Skip to content

Commit

Permalink
\Delta_{CT} is replaced by \Delta_{FB}
Browse files Browse the repository at this point in the history
  • Loading branch information
aziziph authored Oct 3, 2023
1 parent 3df7ff1 commit b1602bc
Showing 1 changed file with 2 additions and 2 deletions.
4 changes: 2 additions & 2 deletions JOSS/paper.md
Original file line number Diff line number Diff line change
Expand Up @@ -135,7 +135,7 @@ $\gamma_k$, $\delta_{kj}$, $\eta_{jk}$, $\lambda_{kj}$ depend on the energy gap
GX-TimeFrequency requires as input the grid size $n$, the minimal eigenvalue difference $\Delta_{\text{min}}$, and the maximal eigenvalue difference $\Delta_{\text{max}}$. For the output parameters, see Table \ref{tab:output}. The library component retrieves tabulated minimax parameters $\{\tau_j(R)\}_{j=1}^n$, $\{\sigma_j(R)\}_{j=1}^n$, $\{\omega_k(R)\}_{k=1}^n$, $\{\gamma_k(R)\}_{k=1}^n$ of the requested grid size $n$ for the smallest range $R$ that satisfies $R \ge \Delta_\text{max}/\Delta_\text{min}$. GX-TimeFrequency then rescales the retrieved minimax parameters according to Equation 6 with $\Delta_\text{min}$ and prints the results $\{\tau_j^\text{mat}\}_{j=1}^n$, $\{\sigma_j^\text{mat}\}_{j=1}^n$, $\{\omega_k^\text{mat}\}_{k=1}^n$, $\{\gamma_k^\text{mat}\}_{k=1}^n$. Fourier integration weights are computed on-the-fly via least-squares optimization. The precision of a global forward cosine transformation followed by backward cosine transformations, is measured from
\fontsize{8}{10}\selectfont
\begin{eqnarray}
\Delta_\text{CT}=\max_{j,j'\in\{1,2,\ldots,n\}} \left| \sum_{k=1}^n \eta_{j'k} \cos(\tau_{j'}\omega_k) \cdot \delta_{kj} \cos(\omega_k\tau_j) - (\mathbb{I})_{j'j}\right|
\Delta_\text{FB}=\max_{j,j'\in\{1,2,\ldots,n\}} \left| \sum_{k=1}^n \eta_{j'k} \cos(\tau_{j'}\omega_k) \cdot \delta_{kj} \cos(\omega_k\tau_j) - (\mathbb{I})_{j'j}\right|
\end{eqnarray}\normalsize
with $\mathbb{I}$ being the identity matrix. Inputs and outputs are in atomic units.

Expand All @@ -148,7 +148,7 @@ with $\mathbb{I}$ being the identity matrix. Inputs and outputs are in atomic un
|$\{\delta_{kj}\}_{k,j=1}^n$ | Fourier weights | ls RPA, ls \textit{GW} | on-the-fly L2 opt
| $\{\eta_{jk}\}_{k,j=1}^n$ | Fourier weights | ls \textit{GW} | on-the-fly L2 opt|
|$\{\lambda_{kj}\}_{k,j=1}^n$ | Fourier weights | ls \textit{GW} | on-the-fly L2 opt
| $\Delta_\text{CT}$ | duality error | ls \textit{GW} | on-the-fly |
| $\Delta_\text{FB}$ | duality error | ls \textit{GW} | on-the-fly |
: Output returned by the GX-TimeFrequency component of GreenX. We abbreviate low-scaling as ls, and least-squares optimization as L2 opt.\label{tab:output}

# Acknowledgements
Expand Down

0 comments on commit b1602bc

Please sign in to comment.