TAILIEUCHUNG - A textbook of Computer Based Numerical and Statiscal Techniques part 29

A textbook of Computer Based Numerical and Statiscal Techniques part 29. 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. | 266 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES H x Hermite s interpolation formula is 1 - 2 x - x L x L x 2 y g x - x L x f yi 1-2 x - x0 Lo x0 Lo x 2 y0 1 - 2 x - x L xi Li x 2yi x - xo Lo x 2 y o x - xi Li x 2 yi . 1 Now L0 x x x1 x b xo xi a b Li x L x x x0 x a x- xo b a 1 ----- and L x a b 1 1 b a 1 1 7 bi and Lt x. t - o Therefore from equation 1 x x - a H x 1 - 2 ----A a - b Hence L0 x0 2 I f a 1 2 x b b a 2 x a y f b x b a b b a i x-b y2 i x-n y2 x a t f a x b k f b ia b i b a b h i a b l 2 y2 a 1 2 ---- a b a b f a a b 1 2 . a b _ 1 a l 2 b 2__ a b y2 f b b a b a a b a 2__ b 1 7 y2 f a 4 b f b 11 2 f a 2 f b fv - f b 88 f a f b b a f a f b -----1---------. Hence Proved. 2 8 PROBLEM SET 1. Apply Hermite formula to find a polynomial which meets the following specifications xk 0 1 2 Vk 0 1 0 y k 0 0 0 Ans. x4 - 4x3 4x2 INTERPOLATION WITH UNEQUAL INTERVAL 267 2. Apply Hermite s interpolation to find given x 1 Ans. 3. Apply Hermite s interpolation to find log given that x log x 1 x Ans. 4. Determine the Hermite polynomial of degree 5 which fits the following data and hence find an approximate value of loge x y logex 1 y -x Ans. 5. Find y f x by Hermite s interpolation from the table xi yi y i -1 1 -5 0 1 1 1 3 7 Computer y2 and y 2. Ans. 1 x - x2 2x4 y2 31 y2 61 6. Compute e by Hermite s formula for the functionf x ex at the points 0 and 1. Compare the value with the value obtained by using Lagrange s interpolation. Ans. 1 3x 1 - x 2 2 - x ex2 268 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 7. Apply Hermite s formula to find a polynomial which meets the following specifications xi yi yi Ans. 1 5x3 - 3x5 -1 0 1 -1 0 1 0 0 0 8. Apply osculating interpolation formula to find a polynomial which meets the following requirements Xi y yi Ans. x4 - 4x3 4x2 0 1 2 1 0 9 0 0 0 9. .

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