4. Use the function f(x)=sin^2(x) on [0,pi] to verify the convergence therorems

Question

4. Use the function f(x)=sin^2(x) on [0,pi] to verify the convergence therorems for piecewise linear and cubic spline interpolation. Using 11, 21, 41, and 81 equally spaced points, calculate the errors in approximations, and then calculate convergence rates. 0.3112 0.0301 0.5285 0.1476 0.1656 0.0045 0.6020 0.2181 0.2630 0.0182 0.6541 0.2798 0.6892 0.3274 0.74820.4188 Extend the ideas of this section for the 'line of best fit' to find the qu best fit'ao+ ax+ azx for this data. Hint: the procedure is similar, and the difference is that now e - elas,ar, a2)h and you will have 3 partials that must be zero. Use the function f(x)-sin' (x) on [0, 4. to verify the convergence theorems for piecewise linear and cubic spline interpolation. Using 11, 21, 41, and 81 equaly spaced points, calculate the errors in the approximations (error as defined in the theorems), and then calculate convergence rates.

Explanation / Answer

x=[11 21 41 81];

y=sin2(X);

c = polyfit(x,y,length(x)-1);

xi = 0:0.1:7; zi = sin2(x);

yi =polyval(c,xi);

subplot(2,1,1)

title('Interpolation')

plot(xi,yi,'-.',xi,zi,x,y,'*');

subplot(2,1,2)

title('Interpolation Error')

plot(xi,zi-yi,x,zeros(1,length(x)),'-*');

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