From 805718880c936d778c99fe0d5cfdb238342a83c7 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 1 Jul 2021 22:21:26 +0200
Subject: Expand knowledge base

---
 content/know/concept/pulay-mixing/index.pdc | 13 ++++++-------
 1 file changed, 6 insertions(+), 7 deletions(-)

(limited to 'content/know/concept/pulay-mixing')

diff --git a/content/know/concept/pulay-mixing/index.pdc b/content/know/concept/pulay-mixing/index.pdc
index 8daa54f..4e7a411 100644
--- a/content/know/concept/pulay-mixing/index.pdc
+++ b/content/know/concept/pulay-mixing/index.pdc
@@ -16,8 +16,8 @@ by generating a series $\rho_1$, $\rho_2$, etc.
 converging towards the desired solution $\rho_*$.
 **Pulay mixing**, also often called
 **direct inversion in the iterative subspace** (DIIS),
-is an effective method to speed up convergence,
-which also helps to avoid periodic divergences.
+can speed up the convergence for some types of problems,
+and also helps to avoid periodic divergences.
 
 The key concept it relies on is the **residual vector** $R_n$
 of the $n$th iteration, which in some way measures the error of the current $\rho_n$.
@@ -113,17 +113,16 @@ $\lambda = - \braket{R_{n+1}}{R_{n+1}}$,
 where $R_{n+1}$ is the *predicted* residual of the next iteration,
 subject to the two assumptions.
 
-This method is very effective.
 However, in practice, the earlier inputs $\rho_1$, $\rho_2$, etc.
 are much further from $\rho_*$ than $\rho_n$,
-so usually only the most recent $N$ inputs $\rho_{n - N}$, ..., $\rho_n$ are used:
+so usually only the most recent $N\!+\!1$ inputs $\rho_{n - N}$, ..., $\rho_n$ are used:
 
 $$\begin{aligned}
 	\rho_{n+1}
-	= \sum_{m = N}^n \alpha_m \rho_m
+	= \sum_{m = n-N}^n \alpha_m \rho_m
 \end{aligned}$$
 
-You might be confused by the absence of all $\rho_m^\mathrm{new}$
+You might be confused by the absence of any $\rho_m^\mathrm{new}$
 in the creation of $\rho_{n+1}$, as if the iteration's outputs are being ignored.
 This is due to the first assumption,
 which states that $\rho_n^\mathrm{new}$ are $\rho_n$ are already similar,
@@ -155,7 +154,7 @@ while still giving more weight to iterations with smaller residuals.
 
 Pulay mixing is very effective for certain types of problems,
 e.g. density functional theory,
-where it can accelerate convergence by up to one order of magnitude!
+where it can accelerate convergence by up to two orders of magnitude!
 
 
 
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