Mathematical Tools for Physics

11—Numerical Analysis 337

have proposed a model that this data is to be represented by a linear combination of some set of functions,fμ(x)

y=

∑M

μ=1

αμfμ(x), (43)

what values ofαμwill represent the observations in the “best” way? There are several answers to this question
depending on the meaning of the word “best.” The most commonly used one, largely because of its simplicity, is
Gauss’s method of least squares. This criterion for best fit is that the sum

∑N

i=1



yi−

∑M

μ=1

αμfμ(xi)





2

=Nσ^2 (44)

be a minimum. The mean square deviation of the theory from the experiment is to be least. This quantityσ^2 is
called the variance.
Some observations to make here: N≥M, for otherwise there are more free parameters than data to fit
them, and almost any theory with enough parameters can be forced to fit any data. Also, the functionsfμmust
be linearly independent; if not, then you can throw away some and not alter the result — the solution is not
unique. A further point: there is no requirement that all of thexi are different; you may have repeated the
measurements at some points.
This is now a problem in ordinary calculus.

∂

∂αν

∑N

i=1



yi−

∑M

μ=1

αμfμ(xi)





2

=− 2

∑

i

[

yi−

∑

μ

αμfμ(xi)

]

fν(xi) = 0

rearrange:

∑

μ

[

∑

i

fν(xi)fμ(xi)

]

αμ=

∑

i

yifν(xi). (45)