Newton Divided Difference polynomial not working at end point for higher values of "n"
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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
2 件のコメント
Muskan
2023 年 3 月 23 日
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
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