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\title{QR Decomposition}
\author{By Jose Divas\'on and Jes\'us Aransay\thanks{This research has been funded 
  by the research grant FPI-UR-12 of the Universidad de La Rioja and by the project MTM2014-54151-P from Ministerio de Econom\'ia y Competitividad
(Gobierno de Espa\~na).}}
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\begin{abstract}
In this work we present a formalization of the QR decomposition, an algorithm which decomposes
a real matrix $A$ in the product of another two matrices $Q$ and $R$, where $Q$ is an orthogonal matrix
and $R$ is invertible and upper triangular. The algorithm is useful for the least squares problem, 
i.e. the computation of the best approximation of an unsolvable system of linear equations. 
As a side-product, the Gram-Schmidt process has also been formalized. A refinement using immutable arrays is presented as well. 
The development relies, among others, on the AFP entry \emph{Implementing field extensions of the form $\mathbb{Q}[\sqrt{b}]$} by Ren\'e Thiemann,
which allows to execute the algorithm using symbolic computations. Verified code can be generated and executed using floats as well.

\end{abstract}

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