# Bild 1 Parallel Generalized Eigenvalue Solver PQZ or QZ Björn Adlerborn adler

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```					Parallel Generalized Eigenvalue
Solver (PQZ or // QZ)

Björn Adlerborn (adler@cs.umu.se)
Joint work with Bo Kågström

Department of Computing Science – Umeå, Sweden

Trogir 2011-10-11
Round 2/3: 2 - 1

PQR PQR    PQZ PQZ

Ax = λ x   Ax = λ B x
Motivation

• Motivation 1: Lots of problems ends
up in finding
eigenvalues/eigenspaces.
– Want to be able to compute them fast
and accurate using HPC.
• Motivation 2: There exists no // QZ.
Goal & Method

• Solve the equation Ax = λBx to find all n
eigenvalues λ of a regular matrix pair
such that det(A - λB) =0
– A, B dense matices
– Method : Compute ortohgonal matrices Q and
Z such that (S,T) = (QTAZ, QTBZ) is in
generalized Schur Form, i.e S i quasi upper
trangular with 1x1 and 2x2 block on the
diagonal, while T is upper-triangular.
– The eigenvalues can easily be extracted just by
looking at the diagonal elements of (S,T).

(A,B)         (HR,T)       (H,T)          (S,T)
Key Features in PQZ (as in PQR)

• Iterative method
• AED
• Tightly coupled chains of bulges chased
down the diagonal of (H,T)
• Delayed updates
• Recursion / Bootstraping (under devl.)
– Adding a 2nd level of recursion soon
• Build in the same maner as and on
ScaLAPACK, LAPACK, PBLAS, BLACS and
BLAS. Contributing to ScalaPACK in a near
future.
PQZ vs PQR

• 2 matrices, twice the work, twice as
slow? Set B = I and we solve the same
type of problem…
• Infinite eigenvalues (elements in diag(T)
are 0 or close to 0).
• AED can fail.
• Inventing the wheel, not making it
rounder.
– No existing code to rely on in the recursive
calls (Compare with PQZ and PDHSEQR).
Future work

• Testing/Evaluation of the newly
developed recursion/bootstraping
code
• Finalize code for ScaLAPACK
contribution
• PHD exam ?

```
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