when I run the code, I only get one plot for f. I need to have a plot for p as well. I need to code this part p(x) =f(x0)l0(x) +f(x1)l1(x) +f(x2)l2(x) to make it works

clear all

n = 3; % the order of the polynomial

a = 2.0; % left end of the interval

b = 3.0; % right end of the interval

h = (b - a)/n; % interpolation grid size

t = a:h:b; % interpolation points

f = 1./t; % f(x) = 1./x, This is the function evaluated at interpolation points

%%%% pn(x) = \sum f(t_i)l_i(x)

hh = 0.01; % grid to plot the function both f and p

x = a:hh:b;

l = zeros(n+1, length(x)); %%%% l(1,:): l_0(x), ..., l(n+1): l_n(x)

for i = 1:n+1

   j = 1:length(x)

       n(i,j) = 1;

       d(i,j) = 1;

       for k = 1:n+1

           if i ~= k

               n(i,j) = n(i,j) * (x(j) - t(k));

               d(i,j) = d(i,j) * (t(i) - t(k));

           end

       end

       l(i,j) = n(i,j)/d(i,j);
end
sum = 0;
for i=1:n+1
    sum = sum + f(i)*L(i,j);
end

plot(t,f,'-b')
hold on 
plot(x,sum,'--r' )

[ebtisam almehmadi]

sum = 0;

for i=1:n+1

sum = sum + f(i)*L(i,j);

end

I think there is something wrong in this part. Please help

Answer 1

[Jogesh Kumar Mukala]

Hi,

The lines "n(i,j) = n(i,j) * (x(j) - t(k))" & "d(i,j) = d(i,j) * (t(i) - t(k))" are leading to the error in the code. The number of columns in the first matrix doesnot match with the number of rows in the second matrix which will lead to error in matrix multiplication. There seems to be a for loop missing which you may refer below.

for i = 1:n+1
    for j = 1:length(x)
       n(i,j) = 1;
       d(i,j) = 1;
       for k = 1:n+1
           if i ~= k
               n(i,j) = n(i,j) * (x(j) - t(k));
               d(i,j) = d(i,j) * (t(i) - t(k));
           end
       end
       l(i,j) = n(i,j)/d(i,j);
    end
end

[ebtisam almehmadi]

I have corrected it but I still get on plot for f. I need to have a plot for p as well. I'm not fimiliar with MATLAB, but I think there is somthing wrong with the last for loop.

[Jogesh Kumar Mukala]

As far as I see your code clearly, You are using same variables at two instances i.e. for order of polynomial(n) and numerator(n(i,j)). This is causing an issue in the final 'for' loop where it is considering 'n' as matrix. Also you could use Element-wise multiplication operator instead of 'for' loop which i mentioned in my above answer. Please refer below code for solving the issues.

clear all
n = 3; % The order of the polynomial
a = 2.0; % Left end of the interval
b = 3.0; % Right end of the interval
h = (b - a)/n; % Interpolation grid size
t = a:h:b; % Interpolation points
f = 1./t; % f(x) = 1./x, This is the function evaluated at interpolation points
hh = 0.01; % Grid to plot the function both f and p
x = a:hh:b;
L = zeros(n+1, length(x)); 

for i = 1:n+1
    j = 1:length(x);
        L(i,j) = 1;
        d(i,j) = 1;
        for k = 1:n+1
            if i ~= k
            L(i,j) = L(i,j) .* (x(j) - t(k));
            d(i,j) = d(i,j) .* (t(i) - t(k));
            end
        end
L(i,j) = L(i,j)/d(i,j);
end
sum = 0;
for i=1:n+1
    sum = sum + f(i)*L(i,j);
end
plot(t,f,'-b')
hold on 
plot(x,sum,'--r' )