Quadratic fit
Quadratic curve formula:
y = (a * x2) + (b * x) + c
At least three calibration points are required for the quadratic fit. Two points are required if the origin is included or forced.
Calculation of coefficients for quadratic fit
The coefficients result from the below simultaneous linear equations. Crout's algorithm is used to solve the corresponding normal matrix equation (ATAx = ATy). In the given formula, sums are abbreviated as:
In order to avoid overflow, the x-values are normalized before entering calculation:
Norm = Σ(x)
x = x / Norm
Normal equations for quadratic curve:
Or written as matrix equation:
Crout's decomposition:
With value abbreviations:
Finally, the normalization must be reversed:
Force Origin
If the force origin option is selected, the offset term c is set to zero when creating the normal equations.
Include origin
If the origin is included, the point (0,0) is added to the other points and weighted by the mean value of the weights of the other points, that is, the Σ(wt) term is increased by the mean value of the weights of the other points.