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Problems 3 and 4: More complex equations

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

Equation 24

Equation 25

Equation 26

leads eventually to

Equation 27

(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 ident x2, t2 ident x, and t3 ident 1 then the coefficients of the best-fitting parabola

Equation 28

are easily seen to be

Equation 29

Or if

Equation 30

you can substitute

Equation 31

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, . . .

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