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Unformatted text preview: 1 Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: TuTh 11:0012:00 2 2 Lecture 23 Lecture 23 Attention: The last homework HW5 and the last project are due on Tuesday November 24!! Last time: Linear regression versus sample mean. Coefficient of determination Polynomial least squares fit Multiple linear regression General linear squares More on nonlinear models Interpolation (Chapter 15) Today Lagrange interpolating polynomials Splines Cubic splines Searching and sorting. Next Time More on Splines Numerical integration (chapter 17) 3 Newton interpolating polynomial of degree n1 In general, an ( n1) th Newton interpolating polynomial has all the terms of the ( n2) th polynomial plus one extra. The general formula is: where and the f represent divided differences . f n 1 x ( 29 = b 1 + b 2 x x 1 ( 29 + L + b n x x 1 ( 29 x x 2 ( 29 L x x n 1 ( 29 b 1 = f x 1 ( 29 b 2 = f x 2 , x 1 [ ] b 3 = f x 3 , x 2 , x 1 [ ] M b n = f x n , x n 1 , L , x 2 , x 1 [ ] 4 Divided differences Divided difference are calculated as follows: Divided differences are calculated using divided difference of a smaller number of terms: f x i , x j [ ] = f x i ( 29 f x j ( 29 x i x j f x i , x j , x k [ ] = f x i , x j [ ] f x j , x k [ ] x i x k f x n , x n 1 , L , x 2 , x 1 [ ] = f x n , x n 1 , L , x 2 [ ] f x n 1 , x n 2 , L , x 1 [ ] x n x 1 5 6 Lagrange interpolating polynomials Another method that uses shifted value to express an interpolating polynomial is the Lagrange interpolating polynomial . The differences between a simply polynomial and Lagrange interpolating polynomials for first and second order polynomials is: where the L i are weighting coefficients that are functions of x. Order Simple Lagrange 1 st f 1 ( x ) = a 1 + a 2 x f 1 ( x ) = L 1 f x 1 ( 29 + L 2 f x 2 ( 29 2 nd f 2 ( x ) = a 1 + a 2 x + a 3 x 2 f 2 ( x ) = L 1 f x 1 ( 29 + L 2 f x 2 ( 29 + L 3 f x 3 ( 29 7 Firstorder Lagrange interpolating polynomial The firstorder Lagrange interpolating polynomial may be obtained from a weighted combination of two linear interpolations, as shown. The resulting formula based on known points x 1 and x 2 and the values of the dependent function at those points is: f 1 ( x ) = L 1 f x 1 ( 29 + L 2 f x 2 ( 29 L 1 = x x 2 x 1 x 2 , L 2 = x x 1 x 2 x 1 f 1 ( x ) = x x 2 x 1 x 2 f x 1 ( 29 + x x 1 x 2 x 1 f x 2 ( 29 8 Lagrange interpolating polynomial for n points In general, the Lagrange polynomial interpolation for n points is: where L i is given by: f n 1 x i ( 29 = L i x ( 29 f x i ( 29 i = 1 n L i x ( 29 = x x j x i x j j = 1 j i n 9 10 Inverse interpolation Interpolation find the value f ( x ) for some x between given independent data points....
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This note was uploaded on 02/17/2012 for the course EGN 3420 taught by Professor Staff during the Spring '08 term at University of Central Florida.
 Spring '08
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