If one knows an approximate fit for the the curve in the cooling experiment one can use Gauss' method of least squares to improve on the fit. The normal equations were given in his Theoria Motus. The difficulty lies in finding a set of correction equatons. This can be accomplished by substituting a small correction to the set of parameters.
(tV) is a vector whose components are the individual products tkVk. One can then find the changes to the deviations.
One ends up with the following set of correction equatons.
One can use these equations to improve on an approximate fit by feedback of the corrected parameters.
With the same data as before,
One gets the following fit.
Supplemental (Jun 17): Gauss introduced the square bracket notation for the normal equations.
No comments:
Post a Comment