Shift-and-Invert QEP.

Combining the above shift-and-invert spectral transformations (SI), the so-called shift-and-invert QEP becomes

$$\left(\mu^2 \widehat{M} + \mu \widehat{C} + \widehat{K} \right) x = 0,$$ (261)

where

$$\mu=\frac{1}{\lambda-\sigma},\quad$$

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

$$\sigma + \frac{1}{\mu}.$$

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

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

or

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

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