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A textbook of Computer Based Numerical and Statiscal Techniques part 31

A textbook of Computer Based Numerical and Statiscal Techniques part 31. 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. | 286 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 1. S x . f xi i 0 1 2 .n 2. On each subinterval xi-1 xi 1 i n S x is a polynomial in n of degree at most n . 3. S x and its n -1 derivatives are continuous on a b . 4. S x is a polynomial of degree one for x a and x b. The process of constructing such type of polynomial is called spline interpolation. 5.7.4 Cubic Spline Interpolation for Equally and Unequally Spaced Values According to the idea of draftsman spline it is required that both and the curvature are dx dx the same for the pair of cubic that join at each point. The cubic spline have possess the following properties 1. S xi f i 0 1 2 . n. 2. The cubic and their first and second derivatives are continuous i.e. S x S1 x and S11 x and continuous on a b 3. On each subintervals xi-1 xi 1 i n S x is a third degree polynomial. 4. The third derivatives of the cubics usually have jumps discontinuities at the ducks or the junction points. FIG. 5.4 Where x for i 0 1 2. n may or may not be equally spaced. Let a cubic polynomial for the ith interval is S xi a x - x b x - x 2 ci x - x d . 1 Since this polynomial is valid for both the points x and xi 1 therefore S x . a x - x . 3 b x - x . 2 c x - x . di . 2 S x d S x 1 a x 1 - x . 3 b x 1 - x . 2 c x 1 - x . d S xi 1 aiC1 bih2i 1 cA l di . 3 where h 1 x 1 - x . INTERPOLATION WITH UNEQUAL INTERVAL 287 Now Twice differentiate Equation 1 we get S x 3ai x - x 2 2b x - x c S xi 6ai x - x . 4 2bi . 4 . 5 Now Let pi S x then equation 5 becomes pi 6a x - x 2bi at x xi Pi p. 2bi hi -2 . 6 at x xi 1 Pi 1 6ai xi 1 - xi 2bi Pi 1 6ai xi 1 - xi Pi Pi 1 6ah 1 P Pi 1 - Pi using 6 ai 6hi 1 . 7 Now substituting the values of d ai and b from 2 6 and 7 in 3 1 . P .O S xM Pi 1 - Pi hM -2 hi 1 Cihi 1 s Xi S xi i pi i - pi P h2 1 cihi i s xi 6 2 P. _p p A n S xi 1 S xi l - - - h i 1 cihi 1 I 6 2 . h2u. i S x 41 S x - Pi 1 - Pi 3Pi Cihi 1 6 ci S xi 1 -S xi 2 P- 1 - Pi 3P- 6 hi 1 S xi 1 -S xi h p- 1 2P . 8 c ci hi 1 Now the slope at the .