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Parallel Generalized Eigenvalue Solver (PQZ or // QZ)

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. PQZ. PQR. PQZ. PQR. Round 2/3: 2 - 1. Ax = λ x. Ax = λ B x. Motivation.

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Parallel Generalized Eigenvalue Solver (PQZ or // QZ)

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  1. 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

  2. PQZ PQR PQZ PQR Round 2/3: 2 - 1 Ax = λ x Ax = λ B x

  3. 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.

  4. 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)

  5. 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.

  6. 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).

  7. Future work Testing/Evaluation of the newly developed recursion/bootstraping code Finalize code for ScaLAPACK contribution PHD exam ?

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