From: Bjørn Rustad <rustadbjornen@gmail.com>
Date: Mon, 3 Oct 2011 22:05:54 +0000 (+0200)
Subject: Finishing up, report handed in
X-Git-Url: http://git.rustad.me/?a=commitdiff_plain;h=refs%2Fheads%2Fmaster;p=nummat1

Finishing up, report handed in
---

diff --git a/main.py b/main.py
index 78c24a4..5e05335 100644
--- a/main.py
+++ b/main.py
@@ -13,11 +13,11 @@ def main():
     b = np.array([-0.3119187,  0.14508516])
     c = np.array([ 0.47736331, 1.48834978])
 
-    x0 = np.array([-50, 50])
+    x0 = np.array([-1.28, -1.2])
     
     function_values = []
     
-    alpha = 0.001
+    alpha = 0.1
     tolerance = 0.000000001
 
     start = time.time()
@@ -47,7 +47,7 @@ def main():
 
     mlab.figure(figure="Surface", bgcolor=(1,1,1), fgcolor=(0,0,0))
 
-    plen = 15
+    plen = 200
     for point in optimal_points[:plen]:
         z = function_g(point, H, b, c)
         f_optimal.append(z)
diff --git a/rapport.tex b/rapport.tex
index 6634a4f..5c30469 100644
--- a/rapport.tex
+++ b/rapport.tex
@@ -1,6 +1,5 @@
 % Fag: Numerisk matematikk 
 % Project 1
-% Utført av: Bjørn, Mats og Edvard
 
 \documentclass[12pt, a4paper]{article}
 \usepackage[latin1]{inputenc}
@@ -61,7 +60,7 @@
 The function analyzed in this project was
 \begin{equation}
   \label{eq:func_g}
-  g(\mathbf{x}) := -\mathbf{b}^{T}\mathbf{x} +
+  g(\mathbf{x}) = -\mathbf{b}^{T}\mathbf{x} +
   \frac{1}{2}\mathbf{x}^{T}H\mathbf{x} +
   \frac{1}{12}\mathbf{x}^{T}C(\mathbf{x})\mathbf{x},
 \end{equation}
@@ -73,13 +72,13 @@ this purpose we used the following randomly generated data:
   \begin{bmatrix}
     1.8580682 & -1.01197101 \\
     -1.01197101 & 0.63456543 
-  \end{bmatrix}
-  ,  \mathbf{b} =
+  \end{bmatrix},
+  \mathbf{b} =
   \begin{bmatrix}
     -0.3119187 \\
     0.14508516
-  \end{bmatrix}
-  ,  \mathbf{c} =
+  \end{bmatrix},
+  \mathbf{c} =
   \begin{bmatrix}
     0.47736331 \\
     1.48834978
@@ -105,6 +104,7 @@ From equation \eqref{eq:func_g} we get
 \end{eqnarray}
 The three functions, \eqref{eq:func_g}, \eqref{eq:grad_g} and \eqref{eq:hess_g}, are
 implemented in the Python code below.
+\newpage
 
 \lstinputlisting[language=Python]{bj_function.py}
 
@@ -143,9 +143,9 @@ Consider the equation
   \mathbf{r}^{(0)}.
 \end{equation}
 One can view the function $\mathbf{r}(t)$ as a parameterisation of the curve
-going through the starting point to the minima, always going in the direction of
-the steepest descent. By applying the forward Euler method to this differential
-vector equation, we get the following iteration:
+going through the starting point and to the minima, always going in the
+direction of the steepest descent. By applying the forward Euler method to this
+differential vector equation, we get the following iteration:
 \begin{equation}
   \label{eq:forward_iter}
   \mathbf{r}^{(k+1)} = \mathbf{r}^{(k)} - \alpha \nabla g\left(\mathbf{r}^{(k)}\right)
@@ -154,8 +154,9 @@ where $t_{n+1} = t_n + \alpha$. Looking at this result, we see that it is
 equivalent to the steepest descent iteration with constant step size $\alpha$. 
 
 To find the optimal step length, $\alpha^*$, we walk along the line of steepest
-descent until we get to the lowest point. This point can be found analytically
-by solving the following equation for $\alpha$:
+descent until we get to the lowest point and it starts going upwards again.
+This point can be found analytically by solving the following equation for
+$\alpha$:
 \begin{eqnarray}
   \label{eq:poly}
   \frac{\partial}{\partial
@@ -171,7 +172,8 @@ Where $A$, $B$,
   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}.
 \end{eqnarray}
-Differentiating the polynomial results in:
+Differentiating the polynomial and using the fact that the Hessian is positive
+definite we get the following:
 \begin{eqnarray}
   \frac{\partial^2}{\partial \alpha^2}\left(g\left(\mathbf{x} +
     \alpha\mathbf{u}\right)\right) &=& \mathbf{u}^T H \mathbf{u} + \mathbf{u}^T
@@ -180,7 +182,7 @@ Differentiating the polynomial results in:
     C\left(\mathbf{u}\right) \mathbf{u}\alpha \nonumber \\
   &=& \mathbf{u}^T H \mathbf{u} + \mathbf{u}^T C\left(\mathbf{x} +
     \alpha\mathbf{u}\right)\mathbf{u} \nonumber \\
-  &=& \mathbf{u}^T\nabla^2 g\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u}.
+  &=& \mathbf{u}^T\nabla^2 g\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u} > 0.
 \end{eqnarray}
 From this we see that the polynomial \eqref{eq:poly} is monotonically increasing
 and therefore only has one real zero. This is the zero we want to use as our
@@ -207,15 +209,17 @@ The Newton method was also implemented to solve $\nabla g\left(\mathbf{x}^*\righ
   \end{center}
   \label{tab:time}
 \end{table}
-We see from table \ref{tab:time} that newtons method uses the least time to
-solve the problem, even though figure \ref{fig:fn_resid} shows that the
-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.
+We see from table \ref{tab:time} that Newtons method uses the least time to
+solve an instance of the problem, even though we adjusted the combination method
+to use as few iterations as possible as shown in figure \ref{fig:fn_resid}.
+This comes from the fact that computing the optimal step size requires solving a
+cubic equation and is computationally expensive. A faster and more approximated
+solution for the optimal step size could make the steepest descent method even
+faster.
 
 \begin{figure}[hb]
   \centering
-  \includegraphics[width=0.9\textwidth]{surface}
+  \includegraphics[width=0.9\textwidth]{snapshot}
   \caption{\sf Surface plot of function with points showing iteration of steepest
   descent method, both with optimal and constant $\alpha$}
   \label{fig:surface}
diff --git a/snapshot.eps b/snapshot.eps
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diff --git a/snapshot.png b/snapshot.png
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diff --git a/steepest.eps b/steepest.eps
index 2cc61ff..f483d87 100644
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+++ b/steepest.eps
@@ -1,9 +1,9 @@
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