tex.tex (1047B)
1 % Time-stamp: <2004/04/06, 16:46:43 (EST), maverick, test.tex> 2 \subsection{Strict diagonal-dominance} 3 Suppose we are given a matrix $A=L+D$, where $L$ is a Laplacian and 4 $D$ is a nonnegative diagonal matrix, for which we seek to construct a 5 preconditioner. 6 7 We may construct a Support Tree Preconditioner, $B = 8 \begin{pmatrix} T & U\\U\TT & W\end{pmatrix}$ for $L$ and to use $B' 9 =\begin{pmatrix} T & U \\U\TT & W+D\end{pmatrix}$ as a preconditioner 10 for $A$. If we let $Q = W - U\TT T\IV U$, by Lemma~\ref{lem:stcg} it 11 suffices to bound $\sigma(A/Q+D)$ and $\sigma(Q+D/A)$. 12 13 \begin{proposition}\label{prop:XZ-YZ} 14 If $X$, $Y$, and $Z$ are spsd matrices of the same size then 15 $\sigma(X+Z/Y+Z) \leq \max\{\sigma(X/Y),\, 1\}$. 16 \end{proposition} 17 18 \Proof We have $\sigma(X+Z/Y+Z) = 19 \min\{\tau \mid \forall\vv{x},\, \tau\cdot \vv{x}\TT (Y+Z)\vv{x} \geq 20 \vv{x}\TT(X+Z)\vv{x}\} = 21 \min\{\tau \mid \forall\vv{x},\, (\tau-1)\cdot \vv{x}\TT Z\vv{x} + 22 \tau \cdot\vv{x}\TT Y\vv{x} \geq \vv{x}\TT X\vv{x}\} \leq 23 \max\{1,\,\sigma(X/Y)\}$.\QED 24