\Delta_{CT} is replaced by \Delta_{FB}

JOSS/paper.md

@@ -107,7 +107,7 @@ with $F^\text{even}(x)=F^\text{even}(-x)$ and $F^\text{odd}(x)=-F^\text{odd}(-x)
   \begin{eqnarray}
    F^\text{odd}(i\omega_k) &=& i\sum_{j=1}^{n} \lambda_{kj} \mathrm{sin}(\omega_k\tau_j)\hat{F}^\text{odd}(i\tau_j)
     \\
-    \hat{F}^\text{odd}(i\tau_j) &=& -i \sum_{k=1}^{n} \zeta_{jk} \mathrm{sin}(\tau_j\omega_k)F^\text{odd}(i\omega_k)
+    \hat{F}^\text{odd}(i\tau_j) &=& -i\sum_{k=1}^{n} \zeta_{jk} \mathrm{sin}(\tau_j\omega_k)F^\text{odd}(i\omega_k)
   \end{eqnarray}
 \end{minipage}
 where $\{\tau_j\}_{j=1}^n, \tau_j\,{>}\,0$ are again the time grid points, $\{\omega_k\}_{k=1}^n,\omega_k\,{>}\,0$ frequency grid points and $\{\delta_{kj}\}_{k,j=1}^n$, $\{\eta_{jk}\}_{k,j=1}^n$,$\{\lambda_{kj}\}_{k,j=1}^n$, $\{\zeta_{jk}\}_{k,j=1}^n$ the corresponding Fourier integration weights. $\hat{\chi}^0(i\tau)$ is an even function: the transform defined in Equation 2 yields $\chi^0(i\omega)$. The screened Coulomb interaction is also even and Equation 3 converts $W(i\omega)$ to $\widehat{W}(i\tau)$. The self-energy, neither odd nor even, is treated with Equation 1 in combination with Equations 2 and 4 to transform $\widehat{\Sigma}(i\tau)$ to $\Sigma(i\omega)$ [@liu2016cubic]. The transformation defined in Equation 5, not required for the methods summarized in Fig. \ref{fig:flowchart}, is added for completeness.
@@ -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.
 
@@ -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  |
+|  $\Delta_\text{FB}$ | forward-backward 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