Pn.m code: function P = Pn(X,x,y) n = length(x)-1; dX = ones(n+1,1)*X - x\'*ones

Question

Pn.m code:

function P = Pn(X,x,y)
  
n = length(x)-1;
dX = ones(n+1,1)*X - x'*ones(1,length(X));

for k = 1:n+1
i = [ 1:(k-1) (k+1):n+1];
L(k,:) = prod(dX(i,:)./((x(k)-x(i))'*ones(1,length(X))));
end

P = sum(y'*ones(1,length(X)).*L);

end

Legendre Polynomials and Chebyshev Polynomials You are asked to submit the following five functions: 1. function y f (x) 2. function P n (x,x,y) 3. function y LegPoly(n,x) 4. function L LegPolyApprox CX,n 5. function TestPoly (n) The first function f.m should be function y f (x) y exp(sin (x 5)); end The second function Pn.m can be copied from homework 3 solution. 1 overall Requirements We will run your code from matlab console by typing TestPoly (5) 0.6476 0.4001 0.2753 0.1912 In addition to these numbers, we expect to see two figures which should be essentially the same as Figure 1. We will test the code for any 2 S n S 100. In order to encourage students write reasonably efficient code, the program should not take more than 1 minute (which is much more than necessary).

Explanation / Answer

function TestPoly(n)

for X= -1: 0.001 : 1

y = f(X)

x =X

C(x)= Pn(X,x,y,n)

L(x) = LogpolyApprox(X,y,n)

err(x) = //as given in question

end

plot(C,X)

plot(L,X)

print(err)

end

function P =Pn(..///given in question add n as argument

function LogpolyApprox(X,y,n)

sum = 0

for k = 1:n

b =integral(PF,-1,1,X,yk-1)

d= (2k-1)*b/2

sum = sum + d*P(X,x,y,k-1)

end

return sum

function PF(X,x,y,n)

return f(x) *Pn(X,x,yn)

function f(x)

return exp(sin(x*5))

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