### write sth. for the documentation

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 ... ... @@ -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} }
 ... ... @@ -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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