0.63 0.64 0.65 0.66 0.67 ex 1.840431 1.858928 1.877610 1.896481 1.915541 1.934792 1.954237 Sol. Given h 0.01 take it origin as 0.64 x a _ 0.644 0.64 _ 0.004 h 0.01 0.01 u 0.4 Difference table for the given data is as x f x A A2 A3 A 4 A 5 A6 0.61 1.840431 0.018497 0.62 1.858928 0.018682 0.000165 0.000004 0.63 1.877610 0.18871 0.000189 0 0.000004 0.000006 0.64 1.896481 0.1906 0.000189 0.000002 0.000002 0.000001 0.000007 0.65 1.915541 0.019251 0.000191 0.000003 0.000001 0.66 1.934792 0.019445 0.000144 0.67 1.954237 208 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Bessel s formula f u f 0 f 1 u - 1 Af 0 u u-1 iA2f 0 A2f -1 . 2 A3 f 11 TI 2 f 3 A f -1 u 1 u u- 1 u-2 f A4 f -1