Chemistry Reference
In-Depth Information
(If.only.eigenvalues.are.required.then.the.routines.offer.identical.performance.).The.
two. routines. mentioned. above. differ. in. the. algorithm. used. to. produce. the. eigen-
vectors..The.routine.DSYGVD.uses.a.“divide.and.conquer”.approach,.signiicantly.
reducing.the.time.required.to.complete.the.solution.to.the.eigenvalue.problem..Both.
of.the.algorithms.discussed.in.this.section.scale.(in.number.of.operations).as. O ( n 3 ).
where. n .is.the.dimension.of.the.matrix..In.practice,.the.divide.and.conquer.approach.
has. a. smaller. prefactor. m . where. performance. scales. as. mn 3 . and. hence. has. better.
performance.
The. procedure. for. solving. the. generalized. symmetric. eigenvalue. problem. pro-
ceeds. in. three. major. steps:.(1). the. generalized. eigenvalue. problem. is. reduced. to. a.
standard.eigenvalue.problem,.(2).the.matrix.is.reduced.to.tridiagonal.form,.and.(3).
the.eigenvalues.and.eigenvectors.are.produced..Recently.a.new.algorithm.known.as.
MRRR. [68]. (multiple. relatively. robust. representations). for. calculating. eigenvalues.
from.a.tridiagonal.matrix.was.introduced..The.scaling.of.this.algorithm.is. O ( n 2 ).and.
thus.should.be.faster.than.the.two.algorithms.mentioned.above..However,.because.
this. algorithm. only. reduces. the. scaling. for. one. of. the. three. steps. in. the. solution.
of. the. generalized. symmetric. eigenvalue. problem,. the. overall. scaling. of. the. algo-
rithm.remains. O ( n 3 )..The.current.release.of.LAPACK.does.not.include.a.routine.for.
solving.the.generalized.symmetric.eigenvalue.problem.that.incorporates.the.MRRR.
algorithm..However,.it.is.straightforward.to.modify.an.existing.LAPACK.to.use.the.
MRRR. algorithm.. We. have. done. this. and. call. the. resulting. subroutine. DSYGVR.
in.accordance.with.the.LAPACK.naming.scheme..In.practice,.we.have.found.that.
the. performance. of. DSYGVD. and. DSYGVR. are. very. close.. Thus. we. plot. results.
only. for. DSYGVR,. which.are.representative. of. the. performance. characteristics. of.
DSYGVD..This.result.is.surprising.in.light.of.the.discussion.above,.and.illustrates.
the.importance.of.testing.these.algorithms.under.realistic.conditions.
A. plot. comparing. the. performance. of. the. DSYGVX. and. DSYGVR. routines.
is. given. in. Figure. 8.7.. Note. that. the. DSYGVR. routine. is. 2.5. times. faster. than.
the. DSYGVX. routine. at. the. largest. problem. size. (8307. orbitals).. In. general,. the.
DSYGVD.and.DSYGVR.routines.are.appreciably.faster.than.DSYGV(X).and.their.
use.is.recommended.
In. closing,. we. emphasize. the. importance. of. optimized. BLAS. libraries. as. the.
key.to.achieving.good.performance.on.a.wide.variety.of.scientiic.problems..These.
libraries.are.available.from.vendors.[86].(e.g.,.ACML.[86],.ESSL.[87],.MKL.[88]).
and.from.other.projects.(e.g.,.ATLAS.[89],.GotoBLAS.[90])..In.the.present.work,.we.
use.the.Goto.BLAS.library.(version.2).
8.6.2  P arallelization
Again,.we.primarily.consider.solution.of.the.eigenvalue.problem.as.this.dominates.
the.time.required.for.calculation.of.the.TB.energy..As.before,.the.choice.of.algorithm.
will.have.a.strong.effect.on.the.eficiency.of.this.step..In.the.case.of.a.parallel.algo-
rithm,.eficiency.will.also.refer.to.speedup.on.multiple.processors.compared.to.the.
single.processor.performance.
Modern.parallel.computers.generally.fall.into.two.categories..Symmetric.mul-
tiprocessor.(SMP).machines.are.characterized.by.having.more.than.one.CPU.(or.
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