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Shift-and-Invert QEP.

Combining the above shift-and-invert spectral transformations (SI), the so-called shift-and-invert QEP becomes
\begin{displaymath}
\left(\mu^2 \widehat{M} + \mu \widehat{C} + \widehat{K} \right) x = 0,
\end{displaymath} (261)

where

\begin{displaymath}
\mu=\frac{1}{\lambda-\sigma},\quad
\end{displaymath}

and $\widehat M= \sigma^2 M + \sigma C + K$, $\widehat{C} = C + 2\sigma M$, and $\widehat{K} = M$. The exterior eigenvalues $\mu$ of the QEP (9.17) approximate the eigenvalues $\lambda$ of the original QEP (9.1) closest to the shift $\sigma$. These eigenvalues $\lambda$ are given by

\begin{displaymath}
\sigma + \frac{1}{\mu}.
\end{displaymath}

Again, the corresponding generalized ``linear'' eigenvalue problem in terms of $\lambda$, rather than $\mu$, is

\begin{displaymath}
\twobytwo{-\widehat{C}}{-\widehat{K}}{I}{0}
\twobyone{x}{(\l...
...twobytwo{\widehat{M}}{0}{0}{I} \twobyone{x}{(\lambda-\sigma)x}
\end{displaymath}

or

\begin{displaymath}
\twobytwo{\widehat{C}}{\widehat{K}}{\widehat{K}}{0}
\twobyon...
...idehat{M}}{0}{0}{\widehat{K}} \twobyone{x}{(\lambda-\sigma)x},
\end{displaymath}

if the Hermitian of the matrix triplet $\{M,K,C\}$ wants to be preserved.



Susan Blackford 2000-11-20