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of.their.dense.variants..This.is.a.consequence.of.the.signiicantly.smaller.number.of.
operations.that.must.be.performed..Ultimately,.sparse.matrix.methods.can.scale.as.
O ( n ).which.is.highly.desirable.
Sparse.matrix.methods.hold.a.great.deal.of.promise.for.the.future.of.many.TB.
applications..The.DFTB+.code.has.recently.implemented.sparse.storage.techniques.
and. plans. to. replace. dense. eigenvalue. solvers. with. their. sparse. equivalent. in. the.
near. future. [79].. Zhang. et. al.. [101]. have. implemented. a. shift-and-invert. spectral.
transformation.(SIPs).algorithm.which.scales.as. O ( n 2 ).and.demonstrated.its.use.for.
systems. up. to. dimension. 64,000. with. a. signiicant. increase. in. performance. over.
ScaLAPACK.solvers..It.is.important.to.note.that.the.SIPs.algorithm.is.capable.of.
computing. most. or. all. of. the. eigenvalues. (and. associated. eigenvectors). that. are.
required.by.the.electronic.structure.problem..A.sparse,.parallel.implementation.of.
TB.has.been.reported.by.Colombo.and.Sawyer.[102]..If.only.a.small.portion.of.the.
eigenvalue.spectrum.is.needed,.the.matrix.dimension.may.go.into.the.millions.as.
demonstrated.by.Naumov.et.al..[103].using.an.algorithm.implemented.in.NEMO-3D.
[104]..Computations.of.optical.spectra,.as.discussed.in.Section.5.2,.may.beneit.from.
such.algorithms.
8.7  CONCLUSION
In.this.chapter,.we.have.presented.a.summary.of.some.of.the.important.historical.
developments.in.the.theory.of.TB.and.have.presented.several.modern.applications.of.
this.method..We.have.discussed.some.of.the.most.important.aspects.of.the.perfor-
mance.of.TB.algorithms..Particular.attention.has.been.given.to.parallelization.and.
sparse.matrix.methods.
REFERENCES
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