trying to create a function for simple Newton polynomial interpolation

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I am trying to create a function for simple Newton polynomial interpolation.
This is the recurrance relation I am trying to create on matlab, where the data points (x_i,y_i) are specified by the input vectors x and y, and the points at which I want to evaluate the polynomial at are given by the input vector t. I then want to store the evaluated points in a vector p and display this vector.
I have been inputting two vectors of the same size x and y and then letting the vector t = x so that if everything went right I should get the vector y back out, but I don't and I'm not sure why? Any Help.
nt = length ( t );
%Number of data points
n = length ( x );
%Initial polynomial approximation
p = y(1);
for k = 2:n
%initialise product in the recursive formula
for j = 1:k-1
p = p + (y(k)-p(x(k)))*(t-x(j))/(x(k)-x(j));
end
end
p = p (1:nt);
for l = 1:nt
p(nt) = p(t(nt));
end
display(p);
end

Respuesta aceptada

Shashi Kiran
Shashi Kiran el 5 de Nov. de 2024
Hi @Ole,
To create a function that performs Newton polynomial interpolation based on the recurrence relation given these steps can be followed.
  • Start with the base polynomial where ​ is the first y-value.
  • This base value will serve as the initial polynomial value at each point in the vector t where we want to evaluate the polynomial.
function p = newton_interpolation_algorithm(x, y, t)
n = length(x);
nt = length(t);
p = zeros(1, nt); % Initialize output vector for evaluated points
for j = 1:nt
p(j) = y(1); % Initialize p(t(j)) as y_0 for each evaluation point t(j)
end
  • For each evaluation point , calculate a product term that captures the influence of all previous values:
  • Use this product term to update with a new term derived from
  • After computing the product term, add the new term to :
% Loop for each additional data point
for k = 2:n
for j = 1:nt
% Calculate product term for each t(j)
product_term = 1;
for i = 1:k-1
product_term = product_term * (t(j) - x(i)) / (x(k) - x(i));
end
% Update polynomial value at t(j)
p(j) = p(j) + (y(k) - p(j)) * product_term;
end
end
  • Once the recursion is complete, p contains the evaluated polynomial values at each point in t.
disp('Evaluated polynomial values at points in t:');
disp(p);
end
x = [1, 7, 3, 4];
y = [1, 49, 9, 16];
t = x; % Points to evaluate the polynomial (for testing)
p = newton_interpolation_algorithm(x, y, t); % Should return values close to y if correct
Evaluated polynomial values at points in t: 1 49 9 16
Hope this helps.​

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