References DGEMM

stochastic_triples.tex

@@ -130,6 +130,26 @@ In the following sections of this paper, we will provide a brief introduction to
 %c. Challenges and limitations of traditional approaches
 
 
+The perturbative triples correction,
+\begin{equation}
+E_{(T)}  =  \sum_{ijkabc} E_{ijk}^{abc} 
+\end{equation}
+is a sum of $N=N_o^3 \times N_v^3$ terms,
+\begin{equation}
+ E_{ijk}^{abc}  =  \frac{(4 W_{ijk}^{abc} +
+              W_{ijk}^{bca} + W_{ijk}^{cab})
+              (V_{ijk}^{abc} - V_{ijk}^{cba})}{\epsilon_i + \epsilon_j + \epsilon_k -
+\epsilon_a - \epsilon_b - \epsilon_c} 
+\end{equation}
+which depend on the canonical orbital energies $\epsilon$, and the tensors $W$ and $V$.
+The indices $i,j,k$ and $a,b,c$ denote respectively occupied and virtual orbitals.
+
+The bottleneck is the computation of $W$, which requires $\order{N_o^3 \times
+N_v^4}$ operations. However, most of the operations involved in the computation of $W$
+can be recast into matrix multiplications, which are among the most efficient
+operations than can be executed on modern CPUs and
+accelerators.\cite{ma_2011,haidar_2015,dinapoli_2014,springer_2018}
+
 %=================================================================%
 \section{Semi-Stochastic Algorithm}
 \label{sec:algorithm}
@@ -145,6 +165,12 @@ In the following sections of this paper, we will provide a brief introduction to
 %c. Discussion of the algorithm's advantages and potential trade-offs
 %d. Detailed pseudocode or algorithmic steps, if applicable
 
+\subsection{Stochastic formulation}
+
+\subsection{Test code}
+\label{subsec:test_code}
+% Include the test code here, if applicable.
+
 
 %=================================================================%
 \section{Implementation Details}

triples.org

@@ -2,7 +2,7 @@
 
 * Stochastic formulation
 
-The perturbative correction is expressed as:
+The perturbative correction reads:
 
   \[
 E_{(T)} = \sum_{ijkabc} E_{ijk}^{abc}  = \sum_{ijkabc} \frac{(4 W_{ijk}^{abc} +
@@ -12,14 +12,19 @@ E_{(T)} = \sum_{ijkabc} E_{ijk}^{abc}  = \sum_{ijkabc} \frac{(4 W_{ijk}^{abc} +
 \epsilon_a - \epsilon_b - \epsilon_c}
   \]
 
-  
+where the indices $i,j,k$ run over the $N_o$ occupied orbitals, and
-If $E_{(T)}$ is computed as
+the indices $a,b,c$ run over the $N_v$ virtual orbitals.
+
+If $E_{(T)}$ is re-expressed as
   \[
 E_{(T)} = \sum_{ijkabc} \frac{1}{N} \left( E_{ijk}^{abc} \times N \right)
   \]
-where $N$ is the total number of terms, it can be computed by drawing
+where $N=N_o^3 N_v^3$ is the total number of terms, it can be computed by drawing
-uniformly samples $E_{ijk}^{abc}$ and computing the statistical
+uniformly samples $E_{ijk}^{abc}$ and using the statistical average as
-average.
+
+  \[
+E_{(T)} = N \times \langle E_{ijk}^{abc} \rangle
+  \]
 
 The stochastic calculation is unbiased, but the rate of convergence of
 the statistical error is be governed by the variance of the
@@ -115,8 +120,7 @@ $E_B$ stochastically and we can separate the buckets into stochastic
 
 \[
 E_{(T)} = \sum_{B \in \mathcal{D}} E_B + \frac{1}{|\mathcal{S}|} \sum_{B \in \mathcal{S}}
-\left \langle E^B_{abc} \times \frac{\epsilon_{\text{occ}} -
+\left \langle E^B_{abc} \times \frac{- \epsilon_a - \epsilon_b - \epsilon_c}{\mathcal{N}} \right \rangle_{P(a,b,c), (a,b,c) \in B}
-\epsilon_a - \epsilon_b - \epsilon_c}{\mathcal{N}} \right \rangle_{P(a,b,c), (a,b,c) \in B}
 \]
 
 All the buckets don't have the same size: the number of triplets per