generating polynomial using newton divided difference

Question

michael 2023-6-27

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1988758-generating-polynomial-using-newton-divided-difference

编辑： Vinayak Agrawal 2023-6-28

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

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

%

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 =

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Answer 1

Vinayak Agrawal 2023-6-27

0
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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1988758-generating-polynomial-using-newton-divided-difference#answer_1263488

编辑：Vinayak Agrawal 2023-6-28

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

% 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.