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18.02 Multivariable Calculus
Fall 2007
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View Full DocumentLS. Least Squares Interpolation
1.
The leastsquares line.
Suppose you have a large number n of experimentally determined points, through which
you want to pass a curve. There is a formula (the Lagrange interpolation formula) producing
a polynomial curve of degree n

1which goes through the points exactly. But normally one
wants to find a simple curve, like a line, parabola, or exponential, which goes approximately
through the points, rather than a highdegree polynomial which goes exactly through them.
The reason is that the location of the points is to some extent determined by experimental
error, so one wants a smoothlooking curve which averages out these errors, not a wiggly
polynomial which takes them seriously.
In this section, we consider the most common case

finding a line which
goes approximately through a set of data points.
I
.
Suppose the data points are
and we want to find the line
which "best" passes through them. Assuming our errors in measurement are distributed
randomly according to the usual bellshaped curve (the socalled "Gaussian distribution"),
it can be shown that the right choice of
a
and b is the one for which the sum D of the
squares of the deviations
I
i=
l
is a
minimum.
In the formula (2), the quantities in parentheses (shown by
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 Spring '08
 Auroux
 Multivariable Calculus, Least Squares

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