Question

Writing Matlab code for the Least Squares Apporximation

Can I please get help with writng the matlab code for the least squares approximation and answering any theoretical questions that may come with it

7. [Least Squares Approximation] Approximate using least squares approximation with polynomials of order 5, 10, 20 on [-5,5]. Does the approximation get better higher order polynomials? Does this seem better than the result from Problem 4, 5, and 6?

Answer 1

function least_squares(x,n)

%least_squares(x,n) fits a least-squares polynomial of order n through
%a set of data where x and n are the coordinates.

%The output to least_squares(x,n) is the set of coefficients c0, c1,...,cm
%for the least-square polynomial Pm(x) = c0 + c1*x + c2*x^2 + ...+ cm*x^m
%Hence, there will be m+1 values in the vector c.
%A graph displays the set of points and the fitted polynomial

n = size(x,1);
if n == 1
   n = size(x,2);
end

b = zeros(m+1,1);
for i = 1:n
   for j = 1:m+1
      %right-hand side column vector consisting of sums of
      %powers of x multiplied by y's
      b(j) = b(j) + y(i)*x(i)^(j-1);
   end
end


p = zeros(2*m+1,1);
for i = 1:n
   for j = 1:2*m+1
      %sums of powers of x
      p(j) = p(j) + x(i)^(j-1);
   end
end


for i = 1:m+1
   for j = 1:m+1
      %distributing the sums of powers of x in a matrix A
      A(i,j) = p(i+j-1);
   end
end

c = A\b

%creating an x-axis and evaluating the least-ssquare polynomial
%this is only needed for data visualization
t = min(x):0.05:max(x);
n = size(t,2);
for i = 1:n
   f(i) = c(m+1);
   for j = m:-1:1
      f(i) = c(j) + f(i)*t(i);
   end
end

%data visualization
figure
plot(t,f)       %the least_squares polynomial
hold on
plot(x, n, 'r*')        %the data points
grid on