Newton Divided Difference polynomial not working at end point for higher values of "n"

The code below is for generating a Newton Divided difference polynomial for a Runge's function 1/1+36x^2 between x = -1 to 1. I have generated the code which is working fine for n=6,11,21 with proper symmetry of oscillations about y axis, most importantly passing through the given data points. But at higher values of n, say 101, the polynomial does not pass through the point at x =1. Please spot the error in the code as I do not understand why its not working only for a certain n value.

n3 = 101;

X3 = vpa(linspace(-1,1,n3)); % x values of data points

Y3 = 1./(1.+(36.*(X3.^2))); % y values of data points

F3 = NaN(n3,n3+1);

for i = 1:n3

F3(i,1) = X3(1,i);

end

for i = 1:n3

F3(i,2) = Y3(1,i);

end

for i= 3:n3+1

for j = 1:n3-1

if (j<=n3-(i-2))

F3(j,i) = (F3(j+1,i-1) - F3(j,i-1))/(F3(j+i-2,1)- F3(j,1)); % generating newton divided diff. table

end

end

end

syms x

phi3 = sym(ones (1,n3));

for j = 2:n3

phi3(1,j) = phi3(1,j-1)*(x-X3(1,j-1)); % generating coefficients of newton polynomial

end

N3 = 0;

for i = 2:n3+1

N3 = N3 + F3(1,i)*phi3(1,i-1); % generating newton polynomial

end

N3

plot (X3,Y3,'o')

axis([-1.5 1.5 -0.2 1.25])

hold on

fplot(N3,'b-')

hold on

eval(subs(N3,vpa(x),1)) % checking whether the polynomials shows the correct values of y at x =1

[Sulaymon Eshkabilov]

It is working ok, e.g. n=105.

[Muskan]

Hi Vishnu,

I tried running your code in MATLAB R2022b, and for the higher value of n, which you mentioned “n=101”, it seemed to work fine. You can try clearing out your workspace and then running the code.

Let me know if there are any other issues.

Thanks,

Muskan