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A textbook of Computer Based Numerical and Statiscal Techniques part 24

A textbook of Computer Based Numerical and Statiscal Techniques part 24. 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. | 216 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Difference table is u f u Af u A2f u A3 f u -1 2854 308 0 3162 3926 3618 -6648 1 7088 896 -3030 2 7984 By Everett s formula f .25 0.25 7088 125 X-075 -3030 . 0.75 3162 SX0.75 -25 3618 .1 I 3 JI 5 J 4064 Hence f 30 4064. Example 3. Apply Laplace Everett s formula to find the value of log 2375from the data given below x 21 22 23 24 25 26 log x 1.3222 1.3424 1.3617 1.3802 1.3979 1.4150 Sol. Here h 1 We take origin at 23. Now difference table is given by x log x A A2 A3 A4 A5 -2 21 1.3222 0.0202 -1 22 1.3424 0.0193 -0.0009 0.0001 0 23 1.3617 0.0185 -0.0008 0 -0.0001 0.0003 1 24 1.3802 0.0171 -0.0008 0.0002 0.0002 2 25 1.3979 0.0171 -0.0006 3 26 1.4150 INTERPOLATION WITH EQUAL INTERVAL 217 Here h 1 . u x-a 23-75 - 23 0.75 h 1 w 1 - 0.75 0.25 From Laplace Everett formula we have f u Lf 1 1 -1 A2 f 0 2 u 1 u u- 1 u - 2 A4 f -1 .1 3 5 Lf 0 W 1 W W -1 A 2 f -1 w 2 w 1 w w - 1 w - 2 A 4 f -2 I 0.75 z 1.3802 5025 x -0.0008 L75 M0 -025 -1-25 x 0.0002 0.25 1.3617 58 5 1.25 0.25 -0.75 -l.75 x -0.0001 1 6 120 _ 1.035419 0.340455 1.375874 log 2375 log 23.75 x 100 log 23.75 log 100 log 2375 1.375872 2 3.375872 Example 4. Find the value of ex when x 1.748 from the following data x 1.72 1.73 1.74 1.75 1.76 1.77 e x 0.1790 0.1773 0.1755 0.1738 0.1720 0.1703 Sol. Here h 0.01 take origin as 1.74. The difference table for the given data is as x e x A A2 A3 A 4 A 5 1.72 0.1790 -0.0017 1.73 0.1773 -0.0018 -0.0001 0.0002 1.74 0.1755 -0.0017 0.0001 -0.0002 -0.0004 0.0008 1.75 0.1738 -0.0018 -0.0001 -0.0002 0.0004 1.76 0.1720 -0.0017 0.0001 1.77 0.1703 218 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES u 1748 -174 0.8 0.01 w 0.2 f 0.8 0.8 0.1738 0.8 1.8 -0.2 -0.00017 2.8 1 8 0.8 0.0004 0.2 0.1755 -0.8 062 1.2 x 0.0001 1.2 2.2 0i2 o -0.8 -1.8 x -0.0004 0.13904 0.0000816 0.000003225 0.0351 - 0.0000032 0.000002534 0.174224. Example 5. Prove that if third differences are assumed to be constant yx xy1 A2y0 uy0 31 A2y-1 where u 1 - x. .