| \end{Verbatim} |
| where \texttt{type} can be one of |
| \texttt{STSHIFT}, \texttt{STFOLD}, |
| \texttt{STSINV}, \texttt{STCAYLEY} or |
| \texttt{STSHELL}. |
| \texttt{STSINVERT}, \texttt{STCAYLEY}, |
| or \texttt{STSHELL}. |
| The \ident{ST} type can also be set with the command-line option \Verb!-st_type! followed by the name of the method (see Table \ref{tab:transforms}). The first four spectral transformations are described in detail in the rest of this section. The last possibility, \texttt{STSHELL}, uses a specific, application-provided spectral transformation. Section \ref{sec:shell} describes how to implement one of these transformations. |
| \begin{table} |
| Spectral Transformation & \ident{STType} & {\footnotesize Name} & Operator\\\hline |
| Shift of Origin & \texttt{STSHIFT} & \texttt{shift} & $B^{-1}A+\sigma I$\\ |
| Spectrum Folding & \texttt{STFOLD} & \texttt{fold} & $(B^{-1}A-\sigma I)^2$\\ |
| Shift-and-invert & \texttt{STSINV} & \texttt{sinvert} & $(A-\sigma B)^{-1}B$\\ |
| Cayley & \texttt{STCAYLEY} & \texttt{cayley} & $(A-\sigma B)^{-1}(A+\nu B)$\\ |
| Shift-and-invert & \texttt{STSINVERT}& \texttt{sinvert} & $(A-\sigma B)^{-1}B$\\ |
| Generalized Cayley & \texttt{STCAYLEY} & \texttt{cayley} & $(A-\sigma B)^{-1}(A+\nu B)$\\ |
| Shell Transformation & \texttt{STSHELL} & \texttt{shell} & \emph{user-defined}\\\hline |
| \end{tabular} } |
| \caption{\label{tab:transforms}Spectral transformations available in the \ident{ST} package.} |
| \subsection{Shift-and-invert} |
| The shift-and-invert spectral transformation (\texttt{STSINV}) is used to enhance convergence of eigenvalues in the neighborhood of a given value. In this case, the solver deals with the expressions |
| The shift-and-invert spectral transformation (\texttt{STSINVERT}) is used to enhance convergence of eigenvalues in the neighborhood of a given value. In this case, the solver deals with the expressions |
| \begin{eqnarray} |
| (A-\sigma I)^{-1}x=\theta x\;\;,\\ |
| (A-\sigma B)^{-1}B x=\theta x\;\;, |