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

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Bild 1 Parallel Generalized Eigenvalue Solver PQZ or QZ Björn Adlerborn adler Powered By Docstoc
					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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