TAILIEUCHUNG - A textbook of Computer Based Numerical and Statiscal Techniques part 26

A textbook of Computer Based Numerical and Statiscal Techniques part 26. 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. | 236 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 10. Iffix 1 a - x show that f XQ xi X2 X3 . xn 1 7 and f x0 xi x2 x3 . xn x a xo a xi . a xn 1__ a - x0 a - x1 . a - xn a - x 11. Certain corresponding values of x and log10 x are given as x 300 304 305 307 log10 x Find the log10301 by Lagrange s formula. Ans. 12. The following table gives the normal weights of babies during the first 12 months of life Age in Months 0 2 5 8 10 12 Weight in lbs 15 16 18 21 Find the weight of babies during 5 to months of life. Ans. 13. Find the value of tan 33 by Lagrange s formula if tan 30 tan 32 tan 35 tan 38 . Ans. 14. Apply Lagrange s formula to find 5 and 6 given that 2 4 1 2 3 8 f 7 128. Explain why the result differs from those obtained by completing the series of powers of 2 Ans. 74 2x is not a polynomial ERRORS IN POLYNOMIAL INTERPOLATION Let the function y x defined by the n 1 points x y i 0 1 2 .n be continuous and differentiable n 1 times and let y x be approximated by a polynomial n x of degree not exceeding n such that n xi yi i 1 2 .n. . 1 Now use n x to obtain approximate value of y x at some points other than those defined by 1 . Since the expression y x - n x vanishes for x x0 x1 . xn we put y x - n x L n n 1 x . 2 where n n 1 x x - x0 x - x1 . x - xn . 3 and L is to be determined such that equation 2 holds for any intermediate value of x say x x x0 x xn clearly L y x -t x n. 1 x We construct a function F x such that F x y x - n x - L. nn 1 x . 5 INTERPOLATION WITH UNEQUAL INTERVAL 237 where L is given by 4 . It is clear that F x0 F x1 . F Xn F x 0 that is F x vanished n 2 times in the interval x0 x xn consequently by the repeated application of Rolle s theorem F1 x must vanish n 1 times F11 x must vanish n times etc. in the interval x0 x xn . In particular F n 1 x must vanish once in the interval. Let this point be given by x x0 xn.

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