Question

Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in MATLAB; see "help spline". Use m=10 and m=20. Compute splines that interpolate at equidistant nodes and at Chebyshev nodes. Provide tables of the errors and plots of the function f and the interpolating polynomials and splines.

Answer 1

%%Matlab function for Lagrange Interpolation
clear all
close all
%first function
f1=@(x) 1./(1+25.*x.^2);
%loop for Lagrange polynomial for all n
a=-1;b=1; k=0;
for n=10:10:20
    k=k+1;
    %Equidistance data points
    x1=linspace(a,b,n);
    y1=double(f1(x1));
  
    %Chebyshev data points
    for i=1:n
        x2(i)=(1/2)*(a+b)+(1/2)*(b-a)*cos(((2*i-1)*pi)/(2*n));
    end

    y2=double(f1(x1));
  
  
    syms x
    %x1=independent variable;y1=dependent variable;x=value at which we have to
    %find the dependent variable;y=corresponding value of dependent variable at x;
    p1=polyfit(x1,y1,n-1);
    p2=polyfit(x2,y2,n-1);
  
    %the interpolated data polyfit
    xx1=linspace(a,b,200);
    %polynomial fit for Equidistance points
    yy1=polyval(p1,xx1);
    %polynomial fit for Chebyshev points
    yy2=polyval(p2,xx1);
    %cubic spline
    yy3_eq=spline(x1,y1,xx1);
    yy3_ch=spline(x2,y2,xx1);
  
    %plotting of function
        figure(k)
        hold on
        plot(xx1,yy1,'Linewidth',2)
        plot(xx1,yy2,'Linewidth',2)
        plot(xx1,yy3_eq,'Linewidth',2)
        plot(xx1,yy3_ch,'Linewidth',2)
        plot(xx1,f1(xx1),'Linewidth',2)
      
        xlabel('x')
        ylabel('f(x)')
        legend('Equidistance','Chebyshev','Spline equidistance','Spline equidistance','Actual')
        title(sprintf('Interpolating polynomial for n=%d',n))
  
        fprintf('\tError in Equidistance interpolation for n=%d is %e.\n',n,norm(yy1-f1(xx1)))
        fprintf('\tError in Chebyshev interpolation for n=%d is %e.\n',n,norm(yy2-f1(xx1)))
        fprintf('\tError in Equidistance spline interpolation for n=%d is %e.\n',n,norm(yy3_eq-f1(xx1)))
        fprintf('\tError in Chebyshev spline interpolation for n=%d is %e.\n\n',n,norm(yy3_ch-f1(xx1)))
      

end

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