14 - 14. Curve Fitting: 14. Polynomial Interpolation...

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14. Curve Fitting: 14. Curve Fitting: Polynomial Interpolation Polynomial Interpolation
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Nov 5, 2006 10:40 ECOR2606 -- Hassan & Holtz 2 Chapter Objectives Introduction to polynomial interpolation: know how to determine polynomial coefficients using simultaneous eqns. recognize that this process leads to ill-conditioned equations. know how to determine polynomial cofficients and interpolate using MATLAB polyfit() and polyval() . know to perform interpolation with Newton's polynomial. how how to perform interpolation with a Lagrange polynomial. know how to solve an inverse interpolation problem by recasting it as a roots problem. (very briefly) know the dangers of extrapolation. (very briefly) recognize that higher-order polynomials can have large oscillations.
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Nov 5, 2006 10:40 ECOR2606 -- Hassan & Holtz 3 Interpolation given small number of discrete points ( x i ,y i pairs) predict y values at intermediate points i.e., at x values not in the original set e.g.: in tables of data e.g.: pressure of fixed amount & volume of nitrogen, vs temp. what is pressure at T = 100? also, to help solve other problems: replace complex functions with something simpler e.g.: interpolating polynomials for use in numerical integration (Ch. 16) also, in computer graphics: 2-D splines to draw smooth curves 3-D spline surfaces to render realistic CGI (Computer Generated Imagery) -40 0 40 80 120 160 6900 8100 9300 10 550 11 700 12 900 T , ºC p , N/m 2
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Nov 5, 2006 10:40 ECOR2606 -- Hassan & Holtz 4 14.1 Introduction Polynomials: Commonly written: MATLAB: both of these are for an order n-1 polynomial and are equivalent. will use MATLAB notation at first you will also see, from time-to-time, for a n th order polynomial: or Not much consistency. Sorry No real difference between them, anyway. the important thing is that an order m polynomial has m+1 coefficients. f x  = a 1 a 2 x a 3 x 2 ⋯ a n x n 1 (14.1) f x  = p 1 x n 1 p 2 x n 2 p n 1 x p n (14.2) f x  = a 1 a 2 x a 3 x 2 a n 1 x n f x  = p 1 x n p 2 x n 1 p n x p n 1
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Nov 5, 2006 10:40 ECOR2606 -- Hassan & Holtz 5 Introduction (2) Examples of interpolating polynomials: Figure 14.1 a) linear (or 1 st order, or order 1) b) quadratic (or 2 nd order, or order 2) c) cubic (or 3 rd order, or order 3)
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Nov 5, 2006 10:40 ECOR2606 -- Hassan & Holtz 6 14.1.1 Determining Polynomial Coefficients Given 3 points: Determine coefficients of an order 2 interpolating polynomial: Write equations, with coefficients as unknowns: i 1 3 -1 2 5 3 3 6 8 x i y i =f(x i ) f 2 x  = p 1 x 2 p 2 x p 3 1) p 1 × 3 2 p 2 × 3 p 3 × 1 = − 1 2) p 1 × 5 2 p 2 × 5 p 3 × 1 = 3 3) p 1 × 6 2 p 2 × 6 p 3 × 1 = 8
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Nov 5, 2006 10:40 ECOR2606 -- Hassan & Holtz 7 Determining Polynomial Coefficients (2) Or, in matrix form: Solving: >> A = [ 9 3 1; 25 5 1; 36 6 1]; >> y = [-1 3 8]'; >> p = A\y p = 1.00000 -6.00000 8.00000 Thus: [ 9 3 1 25 5 1 36 6 1 ] { p 1 p 2 p 3 } = { 1 3 8 } y = f 2 x  = x 2 6 x 8
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Nov 5, 2006 10:40 ECOR2606 -- Hassan & Holtz 8 Determining Polynomial Coefficients (3) The matrix of previous page is a
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This note was uploaded on 11/13/2011 for the course CIVE 2*** taught by Professor - during the Spring '11 term at Carleton CA.

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14 - 14. Curve Fitting: 14. Polynomial Interpolation...

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