LeastSquareFitAlgorithm.txt

Given n points (x1, y1) ... (xn, yn)
We consider this a tabulation of function values yi = f(xi)

Compute coefficients  ci  i=1..r such that
the function  g = cr x^r + cr-1 x^r-1 + ... + c1 x + c0
minimizes sum(yi-g(xi)) is a Least Square Fit over the points.

Algorithm: build matrix A, vector Y and solve for vector C
A*C=Y  where all sum's are over i=0,n-1 the x,y points
r is the order of the approximating polynomial, max(r)=n
//
    | sum(1.0)      sum(x[i]^1) sum(x[i]^2)   ... sum(x[i]^r-1)  |
    | sum(x[i]^1)   sum(x[i]^2) sum(x[i]^3)   ... sum(x[i]^r)    |
A = | sum(x[i]^2)   sum(x[i]^3) sum(x[i]^4)   ... sum(x[i]^r+1   |
    | ...           ...         ...           ... ...            |
    | sum(x[i]^r-1) sum(x[i]^r) sum(x[i]^r+1) ... sum(x[i]^2r-1) |
    
    | sum(y[i])            |
    | sum(x[i]^1 * y[i])   |
Y = | sum(x[i]^2 * y[i])   |
    | ...                  |
    | sum(x[i]^r-1 * y[i]) |

See "Evaluate" and "Integrate" for the uses of the coefficients.