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Homelinear least squares
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linear least squares
Let $A$ be an $m\times n$ matrix with $m\geq n$ and $b$ an $m\times 1$ matrix. We want to consider the problem
$Ax\approx b$ 
where $\approx$ stands for the best approximate solution in the least squares sense, i.e. we want to minimize the Euclidean norm of the residual $r=Axb$
$Axb_{2}=r_{2}=\left[\sum_{{i=1}}^{m}r_{i}^{2}\right]^{{1/2}}$ 
Among the different methods to solve this problem, we mention Normal Equations, sometimes illconditioned, QR Decomposition, and, most generally, Singular Value Decomposition. For further reading, [Golub89], [Branham90], [Wong92], [Press95].
Example: Let us consider the problem of finding the closest point (vertex) to measurements on straight lines (e.g. trajectories emanating from a particle collision). This problem can be described by $Ax=b$ with
$A=\begin{bmatrix}a_{{11}}&a_{{12}}\\ \vdots&\vdots\\ a_{{m1}}&a_{{m2}}\end{bmatrix};x=\begin{bmatrix}u\\ v\end{bmatrix};b=\begin{bmatrix}b_{1}\\ \vdots\\ b_{m}\end{bmatrix}$ 
This is clearly an inconsistent system of linear equations, with more equations than unknowns, a frequently occurring problem in experimental data analysis. The system is, however, not very inconsistent and there is a point that lies “nearly” on all straight lines. The solution can be found with the linear least squares method, e.g. by QR decomposition for solving $Ax=b$:
$QRx=b\rightarrow x=R^{{1}}Q^{T}b$ 
References

Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)
 Wong92
S.S.M. Wong, Computational Methods in Physics and Engineering, Prentice Hall, 1992.
 Golub89
Gene H. Golub and Charles F. van Loan: Matrix Computations, 2nd edn., The John Hopkins University Press, 1989.
 Branham90
R.L. Branham, Scientific Data Analysis, An Introduction to Overdetermined Systems, Springer, Berlin, Heidelberg, 1990.
 Press95
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