Lecture23 - 1 Engineering Analysis ENG 3420 Fall 2009 Dan...

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Unformatted text preview: 1 Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00 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 non-linear 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 n-1 In general, an ( n-1) th Newton interpolating polynomial has all the terms of the ( n-2) 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 First-order Lagrange interpolating polynomial The first-order 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.

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Lecture23 - 1 Engineering Analysis ENG 3420 Fall 2009 Dan...

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