st_squares_and_the_orthogonality_principle">orthogonality principle</a>. In brief, we want the squared difference between the polynomial and the points to be as little as possible, so this is a minimization problem. But according to the orthogonality principle, the minimum for the sum of squared residuals is the same point where all the residuals, put in a single vector, are orthogonal to the vector of polynomial values.
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There is a single solution, and there is no need to run numeric methods to f