Corrected a typo in GD

This commit is contained in:
mhjensen
2018-09-20 06:03:42 +02:00
parent c92575ce3b
commit 93390e5dae
46 changed files with 9314 additions and 1408 deletions
-39
View File
@@ -630,45 +630,6 @@ Subtracting this equation from that of $\hat{x}_{i+1}$ we have
!et
!eblock
!split
===== Simple example and demonstration =====
!bblock
The function $f$ can be either the energy or the variance. If we choose the energy then we have
!bt
\begin{equation*}
\hat{\alpha}_{i+1}-\hat{\alpha}_i=\hat{A}^{-1}(\nabla E(\hat{\alpha}_{i+1})-\nabla E(\hat{\alpha}_i)).
\end{equation*}
!et
In the simple harmonic oscillator model, the gradient and the Hessian $\hat{A}$ are
!bt
\begin{equation*}
\frac{d\langle E_L[\alpha]\rangle}{d\alpha} = \alpha-\frac{1}{4\alpha^3}
\end{equation*}
!et
and a second derivative which is always positive (meaning that we find a minimum)
!bt
\begin{equation*}
\hat{A}= \frac{d^2\langle E_L[\alpha]\rangle}{d\alpha^2} = 1+\frac{3}{4\alpha^4}
\end{equation*}
!et
!eblock
!split
===== Simple example and demonstration =====
!bblock
We get then
!bt
\begin{equation*}
\alpha_{i+1}=\frac{4}{3}\alpha_i-\frac{\alpha_i^4}{3\alpha_{i+1}^3},
\end{equation*}
!et
which can be rewritten as
!bt
\begin{equation*}
\alpha_{i+1}^4-\frac{4}{3}\alpha_i\alpha_{i+1}^4+\frac{1}{3}\alpha_i^4.
\end{equation*}
!et
!eblock
!split
===== Conjugate gradient method =====
+2 -1
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@@ -44,11 +44,12 @@ system doconce split_html $html.html --method=space10
# Bootstrap style
html=${name}-bs
system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt
#system doconce split_html $html.html --method=split --pagination --nav_button=bottom
system doconce split_html $html.html --method=split --pagination --nav_button=bottom
# IPython notebook
system doconce format ipynb $name $opt
# Ordinary plain LaTeX document
rm -f *.aux # important after beamer
system doconce format pdflatex $name --minted_latex_style=trac --latex_admon=paragraph $opt