#
Quantum Chromodynamics

One research area that makes use of matrix functions is the field of
lattice quantum chromodynamics (QCD). QCD computations in
chemistry and physics
currently occupy large amounts of supercomputer time,
with a single simulation taking of the order of days,
and computing requirements in the subject are estimated
to be of the order of 10 teraflops by 2003.
Underlying many of these computations are matrix function evaluations.
Much recent work has focused on QCD simulations with the overlap-Dirac
operator, which has the form

###
* D = 1/2 ( 1+ *_{5}
(H_{}) ),

where _{5} =
diag( 1),
*H*_{ }
is the Hermitian Wilson-Dirac operator, and
*(A) =
A(A*^{*}A)^{-1/2}
is the unitary polar factor of *A*
(referred to by physicists as the ``step function'').
Calculations of quark propagators
require the computation of matrix-vector products involving *D*,
so the key computational requirement is to form
*
(H*_{})b
efficiently, for many *b*. Since
*H*_{}
is large and sparse, this must be done without computing
*
(H*_{}) explicitly.
Mathematically, the problem is to compute
*x = f(A)b* without explicitly forming *f(A)*,
with * f(A) = A(A*^{*}A)^{-1/2}
and *A* Hermitian.
As physicists refine their models we can expect other *f* to arise.

HEP-LAT(**H**igh **E**nergy
**P**hysics **LAT**tice gauge theory)
preprint archive server.
QCD Matrices
**S**tanford **L**inear
**A**ccelerator **C**enter (SLAC),
**S**tanford **P**ublic **I**nformation
**RE**trieval **S**ystem (SPIRES).
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