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====== Math 753/853 QR and the least-squares problem ====== The QR decomposition is useful for solving the //linear least-squares problem//. Briefly, suppose you have an $Ax=b$ system with an oblong matrix $A$, i.e. A is an $m \time n$ matrix with $n<m$. Each of the $m$ rows of $A$ corresponds to a linear equation in the unknown $n$ variables that are the components of $x$. But with $n<m$, that means we have more equations than unknowns. In general, a system with more equations than unknowns does not have a solution! So, instead of looking for an $x$ such that $Ax=b$, we look for an $x$ such that $Ax-b$ is small. Since $Ax-b$ is a vector, we measure its size with a norm. That means we are looking for the $x$ that minimizes $\|Ax-b\|$. http://www.math.utah.edu/~pa/6610/20130927.pdf
