171 lines
2.9 KiB
Plaintext
171 lines
2.9 KiB
Plaintext
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#LyX 1.6.7 created this file. For more info see http://www.lyx.org/
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\lyxformat 345
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\begin_document
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\begin_header
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\textclass article
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\use_default_options true
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\language english
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\graphics default
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\use_geometry false
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\use_amsmath 1
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\cite_engine basic
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\use_bibtopic false
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\paperorientation portrait
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\secnumdepth 3
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\tocdepth 3
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\paragraph_separation indent
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\defskip medskip
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\quotes_language english
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\papercolumns 1
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\papersides 1
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\paperpagestyle default
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\tracking_changes false
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\output_changes false
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\author ""
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\author ""
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\end_header
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\begin_body
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\begin_layout Section
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Basic solving with Cholesky
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\end_layout
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\begin_layout Standard
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Solving a linear least-squares system:
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\begin_inset Formula \[
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\arg\min_{x}\left\Vert Ax-b\right\Vert ^{2}\]
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\end_inset
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Set derivative equal to zero:
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\begin_inset Formula \begin{align*}
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0 & =2A^{T}\left(Ax-b\right)\\
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0 & =A^{T}Ax-A^{T}b\end{align*}
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\end_inset
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For comparison, with QR we do
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\begin_inset Formula \begin{align*}
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0 & =R^{T}Q^{T}QRx-R^{T}Qb\\
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& =R^{T}Rx-R^{T}Qb\\
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Rx & =Qb\\
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x & =R^{-1}Qb\end{align*}
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\end_inset
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But with Cholesky we do
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\begin_inset Formula \begin{align*}
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0 & =R^{T}RR^{T}Rx-R^{T}Rb\\
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& =R^{T}Rx-b\\
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& =Rx-R^{-T}b\\
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x & =R^{-1}R^{-T}b\end{align*}
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\end_inset
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\end_layout
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\begin_layout Section
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Frontal (rank-deficient) solving with Cholesky
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\end_layout
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\begin_layout Standard
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To do multi-frontal elimination, we decompose into rank-deficient conditionals.
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\begin_inset Formula \[
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\left[\begin{array}{cccccc}
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\cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
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\cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
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\cdot & \cdot & \cdot & \cdot & \cdot & \cdot\end{array}\right]\to\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula \[
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\left[\begin{array}{cc}
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R^{T} & 0\\
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S^{T} & C^{T}\end{array}\right]\left[\begin{array}{cc}
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R & S\\
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0 & C\end{array}\right]=\left[\begin{array}{cc}
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F^{T}F & F^{T}G\\
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G^{T}F & G^{T}G\end{array}\right]\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset space ~
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula \[
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R^{T}R=F^{T}F\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset space ~
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula \begin{align*}
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R^{T}S & =F^{T}G\\
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S & =R^{-T}F^{T}G\end{align*}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset space ~
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula \begin{align*}
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S^{T}S+C^{T}C & =G^{T}G\\
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G^{T}FR^{-1}R^{-T}F^{T}G+C^{T}C & =G^{T}G\\
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G^{T}QRR^{-1}R^{-T}R^{T}Q^{T}G+C^{T}C & =G^{T}G\\
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\textbf{if }R\textbf{ is invertible, }G^{T}G+C^{T}C & =G^{T}G\\
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C^{T}C & =0\end{align*}
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\end_inset
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\end_layout
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\end_body
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\end_document
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