Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 258

600

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0

0 2000 4000 6000 8000

Number of orbitals

FIGURE 8.9 Peformance.of.various.ScaLAPACK.parallel.eigenvalue.solvers.for.gold.clus-
ters. with. sizes. ranging. from. 13. to. 923. atoms. (117-8307. orbitals).. The. solid. line. indicates.
results.for.the.PDSYGVX.routine,.while.the.dashed.line.indicates.results.for.the.PDSYGVR.
routine.. Symbols. indicate,. two. processors. (squares),. four. processors. (triangles),. and. eight.
processors.(diamonds).

give.the.best.performance..We.compare.the.timings.of.PDSYGVX.to.PDSYGVR.in.
Figure.8.9.(again,.the.results.for.PDSYGVD.are.similar.to.the.results.for.PDSYGVR)..
We.irst.note.that.the.timings.for.all.of.the.parallel.routines.are.larger.than.for.the.
corresponding.multithreaded.cases..We.attribute.this.difference.to.overhead.in.the.
parallel.implementation..This.is.an.important.result.indicating.that.the.best.perfor-
mance.may.be.obtained.using.multithreaded.BLAS.when.running.on.a.single.SMP.
node..However,.we.are.concerned.with.the.ScaLAPACK.routines.that.may.be.run.on.
more.than.one.node..From.Figure.8.9,.the.better.performance.of.PDSYGVR.versus.
PDSYGVX.is.clear..At.the.largest.problem.size,.PDSYGVR.is.20%.more.eficient.
than.PDSYGVX..The.parallel.eficiency.is.very.much.different.in.this.case,.where.
PDSYGVX.has.a.parallel.eficiency.of.69%.for.eight.processors.at.the.largest.prob-
lem.size.versus.33%.for.PDSYGVR.

8.6.3 S ParSe m atrix t echniqueS
In.the.previous.examples.of.this.section,.we.have.ignored.memory.usage..Storage.for.
the.Hamiltonian.and.overlap.matrices.requires. n 2 .double.precision.words..Additional.
storage.of. n 2 .is.required.for.several.of.the.algorithms.for.the.eigenvectors..Temporary.
storage.varies.from. O ( n ).to. O ( n 2 )..In.Table.8.2,.we.show.the.memory.requirements.
for. storing. a. single. matrix. associated. with. the. gold. clusters. we. considered. above.
as.well.as.some.larger.clusters.in.the.same.family..It.is.clear.to.see.that.the.storage.
requirements.quickly.become.a.problem.