Newton Forward difference method
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% Confirmed case Sample data points (x, y)
data = [ 1, 88; 2, 49; 3, 47; 4, 8; 5, 34; 6, 762; 7, 98; 8, 40];
% Extract x and y values
x = data(:, 1);
y = data(:, 2);
n = length(x);
% Calculate forward differences
forward_diff = zeros(n, n);
forward_diff(:, 1) = y;
for j = 2:n
for i = 1:n-j+1
forward_diff(i, j) = forward_diff(i+1, j-1) - forward_diff(i, j-1);
end
end
% Construct the Newton Forward Difference polynomial
syms X;
P = y(1);
for j = 1:n-1
term = forward_diff(1, j);
for i = 1:j
term = term * (X - x(i));
end
P = P + term / factorial(j);
end
disp('Newton Forward Difference Polynomial:');
disp(simplify(P));
Result Newton Forward Difference Polynomial: - (725*X^7)/1008 + (1651*X^6)/80 - (173233*X^5)/720 + (70651*X^4)/48 - (91321*X^3)/18 + (146407*X^2)/15 - (1996741*X)/210 + 3658
The problem now is that when I substitute each value of x, I don't get the corresponding value of y as stated in the table. I need help on how to get the correct polynomial equation.
Accepted Answer
Three coding errors:
% Confirmed case Sample data points (x, y)
data = [ 1, 88; 2, 49; 3, 47; 4, 8; 5, 34; 6, 762; 7, 98; 8, 40];
% Extract x and y values
x = data(:, 1);
y = data(:, 2);
n = length(x);
% Calculate forward differences
forward_diff = zeros(n, n);
forward_diff(:, 1) = y;
for j = 2:n
for i = 1:n-j+1
%forward_diff(i, j) = (forward_diff(i+1, j-1) - forward_diff(i, j-1))
forward_diff(i, j) = (forward_diff(i+1, j-1) - forward_diff(i, j-1)) / (x(i+j-1) - x(i));
end
end
% Construct the Newton Forward Difference polynomial
syms X;
P = y(1);
for j = 1:n-1
%term = forward_diff(1, j);
term = forward_diff(1, j+1);
for i = 1:j
term = term * (X - x(i));
end
%P = P + term / factorial(j);
P = P + term;
end
disp('Newton Forward Difference Polynomial:');
disp(simplify(P));
double(subs(P,X,x))
For all data series with deltax = 1 (as the one above), there is one error in your code. But note that this is a special case and the above code holds for arbitrary deltax.
% Confirmed case Sample data points (x, y)
data = [ 1, 88; 2, 49; 3, 47; 4, 8; 5, 34; 6, 762; 7, 98; 8, 40];
% Extract x and y values
x = data(:, 1);
y = data(:, 2);
n = length(x);
% Calculate forward differences
forward_diff = zeros(n, n);
forward_diff(:, 1) = y;
for j = 2:n
for i = 1:n-j+1
forward_diff(i, j) = forward_diff(i+1, j-1) - forward_diff(i, j-1);
end
end
% Construct the Newton Forward Difference polynomial
syms X;
P = y(1);
for j = 1:n-1
%term = forward_diff(1, j);
term = forward_diff(1, j+1);
for i = 1:j
term = term * (X - x(i));
end
P = P + term / factorial(j);
end
disp('Newton Forward Difference Polynomial:');
disp(simplify(P));
double(subs(P,X,x))
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