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 % 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 Result: >> rungepoly(10) 1.2 Interpolation Interpolating points Exact solution
1 0.8 0.6 0.4 0.2 0 -0.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Result: >> rungepoly(20) 2 Interpolation Interpolating points Exact solution 1.5 1 0.5 0 -0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
0.6 0.8 1 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. 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 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 Result: >> rungecheb(10) interpolation at Chebyshev points 1 Interpolation Interpolating points Exact solution 0.9 0.8 0.7 function 0.6 Compare to equally spaced points: 0.5 1.2
0.4 Interpolation Interpolating points Exact solution 1 0.3 0.8 0.2 0.6 0.1 0 -1 0.4 0.2 -0.8 -0.6 -0.4 -0.2 0 0.2
0.4 0.6 0.8 1 0 -0.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Result:
>> rungecheb(20) interpolation at Chebyshev points 1 Interpolation Interpolating points Exact solution 0.9 0.8 0.7 Compare to equally spaced points: 0.6 0.5 2 0.4 0.3 Interpolation Interpolating points Exact solution 1.5 0.2 1 0.1 0 -1
0.5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 -0.5 -1 -0.8 -0.6 -0.4 -0.2
0 0.2 0.4 0.6 0.8 1 Inclass The following data are related to the life expectances of citizens of two European regions. 1975 1980 1985 1990 WE: 72.8 74.2 75.2 76.4 EE: 70.2 70.2 70.3 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.