# Subversion Repositoriesslepc-dev

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Rev 2095 Rev 2103
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Spectral Transformation & \ident{STType}    & {\footnotesize Name}    & Operator\\\hline
Spectral Transformation & \ident{STType}    & {\footnotesize Name}    & Operator\\\hline
Shift of Origin         & \texttt{STSHIFT}  & \texttt{shift}   & $B^{-1}A+\sigma I$\\
Shift of Origin         & \texttt{STSHIFT}  & \texttt{shift}   & $B^{-1}A+\sigma I$\\
Spectrum Folding        & \texttt{STFOLD}   & \texttt{fold}    & $(B^{-1}A-\sigma I)^2$\\
Spectrum Folding        & \texttt{STFOLD}   & \texttt{fold}    & $(B^{-1}A-\sigma I)^2$\\
Shift-and-invert        & \texttt{STSINVERT}& \texttt{sinvert} & $(A-\sigma B)^{-1}B$\\
Shift-and-invert        & \texttt{STSINVERT}& \texttt{sinvert} & $(A-\sigma B)^{-1}B$\\
Generalized Cayley      & \texttt{STCAYLEY} & \texttt{cayley}  & $(A-\sigma B)^{-1}(A+\nu B)$\\
Generalized Cayley      & \texttt{STCAYLEY} & \texttt{cayley}  & $(A-\sigma B)^{-1}(A+\nu B)$\\
Preconditioner          & \texttt{STPRECOND}& \texttt{precond} & $A-\sigma B$\\\hline
Preconditioner          & \texttt{STPRECOND}& \texttt{precond} & $K^{-1}\approx(A-\sigma B)^{-1}$\\\hline
Shell Transformation    & \texttt{STSHELL}  & \texttt{shell}   & \emph{user-defined}\\\hline
Shell Transformation    & \texttt{STSHELL}  & \texttt{shell}   & \emph{user-defined}\\\hline
\end{tabular} }
\end{tabular} }
\caption{\label{tab:transforms}Spectral transformations available in the  \ident{ST} package.}
\caption{\label{tab:transforms}Spectral transformations available in the  \ident{ST} package.}
\end{table}
\end{table}


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        \texttt{cayley}
        \texttt{cayley}
        & $\sigma\not=0,\nu=0$ & $(A-\sigma I)^{-1}A$  & $(A-\sigma B)^{-1}A$ \\
        & $\sigma\not=0,\nu=0$ & $(A-\sigma I)^{-1}A$  & $(A-\sigma B)^{-1}A$ \\
        & $\sigma=0,\nu\not=0$     & $I+\nu A^{-1}$      & $I+\nu A^{-1}B$ \\
        & $\sigma=0,\nu\not=0$     & $I+\nu A^{-1}$      & $I+\nu A^{-1}B$ \\
        & $\sigma\not=0,\nu\not=0$ & $(A-\sigma I)^{-1}(A+\nu I)$  & $(A-\sigma B)^{-1}(A+\nu B)$ \\ \hline
        & $\sigma\not=0,\nu\not=0$ & $(A-\sigma I)^{-1}(A+\nu I)$  & $(A-\sigma B)^{-1}(A+\nu B)$ \\ \hline
        \texttt{precond}
        \texttt{precond}
        & $\sigma=0$     & $A$           & $A$          \\
        & $\sigma=0$     & $K^{-1}\approx A^{-1}$ & $K^{-1}\approx A^{-1}$          \\
        & $\sigma\not=0$ & $A-\sigma I$  & $A-\sigma B$ \\ \hline
        & $\sigma\not=0$ & $K^{-1}\approx(A-\sigma I)^{-1}$ & $K^{-1}\approx(A-\sigma B)^{-1}$ \\ \hline
        \end{tabular} }
        \end{tabular} }
        \caption{\label{tab:op}Operators used in each spectral transformation mode.}
        \caption{\label{tab:op}Operators used in each spectral transformation mode.}
        \end{table}
        \end{table}


        The expressions shown in Table \ref{tab:op} are not built explicitly. Instead, the appropriate operations are carried out when applying the operator to a certain vector. The inverses imply the solution of a linear system of equations that is managed by setting up an associated \ident{KSP} object. The user can control the behavior of this object by adjusting the appropriate options, as will be illustrated with examples in \S\ref{sec:lin}.
        The expressions shown in Table \ref{tab:op} are not built explicitly. Instead, the appropriate operations are carried out when applying the operator to a certain vector. The inverses imply the solution of a linear system of equations that is managed by setting up an associated \ident{KSP} object. The user can control the behavior of this object by adjusting the appropriate options, as will be illustrated with examples in \S\ref{sec:lin}.
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In this case, the relation between the eigenvalues of both problems is
In this case, the relation between the eigenvalues of both problems is
\begin{equation}\theta=(\lambda+\nu)/(\lambda-\sigma)\;\;.\end{equation}
\begin{equation}\theta=(\lambda+\nu)/(\lambda-\sigma)\;\;.\end{equation}
Therefore, after the solution process, the operation to be performed in function \ident{STBackTransform} is $\lambda=(\theta\sigma+\nu)/(\theta-1)$ for each of the computed eigenvalues.
Therefore, after the solution process, the operation to be performed in function \ident{STBackTransform} is $\lambda=(\theta\sigma+\nu)/(\theta-1)$ for each of the computed eigenvalues.


