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Lecture 17: Polynomial interpolation. Runge example.

Lecture 17: Polynomial interpolation. Runge example. Download rungepoly.m. Runge phenomenon – polydomial wiggle associated rith high degree polynomial interpolation. function rungepoly(n) % interpolate Runge function with polynomial % estimated at uniformly spaced points

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Lecture 17: Polynomial interpolation. Runge example.

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  1. Lecture 17: Polynomial interpolation. Runge example. Download rungepoly.m Runge phenomenon – polydomial wiggle associated rith high degree polynomial interpolation

  2. function rungepoly(n) % interpolate Runge function with polynomial % estimated at uniformly spaced points % input: n - degree of polynomial % plot runge function xmin=-1; xmax=1; f = inline('1./(1 + 12*x.^2)'); x = linspace(xmin,xmax,200); % interpolation with equidistant points x = linspace(xmax,xmin,n); y = f(x); p=polyfit(x,y,n-1); xvec=linspace(xmin,xmax,500); yvec=polyval(p,xvec); yexact= f(xvec); plot(xvec,yvec,'b',x,y,'ro',xvec,yexact,'r--'); legend('Interpolation','Interpolating points','Exact solution'); end

  3. Result: >> rungepoly(10)

  4. Result: >> rungepoly(20)

  5. Runge example with Chebyshev interpolation. Download rungecheb.m • The nth Chebyshev polynomial is given by • Tn(x) = cos(n arrcos(x)) • All the zeros of Tn(x) lie in [-1,+1] and can be calculated by • xi = cos((2i-1) pi/2n) where odd is an odd integer. • The use of the zeros of the Chebyshev nodes as • interpolation points ensures that the maximum of the • interpolation error will be minimized.

  6. function rungecheb(n) % interpolate Runge function with polynomial % estimated at Chebychev points % input: n - degree of polynomial % plot runge function xmin=-1; xmax=1; f = inline('1./(1 + 12*x.^2)'); x = linspace(xmin,xmax,200); % interpolation with equidistant points x = chebroots(n); y = f(x); p=polyfit(x,y,n-1); xvec=linspace(xmin,xmax,500); yvec=polyval(p,xvec); yexact= f(xvec); plot(xvec,yvec,'b',x,y,'ro',xvec,yexact,'r--'); legend('Interpolation','Interpolating points','Exact solution'); title('interpolation at Chebyshev points') end

  7. function x = chebroots(n) % compute roots of Chebyshev polinomial % % input: n - order of polynomial for k=1:n x(k)=cos( pi*(2*k-1)/(2*n) ); end end

  8. Result: >> rungecheb(10) function Compare to equally spaced points:

  9. Result: >> rungecheb(20) Compare to equally spaced points:

  10. Inclass • The following data are related to the life expectances of citizens of two • European regions. • 1975198019851990 • WE:72.874.275.276.4 • EE:70.270.270.3 71.2 • Interpolate each set of data with a polynomial of degree 3 and • estimate the life expectances in 1970, 1983, and 1988. • Plot each fit along with the data and label your plots.

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