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ALF
ALF
Commits
c6aae37c
Commit
c6aae37c
authored
Dec 03, 2021
by
Florian Goth
Browse files
write sth. for the documentation
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Documentation/doc.bib
View file @
c6aae37c
...
...
@@ -371,3 +371,17 @@ eprint = {https://doi.org/10.1137/17M1117732}
timestamp
=
{2018.10.18}
,
url
=
{https://arxiv.org/abs/1810.06595}
,
}
@Article
{
Lee2013
,
Title
=
{Minimal Split Checkerboard Method for Exponentiating Sparse Matrices and Its Applications in Quantum Statistical Mechanics}
,
Author
=
{Lee, Che-Rung.}
,
Journal
=
{SIAM Journal on Scientific Computing}
,
Year
=
{2013}
,
Number
=
{2}
,
Pages
=
{C143-C171}
,
Volume
=
{35}
,
Doi
=
{10.1137/110838716}
,
Owner
=
{sifff}
,
Timestamp
=
{2020.01.05}
}
Documentation/mscb.tex
View file @
c6aae37c
...
...
@@ -12,7 +12,34 @@
ALF has a module for the automatic decomposition of arbitrary graphs/ hermitian matrices into
an almost minimal number of so-called strictly sparse matrices. We call a hermitian matrix
$
A
$
strictly sparse if it has the property
\begin{equation*}
A
_{
i,j
}
=0
\lrarrow
\left
( e
^
A
\right
)
_{
i,j
}
= 0
\forall
i,j
\end{equation*}
\begin{equation}
A
_{
i,j
}
= 0
\lrarrow
\left
( e
^
A
\right
)
_{
i,j
}
= 0
\forall
i,j
\label
{
eq:strictsparse
}
\end{equation}
To exploit that property for a mtrix,
\eg
the hopping matrix
$
T
$
we need a decomposition of
$
T
$
in the form
\begin{equation}
T=
\sum
_
i
^
N T
^
i
\label
{
eq:sep
}
\end{equation}
with each
$
T
^
i
$
having the property
\eqref
{
strictsparse
}
, because then by an application
of the Lie-Trotter formula a first order approximation of
$
e
^{
\Delta
\tau
T
}$
can be constructed.
Of course the question remains how to find a decomposition
\eqref
{
eq:sep
}
.
For some lattices you have seen an example of this type of decomposition in Figure
\ref
{
fig
_
predefined
_
lattices
}
.
As you might notice from the figure, all edges that are attached to a vertex have a different color.
And indeed, as pointed out by
\cite
{
Lee2013
}
a color decomposition will give a decomposition of
$
T
$
with the required properties. More generally the problem of coloring the edges of a graph
is known as the graph-coloring problem and the number of colors required for a graph
$
G
$
is known as
its chromaticity index
$
\chi
(
G
)
$
. An important structural result is Vizing's theorem
which states that a graph
$
G
$
of degree
$
g
$
(meaning the maximum of the number of edges over
$
G
$
) the following holds
\begin{equation}
g
\leq
\chi
(G)
\leq
g+1
\end{equation}
thereby enabling a classification of graphs into two classes.
Determining the chromaticity index for an arbitrary
$
G
$
is an NP-hard problem.
ALF does not try to solve this challenge but tries to find a decomposition with the help of the
Misra-van-Gries graph coloring algorithm.
This algorithm allows itself the freedom to utilize
$
g
+
1
$
colors, from which it can derive the important
knowledge that at every vertex it encounters during the process it has always one free color.
In ALF we allow to utilize higher-order checkerboard (as e.g. demonstrated in the appendix of
\cite
{
goth2020
}
)
via a config switch.
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