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Appendix A – Overdetermined linear equations systems
Appendix A - OVERDETERMINED LINEAR EQUATIONS
SYSTEMS
The system of linear algebraic equations of the form:
AX = b , (A-1)
m×
n
Where A ∈IR , X ∈IR
n
, and
b∈
IR
n
is said to be an overdetermined system if m>n
i.e., there are more equations than unknowns.
tel-00623090, version 1 - 13 Sep 2011
This type of systems appears as a consequence of experimental errors; in order to
obtain a more accurate result one requires more measurements than the strictly
necessary ones. For example, curve fitting which is the process of constructing a curve
that has the best fit to a series of data points. Given m data points (x i , y j ) for i = 1,…m,
we want to adjust these points to a curve of the form:
n
=
0 0
+
1 1
+ +
n n
=∑ j j
j = 1
( ) ( ) ( ) ( )
Px ( ) au x au x au x au x
(A-2)
Where u j (j=0,…,n) are linearly independent given functions and a j (j=0,…,n)
parameters to be determine.
The linear systems (A-1) only as a solution when b belongs to the column space of A.
However, it is also possible to determine X such that it minimizes some vector norm of
the residual
r = b−AX
(A-3)
i.e., determine X such that the residual vector r is as small as possible.
Due to the differentiability of the Euclidean norm, allowing determining the minimum by
the usual process is in general the used norm to minimize the residual. This is called
least squares method.
The goal of the least squares method is to determine the vector X which minimizes the
sum of the squared residuals, i.e.,
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