Function Approximation and Interpolation

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Sardor Butunboev on 11 Oct 2021
Answered: David Hill on 11 Oct 2021
Given: f(x) = exp(-x^2) on the interval [-1; 1].
Need to:
1. Approximate f(x) by a 9-degree monomial basis polynomial interpolant with equidistant nodes. Proceed as follows:
1.1. create a vector x containing the n = 9 interpolation nodes.
1.2. use the function 'vander' to create the interpolation matrix G.
1.3. compute yi = f(xi) at the n interpolation nodes.
1.4. compute the n basis coefficients c.
2. Evaluate the accuracy of the interpolant, say f1, as follows:
2.1. Use the Matlab function 'polyval' to evaluate f1 for 100 evenly distributed points on [-1; 1].
2.2. Compare these interpolated values with the 'true' values of f.
2.3. Plot the approximation error.
So far, could this. But no idea whether they are correct or not. Don't even understand what should do in 2.2 and 2.3
f = @(x) exp(-x.^2);
n = 9;
y = linspace(-1, 1, n);
z = [];
for i = 1:length(y)
z(i) = feval(f,y(i));
end
v = fliplr(vander(y));
a = v\z';
b = a(end:-1:1)';
%5
c = linspace(-1,1);
d = polyval(b, c);
p = polyfit(c,d);

David Hill on 11 Oct 2021
Something like this.
f = @(x) exp(-x.^2);
x = linspace(-1, 1, 9);
G=vander(x);
y=f(x);
c=G\y';
f1=@(x)polyval(c,x);
t=linspace(-1,1,100);
Error=(f(t)-f1(t))./f(t);
plot(t,Error)