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  /external/ceres-solver/docs/
changes.tex 10 options.ordering_type = ceres::SCHUR
65 \item Change LOG(ERROR) to LOG(WARNING) in \texttt{schur\_complement\_solver.cc}.
71 \item Schur ordering was operating on the wrong object (Ricardo Martin)
142 Schur eliminator.
147 \item Fix how static structure detection for the Schur eliminator logs
244 \item Fixed a strict weak ordering bug in the schur ordering.
solving.tex 290 \left[B - EC^{-1}E^\top\right] \Delta y = v - EC^{-1}w\ . \label{eq:schur}
296 is the Schur complement of $C$ in $H$. It is also known as the {\em reduced camera matrix}, because the only variables participating in~\eqref{eq:schur} are the ones corresponding to the cameras. $S \in \reals^{pc\times pc}$ is a block structured symmetric positive definite matrix, with blocks of size $c\times c$. The block $S_{ij}$ corresponding to the pair of images $i$ and $j$ is non-zero if and only if the two images observe at least one common point.
299 Thus, the solution of what was an $n\times n$, $n=pc+qs$ linear system is reduced to the inversion of the block diagonal matrix $C$, a few matrix-matrix and matrix-vector multiplies, and the solution of block sparse $pc\times pc$ linear system~\eqref{eq:schur}. For almost all problems, the number of cameras is much smaller than the number of points, $p \ll q$, thus solving~\eqref{eq:schur} is significantly cheaper than solving~\eqref{eq:linear2}. This is the {\em Schur complement trick}~\cite{brown-58}.
301 This still leaves open the question of solving~\eqref{eq:schur}. The
318 Sparse direct methods, depending on the exact sparsity structure of the Schur complement,
332 The cost of forming and storing the Schur complement $S$ can be prohibitive for large problems. Indeed, for an inexact Newton solver that computes $S$ and runs PCG on it, almost all of its time is spent in constructing $S$; the time spent inside the PCG algorithm is negligible in comparison. Because PCG only needs access to $S$ via its product with a vector, one way to evaluate $Sx$ is to observe that
342 Equation~\eqref{eq:schurtrick1} is closely related to {\em Domain Decomposition methods} for solving large linear systems that arise in structural engineering and partial differential equations. In the language of Domain Decomposition, each point in a bundle adjustment problem is a domain, and the cameras form the interface between these domains. The iterative solution of the Schur complement then falls within the sub-category of techniques known as Iterative Sub-structuring~\ci (…)
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