Question

1. Consider the Runge function, f:IH 1/1+25r). (a) Use your Lagrange interpolation code (from the previous worksheets) to approximate f using 10, 20, 30, and 40 equispaced points from -1 and 1 (inclusive). Make a (single) plot comparing these four approximations with the (exact) function f. Use a legend to help distinguish the five curves. Intuitively, increasing the number of sample points should give a 'better' approximation. Does it? (A qualitative answer is sufficient.) (b) Repeat Part (a) using piecewise linear interpolation ("connect-the-dots'"), by applying your Lagrange in- terpolation code to each pair of successive points. Are these approximations 'better' than those fronm Part (a)? (e) Repeat Part (b) using Matlab's interp1 or griddedInterpolant routines. (You can use either: interpl evaluates the interpolant at given points, whereas griddedInterpolant returns the interpolant as a func- tion that you can evaluate later.) (d) Repeat Part (e) but using piecewise cubic Hermite interpolation. (Pass 'pchip' as the method argument, rather than '1inear', the default.) Are these approximations 'better' than those from Parts (b) and (e)?

Answer 1

`Hey,

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clc
clear all
xx=linspace(-1,1,1000);
plot(xx,1./(1+25*xx.^2));
s{1}='Original function';
n=[10:10:40];
for i=1:length(n)
x=linspace(-1,1,n(i));
y=1./(1+25*x.^2);
for ii=1:length(xx)
yy(ii) = linterp(x,y,xx(ii));
end
hold on;
plot(xx,yy);
s{i+1}=sprintf('Interpolated using %d points',n(i));
end
legend(s)
function y = linterp(X,Y,x)

sigma =0;

for i=1:numel(X)
pii = 1;
for j=1:numel(X)
if j~=i
pii = pii *(x - X(j))/(X(i)- X(j)) ;
end
end
  
sigma = sigma + Y(i)*pii;
end

y = sigma

end

Figure 1 File Edit View srt Iools Desktop Window Help 16000 Original function 14000 Interpolated using 10 points Interpolated using 20 points Interpolated using 30 points In 12000 10000 8000 6000 4000 2000 2000 -0.8 0.6 0.2 0.2 0.4 0.6 0.8 O Type here to search ENG 1:44 PM IN 04/02/19

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