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generating polynomial using newton divided difference

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Hi good day y'all, i'm making a code for newton divided difference but i'm having a hard time generating the right polynomial, can someone please help me? thank you so much.
y=[1 2 3 4];
x=[4.4 4.3 2 5];
n=size(x,2);
DD=zeros(n,n);
DD(:,1)=y';
for j=2:n
for i=1:(n-j+1)
num=DD(i+1,j-1)-DD(i,j-1);
den = (x(i+j-1)-x(i));
DD(i,j)=num./den;
end
end
array2table(DD)
ans = 4×4 table
DD1 DD2 DD3 DD4 ___ ________ _______ ______ 1 -10 -3.9855 8.4714 2 -0.43478 1.0973 0 3 0.33333 0 0 4 0 0 0
n=length(x);
a(1)=x(1);
for k=1:n-1
d(k,1)=(y(k+1)-y(k))/(x(k+1)-x(k));
end
for j=2:n-1
for k=1:n-j
d(k,j)=(d(k+1,j-1)-d(k,j-1))/(x(k+j)-x(k));
end
end
%
for j=2:n
a(j)=d(1,j-1);
end
yn=vpa(x);
d=vpa(d);
a=vpa(a);
clear x
syms x
%
Df(1)=vpa(1);
c(1)=a(1);
for j=2:n
Df(j)=(x-yn(j-1)).*Df(j-1);
c(j)=a(j).*Df(j);
end
format short
f=simplify(sum(c))
f = 

Accepted Answer

Vinayak Agrawal
Vinayak Agrawal on 27 Jun 2023
Edited: Vinayak Agrawal on 28 Jun 2023
Hi Michael,
an updated version of your code that computes the polynomial expression using symbolic calculations in MATLAB:
% Given data
y = [1 2 3 4];
x = [4.4 4.3 2 5];
n = length(x);
DD = zeros(n, n);
DD(:, 1) = y';
for j = 2:n
for i = 1:(n-j+1)
num = DD(i+1, j-1) - DD(i, j-1);
den = (x(i+j-1) - x(i));
DD(i, j) = num / den;
end
end
% Coefficients of the polynomial
a = diag(DD)';
yn = sym(x);
d = sym(diag(DD));
% Compute the polynomial expression
syms x;
f = a(1);
for j = 2:n
term = 1;
for k = 1:j-1
term = term * (x - yn(k));
end
f = f + a(j) * term;
end
% Simplify the polynomial expression
f = simplify(f);
disp(f);
In this updated code, try running this. I've used symbolic calculations (sym) to generate the polynomial expression based on the computed coefficients. The resulting polynomial expression is simplified using simplify for a more concise form.
When you run the code, it will display the simplified polynomial expression. You can further customize the output format or manipulate the polynomial expression as needed.
I hope this helps you generate the correct polynomial expression using Newton's divided difference interpolation.
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