Small fixing

diff --git a/rapport.tex b/rapport.tex
index 30222c15f8ed76771200c3915e3f275c978ea924..6634a4f0d0aff271f95e54d883795e40325da354 100644 (file)
--- 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}\r
 \end{figure}\r
 \r
-\begin{figure}[b]\r
-  \centering\r
-  \includegraphics[width=0.9\textwidth]{surface}\r
-  \caption{\sf Surface plot of function with points showing iteration of steepest\r
-  descent method, both with optimal and constant $\alpha$}\r
-  \label{fig:surface}\r
-\end{figure}\r
-\r
 The steepest descent method works by always taking a step in the direction\r
 opposite the gradient in the point considered (i.e. the direction of the\r
 steepest descent). The step size can be set to a constant or chosen in a more\r
@@ -170,14 +162,14 @@ by solving the following equation for $\alpha$:
     \mathbf{\alpha}}g\left(\mathbf{x}+\alpha\mathbf{u}\right) &=& 0, \text{where  }\r
     \mathbf{u} = -\nabla g(\mathbf{x} )  \nonumber \\\r
   &\Downarrow& \nonumber \\\r
-  A\alpha^3 + B\alpha^2 + C\alpha + D &=& 0,\r
+  A\alpha^3 + B\alpha^2 + C\alpha + D &=& 0.\r
 \end{eqnarray}\r
 Where $A$, $B$, $C$, and $D$ are given by\r
 \begin{eqnarray}\r
   A &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{u})\mathbf{u} \nonumber \\\r
   B &=& \mathbf{x}^TC(\mathbf{u}) \mathbf{u} \nonumber \\\r
   C &=& \mathbf{u}^TH\mathbf{u}+\mathbf{u}^TC(\mathbf{x})\mathbf{u} \nonumber \\\r
-  D &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{x})\mathbf{x}+\mathbf{u}^TH\mathbf{x}-\mathbf{b}^T\mathbf{u}\r
+  D &=& \frac{1}{3}\mathbf{u}^TC(\mathbf{x})\mathbf{x}+\mathbf{u}^TH\mathbf{x}-\mathbf{b}^T\mathbf{u}.\r
 \end{eqnarray}\r
 Differentiating the polynomial results in:\r
 \begin{eqnarray}\r
@@ -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\r
 computationally expensive.\r
 \r
+\begin{figure}[hb]\r
+  \centering\r
+  \includegraphics[width=0.9\textwidth]{surface}\r
+  \caption{\sf Surface plot of function with points showing iteration of steepest\r
+  descent method, both with optimal and constant $\alpha$}\r
+  \label{fig:surface}\r
+\end{figure}\r
+\r
 \end{document}\r