\subsection{Preconditioner}
\subsection{Preconditioner}

\label{sec:precond}


        As mentioned in the introduction of this chapter, the special type \ident{STPRECOND} is used for handling preconditioners or preconditioned iterative linear solvers, which are used in the context of preconditioned eigensolvers for expanding the subspace. For instance, in the GD solver the so-called correction vector $d_i$ to be added to the subspace in each iteration is computed as
        As mentioned in the introduction of this chapter, the special type \ident{STPRECOND} is used for handling preconditioners or preconditioned iterative linear solvers, which are used in the context of preconditioned eigensolvers for expanding the subspace. For instance, in the GD solver the so-called correction vector $d_i$ to be added to the subspace in each iteration is computed as
\begin{equation}

d_i=K^{-1}(A-\theta_i B)x_i,

\end{equation}

or sometimes as

\begin{equation}
\begin{equation}
d_i=K^{-1}P_i(A-\theta_i B)x_i,
d_i=K^{-1}P_i(A-\theta_i B)x_i,
\end{equation}
\end{equation}
where $(\theta_i,x_i)$ is the current approximation of the sought-after eigenpair, and $P_i$ is a projector involving $x_i$ and $K^{-1}x_i$. In the above expressions, $K$ is a preconditioner matrix that is built from $A-\theta_i B$. However, since $\theta_i$ changes at each iteration, which would force recomputation of the preconditioner, we opt for using
where $(\theta_i,x_i)$ is the current approximation of the sought-after eigenpair, and $P_i$ is a projector involving $x_i$ and $K^{-1}x_i$. In the above expressions, $K$ is a preconditioner matrix that is built from $A-\theta_i B$. However, since $\theta_i$ changes at each iteration, which would force recomputation of the preconditioner, we opt for using
\begin{equation}
\begin{equation}
\label{eq:precon}
\label{eq:precon}
K\approx A-\sigma B.
K^{-1}\approx (A-\sigma B)^{-1}.
\end{equation}
\end{equation}


        Similarly, in the JD eigensolver the expansion of the subspace is carried out by solving the correction equation
        Similarly, in the JD eigensolver the expansion of the subspace is carried out by solving a correction equation similar to
\begin{equation}
\begin{equation}
(I-x_ix_i^*)(A-\theta_i B)(I-x_ix_i^*)d_i=-(A-\theta_i B)x_i,
(I-x_ix_i^*)(A-\theta_i B)(I-x_ix_i^*)d_i=-(A-\theta_i B)x_i,
\end{equation}
\end{equation}
or a similar one, where the system is solved approximately with a preconditioned iterative linear solver. For building the preconditioner of this linear system, the projectors $I-x_ix_i^*$ are ignored, and again it is not recomputed in each iteration. Therefore, the preconditioner is built as in expression \ref{eq:precon} as well.
where the system is solved approximately with a preconditioned iterative linear solver. For building the preconditioner of this linear system, the projectors $I-x_ix_i^*$ are ignored, and again it is not recomputed in each iteration. Therefore, the preconditioner is built as in expression \ref{eq:precon} as well.


        It should be clear from the previous discussion, that \ident{STPRECOND} does not work in the same way as the rest of spectral transformations. In particular, it is not intended to be used on the basis of operator applications with \ident{STApply}, and it does not rely on \ident{STBackTransform} either. It is rather a convenient mechanism for handling the preconditioner and linear solver (see examples in \S\ref{sec:lin}). The expressions shown in Tables \ref{tab:transforms} and \ref{tab:op} are just a reference to indicate from which matrix the preconditioner is built by default.
        It should be clear from the previous discussion, that \ident{STPRECOND} does not work in the same way as the rest of spectral transformations. In particular, it is not intended to be used on the basis of operator applications with \ident{STApply}, and it does not rely on \ident{STBackTransform} either. It is rather a convenient mechanism for handling the preconditioner and linear solver (see examples in \S\ref{sec:lin}). The expressions shown in Tables \ref{tab:transforms} and \ref{tab:op} are just a reference to indicate from which matrix the preconditioner is built by default.


        There is the possibility that the user overrides the default behaviour, that is, to explicitly supply a matrix from which the preconditioner is to be built, with
        There is the possibility that the user overrides the default behaviour, that is, to explicitly supply a matrix from which the preconditioner is to be built, with
        \findex{STPrecondSetMatForPC}
        \findex{STPrecondSetMatForPC}