Interesting problem. The system is essentially this matrix equation:
r = A*[t^n]
where r, t and n are defined by necessity as column vectors and A is the matrix of coefficients. This is the inverse of the usual least-squares problem:
t = 0:0.5:99.5; % Define ‘t’
n = 0:length(t)-1; % Define ‘n’
tn = t.^n; % Define ‘t^n’
r = [1530 1215 3243 1111]'; % Given ‘r’ vector
stn = tn(1:length(r))'; % Truncate length of ‘tv’ to match sample ‘r’
stn(1) = eps; % Replace zero with ‘eps’ in ‘stv’
stni = pinv(stn); % Take pseudo-inverse of ‘t^n’
A = r*stni % Calculate ‘A’ coefficient matrix
rt = A*stn % Verify ‘A’ calculation
At least for the data available, this works!
