The solution of the symmetric eigenproblem PDSYEVX consists
of three phases: (1) reduce the original matrix 
to tridiagonal
form 
where 
is orthogonal and 
is tridiagonal, (2)
find the eigenvalues 
and eigenvectors 
of so that 
,
and (3) form the eigenvector matrix 
of so
.
Phases 1 and 3 are analogous to their LAPACK counterparts,
similarly to LU. However, our current design for phase 2
differs from the serial (or shared memory) design. We have
chosen to do bisection followed by inverse iteration
(like the LAPACK expert driver DSYEVX), but with the
reorthogonalization phase of inverse iteration limited to the
eigenvectors stored in a single process. A straightforward
parallelization of DSYEVX would have led to a serial
bottleneck and significant slowdowns in the rare situation
of matrices with eigenvalues tightly clustered together.
The current design guarantees that phase (2) is inexpensive
compared to the other phases once problems are reasonably large.
An alternative algorithm which completely eliminates the need for
reorthogonalization has recently been discovered by Parlett,
Fernando, and Dhillon [17], and we expect to use this
version of the routine in the near future. This new routine
will guarantee high accuracy and high speed independent of
the eigenvalue distribution.