11—Numerical Analysis 283result — the solution is not unique. A further point: there is no requirement that all of thexiare
different; you may have repeated the measurements at some points.
Minimizing this is now a problem in ordinary calculus withMvariables.
∂
∂αν
∑N
i=1
yi−
∑M
μ=1αμfμ(xi)
2
=− 2∑
i[
yi−
∑
μαμfμ(xi)
]
fν(xi) = 0
rearrange:∑
μ[
∑
ifν(xi)fμ(xi)
]
αμ=
∑
iyifν(xi) (11.50)
These linear equations are easily expressed in terms of matrices.
Ca=b,
whereCνμ=
∑N
i=1fν(xi)fμ(xi) (11.51)
ais the column matrix with componentsαμandbhas components
∑
iyifν(xi).
The solution forais
a=C−^1 b. (11.52)
IfCturned out singular, so this inversion is impossible, the functionsfμwere not independent.
Example: Fit to a straight linef 1 (x) = 1 f 2 (x) =x
ThenCa=bis (
N
∑
∑ xi
xi
∑
x^2 i
)(
α 1
α 2
)
=
( ∑
∑ yi
yixi
)
The inverse is
(α 1
α 2
)
=
1
[
N
∑
x^2 i−
(∑
xi
) 2 ]
( ∑
x^2 i −
∑
xi
−
∑
xi N
)( ∑
∑ yi
xiyi
)
(11.53)
and the best fit line is
y=α 1 +α 2 x
11.7 Euclidean Fit
In fitting data to a combination of functions, the least squares method used Eq. (11.49) as a measure
of how far the proposed function is from the data. If you’re fitting data to a straight line (or plane if
you have more variables) there’s another way to picture the distance. Instead of measuring the distance
from a point to the curveverticallyusing onlyy, measure it as theperpendiculardistance to the line.
Why should this be any better? It’s not, but it does have different uses, and a primary one is data
compression.