From 69b7882b75a4d9ed9e396bbd8ee2a12df01fe2c4 Mon Sep 17 00:00:00 2001
From: =?utf8?q?Bj=C3=B8rn=20Rustad?= <rustadbjornen@gmail.com>
Date: Mon, 3 Oct 2011 14:10:16 +0200
Subject: [PATCH] Small fixing

---
 rapport.tex | 20 ++++++++++----------
 1 file changed, 10 insertions(+), 10 deletions(-)

diff --git a/rapport.tex b/rapport.tex
index 30222c1..6634a4f 100644
--- a/rapport.tex
+++ b/rapport.tex
@@ -130,14 +130,6 @@ that the Hessian, $\nabla^{2}g(\mathbf{x})$, is positive definite.
   \label{fig:fn_resid}
 \end{figure}
 
-\begin{figure}[b]
-  \centering
-  \includegraphics[width=0.9\textwidth]{surface}
-  \caption{\sf Surface plot of function with points showing iteration of steepest
-  descent method, both with optimal and constant $\alpha$}
-  \label{fig:surface}
-\end{figure}
-
 The steepest descent method works by always taking a step in the direction
 opposite the gradient in the point considered (i.e. the direction of the
 steepest descent). The step size can be set to a constant or chosen in a more
@@ -170,14 +162,14 @@ by solving the following equation for $\alpha$:
     \mathbf{\alpha}}g\left(\mathbf{x}+\alpha\mathbf{u}\right) &=& 0, \text{where  }
     \mathbf{u} = -\nabla g(\mathbf{x} )  \nonumber \\
   &\Downarrow& \nonumber \\
-  A\alpha^3 + B\alpha^2 + C\alpha + D &=& 0,
+  A\alpha^3 + B\alpha^2 + C\alpha + D &=& 0.
 \end{eqnarray}
 Where $A$, $B$, $C$, and $D$ are given by
 \begin{eqnarray}
   A &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{u})\mathbf{u} \nonumber \\
   B &=& \mathbf{x}^TC(\mathbf{u}) \mathbf{u} \nonumber \\
   C &=& \mathbf{u}^TH\mathbf{u}+\mathbf{u}^TC(\mathbf{x})\mathbf{u} \nonumber \\
-  D &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{x})\mathbf{x}+\mathbf{u}^TH\mathbf{x}-\mathbf{b}^T\mathbf{u}
+  D &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{x})\mathbf{x}+\mathbf{u}^TH\mathbf{x}-\mathbf{b}^T\mathbf{u}.
 \end{eqnarray}
 Differentiating the polynomial results in:
 \begin{eqnarray}
@@ -221,4 +213,12 @@ combination method results in fewer iterations. This comes from the fact that
 computing the optimal step size requires solving a cubic equation and is
 computationally expensive.
 
+\begin{figure}[hb]
+  \centering
+  \includegraphics[width=0.9\textwidth]{surface}
+  \caption{\sf Surface plot of function with points showing iteration of steepest
+  descent method, both with optimal and constant $\alpha$}
+  \label{fig:surface}
+\end{figure}
+
 \end{document}
-- 
2.47.3