Question

. Write a Matlab function a=N2V(c,x), where c is a column n-vector, x is a column (n − 1)- vector and a is a column n-vector, so that if p(x) = c_1 + c_2(x − x_1) + · · · + c_n*(x − x_1)(x − x − 2)· · ·(x − x_n−1), then p(x) = a_1 + a_2*x + · · · + a_n*x^n . In other words, N2V converts from the Newton representation to the Vandermonde representation.

Answer 1

All the explanation is in the code comments. Hope this helps!

Code:

% sample run

c = [1; -2; 3; 4]

x = [-7; 1; 2]

a = N2V(c, x)

% required function

function a = N2V(c, x)

% value of n

n = size(c, 1);

% use y as a symbol for x

syms y

% calculate the expression p(x)

p = c(1);

% term for series of (x-x_i) in multiplication

term = 1;

for i=2:n

% update the term

term = term * (y-x(i-1));

% add the ith term = c_i(x-x_1)*...(x-x_i-1)

p = p + c(i)*term;

end

% now get all the coefficients for the polynomial p(x)

% also transform a to column vector

a = coeffs(p, y);

a = a';

end

Sample run:

Code screenshots: