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\chapter{Introduction}
\label{chapter:introduction}
\par
The Lanczos eigensolver
finds selected eigenvalues and eigenvectors of
% three types of eigenproblems,
$AX = B X \Lambda$, where $X$ are eigenvectors
and $\Lambda$ is a diagonal matrix whose elements are eigenvalues.
Three types of eigenproblems are supported.
\begin{itemize}
\item
An ``ordinary'' eigenvalue problem
where $A$ is symmetric and $B = I$.
\item
An ``vibration'' eigenvalue problem
where $A$ is symmetric and
$B$ is symmetric positive semidefinite.
\item
A ``buckling'' eigenvalue problem
$A$ is symmetric positive semidefinite and
$B$ is symmetric.
\end{itemize}
For the vibration and buckling problems, there must exist a
$\sigma$ that is not an eigenvalue such that
$A - \sigma B$ is nonsingular,
i.e., $A$ and $B$ cannot share the same null space.
\par
During the computations, the eigensolver requires the following
sparse linear algebra computations.
\begin{itemize}
\item
Sparse factorizations of the form $A - \sigma B$.
\item
Solves of the form $(A - \sigma B) Z = Y$.
\item
Multiplies of the form $Z = B Y$ (for the vibration problem)
or $Z = A Y$ (for the buckling problem).
\end{itemize}
The Lanczos eigensolver has defined a specific interface with an
external linear algebra package to perform these three operations.
The eigensolver currently interfaces with the {\bf BCSLIB-EXT} linear
solver in a serial environment and the {\bf SPOOLES} linear solver
in serial, multithreaded and MPI environments.
\par
This paper documents the {\bf SPOOLES} objects and functions
that interface with the eigensolver.
The three following chapters describe the serial, multithreaded and
MPI objects, their data structures, and their methods.
The appendix contains listings of three driver programs to exercise
the eigensolver using the {\bf SPOOLES} library.
\par
Symmetric permutations of the eigensystem do not change the
eigenvalues, and the eigenvectors can be easily constructed
using the permutation matrix.
$$
A X = B X \Lambda
\longrightarrow
{\widehat A} {\widehat X}
= {\widehat B} {\widehat X} \Lambda
\quad \mbox{where} \quad
{\widehat A} = PAP^T, \quad
{\widehat B} = PBP^T, \quad
\mbox{and} \quad
{\widehat X} = PX
$$
The linear algebra package is free to use any permutation matrix
$P$ to most efficiently perform the factorizations and solves
involving ${\widehat A}$ and ${\widehat B}$.
This permutation matrix $P$ is typically found by ordering the
graph of $A + B$ using a variant of minimum degree or nested
dissection.
The ordering is performed prior to any action by the eigensolver.
This ``setup phase'' includes more than just finding the
permutation matrix, e.g., various data structures must be
initialized.
In a parallel environment, there is even more setup work to do,
analyzing the factorization and solves and specifying which threads
or processors perform what computations and store what data.
In a distributed environment, the entries of $A$ and $B$ must
also be distributed among the processors in preparation for
the factors and multiplies.
\par
For each of the three environments --- serial, multithreaded and
MPI --- the {\bf SPOOLES} solver has constructed a ``bridge''
object to span the interface between the linear system solver
and the eigensolver.
Each of the {\tt Bridge}, {\tt BridgeMT} and {\tt BridgeMPI}
objects have five methods: set-up, factor, solve, matrix-multiply
and cleanup.
The factor, solve and matrix-multiply methods follow the calling
sequence convention imposed by the eigensolver, and are passed
to the eigensolver at the beginning of the Lanczos run.
The set-up method is called prior to the eigensolver, and the
cleanup method is called after the eigenvalues and eigenvectors
have been determined.
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