Problems 3 and 4: More complex equations

The same methods may be used for more complicated equations. For instance, suppose our measured quantities y_i depend upon three different known variables:

leads eventually to

(Prove it!) These matrices have a beautifully simple and symmetric structure. I think that if you stare at them for a minute or two, you'll be able to memorize their form and recall them whenever you need them hereafter.

Notice that the known quantities t₁, t₂, and t₃ do not need to be independent of each other. For instance, if t₁ x², t₂ x, and t₃ 1 then the coefficients of the best-fitting parabola

are easily seen to be

Or if

you can substitute

which gives

See? It's not called "linear" least squares because all it can do is fit straight lines - it can fit all sorts of equations, provided they are linear in the unknown parameters a, b, c, . . .