Lecture 33 - Inverse Interpolation Given xs and f(x)s...

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Inverse Interpolation Given x ’s and f ( x )’s – interpolation let us get new f ( x ) from new x What about new x from new f ( x )? Example: f ( x ) = 1/x 1. Switch x and f ( x ) and do new interpolation. However, non-uniform spacing in [ x vs. f ( x )] often leads to oscillations in the resulting interpolating polynomial 2. Fit an n th -order polynomial to the original data [ f ( x ) vs. x ], then use root-finding techniques to find x .
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Extrapolation Extrapolation should be avoided whenever possible. If you have to, extrapolate out just a little bit out.
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Use of a seventh-order polynomial to make a prediction of U.S. population in 2000 based on data from 1920 through 1990 Reasonable to use interpolation, but not the extrapolation Dangers of Extrapolation
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Difficulties with Polynomial Interpolation Humped and Flat Data 2 x 25 1 1 ) x ( f + = Runge’s function
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Runge’s function Runge’s function » x2=-1:0.2:1 x2 = Columns 1 through 7 -1.0000 -0.8000 -0.6000 -0.4000 -0.2000 0 0.2000 Columns 8 through 11 0.4000 0.6000 0.8000 1.0000 » y2=1./(1+25*x2.^2) y2 = Columns 1 through 7 0.0385 0.0588 0.1000 0.2000 0.5000 1.0000 0.5000 Columns 8 through 11 0.2000 0.1000 0.0588 0.0385 » c=Lagrange_coef(x2,y2) c = Columns 1 through 7 0.1035 -1.5830 12.1102 -64.5875 282.5702 -678.1684 282.5702 Columns 8 through 11 -64.5875 12.1102 -1.5830 0.1035 » x=-1:0.02:1; y=1./(1+25*x.^2); » t=x; p=Lagrange_eval(t,x2,c); » H=plot(x,y,'r',t,p,'b-',x2,y2,'mo'); » set(H,'LineWidth',2.5) » print -djpeg075 poly5.jpg
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Oscillations Oscillations 2 x 25 1 1 ) x ( f + =
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MATLAB Functions : polyfit and polyval >> x = linspace(-1,1,5); y = 1./(1+25*x.^2) >> xx = linspace(-1,1); p = polyfit(x,y,4) p = 3.3156 0.0000 -4.2772 0.0000 1.0000 >> y4 = polyval(p,xx); >> yr = 1./(1+25*xx.^2); >> H=plot(x,y,'o',xx,y4,xx,yr,'--') >> x=linspace(-1,1,11); y = 1./(1+25*x.^2) y = Columns 1 through 8 0.0385 0.0588 0.1000 0.2000 0.5000 1.0000 0.5000 0.2000 Columns 9 through 11 0.1000 0.0588 0.0385 >> p = polyfit(x,y,10) p = Columns 1 through 8 -220.9417 0.0000 494.9095 -0.0000 -381.4338
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Lecture 33 - Inverse Interpolation Given xs and f(x)s...

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