25 lines
1.0 KiB
TeX
25 lines
1.0 KiB
TeX
% Time-stamp: <2004/04/06, 16:46:43 (EST), maverick, test.tex>
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\subsection{Strict diagonal-dominance}
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Suppose we are given a matrix $A=L+D$, where $L$ is a Laplacian and
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$D$ is a nonnegative diagonal matrix, for which we seek to construct a
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preconditioner.
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We may construct a Support Tree Preconditioner, $B =
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\begin{pmatrix} T & U\\U\TT & W\end{pmatrix}$ for $L$ and to use $B'
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=\begin{pmatrix} T & U \\U\TT & W+D\end{pmatrix}$ as a preconditioner
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for $A$. If we let $Q = W - U\TT T\IV U$, by Lemma~\ref{lem:stcg} it
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suffices to bound $\sigma(A/Q+D)$ and $\sigma(Q+D/A)$.
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\begin{proposition}\label{prop:XZ-YZ}
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If $X$, $Y$, and $Z$ are spsd matrices of the same size then
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$\sigma(X+Z/Y+Z) \leq \max\{\sigma(X/Y),\, 1\}$.
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\end{proposition}
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\Proof We have $\sigma(X+Z/Y+Z) =
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\min\{\tau \mid \forall\vv{x},\, \tau\cdot \vv{x}\TT (Y+Z)\vv{x} \geq
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\vv{x}\TT(X+Z)\vv{x}\} =
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\min\{\tau \mid \forall\vv{x},\, (\tau-1)\cdot \vv{x}\TT Z\vv{x} +
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\tau \cdot\vv{x}\TT Y\vv{x} \geq \vv{x}\TT X\vv{x}\} \leq
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\max\{1,\,\sigma(X/Y)\}$.\QED
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