Finishing up, report handed in

diff --git a/main.py b/main.py
index 78c24a4875dcb80d7a247a389fd7f0ed03de0813..5e05335a0c93f3ef4efd9ed84bba1b287e986168 100644 (file)
--- 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 6634a4f0d0aff271f95e54d883795e40325da354..5c30469caca1a4a1f46e7879b9f7fe7739aa92d8 100644 (file)
--- a/rapport.tex
+++ b/rapport.tex
@@ -1,6 +1,5 @@
 % Fag: Numerisk matematikk \r
 % Project 1\r
-% Utført av: Bjørn, Mats og Edvard\r
 \r
 \documentclass[12pt, a4paper]{article}\r
 \usepackage[latin1]{inputenc}\r
@@ -61,7 +60,7 @@
 The function analyzed in this project was\r
 \begin{equation}\r
   \label{eq:func_g}\r
-  g(\mathbf{x}) := -\mathbf{b}^{T}\mathbf{x} +\r
+  g(\mathbf{x}) = -\mathbf{b}^{T}\mathbf{x} +\r
   \frac{1}{2}\mathbf{x}^{T}H\mathbf{x} +\r
   \frac{1}{12}\mathbf{x}^{T}C(\mathbf{x})\mathbf{x},\r
 \end{equation}\r
@@ -73,13 +72,13 @@ this purpose we used the following randomly generated data:
   \begin{bmatrix}\r
     1.8580682 & -1.01197101 \\\r
     -1.01197101 & 0.63456543 \r
-  \end{bmatrix}\r
-  ,  \mathbf{b} =\r
+  \end{bmatrix},\r
+  \mathbf{b} =\r
   \begin{bmatrix}\r
     -0.3119187 \\\r
     0.14508516\r
-  \end{bmatrix}\r
-  ,  \mathbf{c} =\r
+  \end{bmatrix},\r
+  \mathbf{c} =\r
   \begin{bmatrix}\r
     0.47736331 \\\r
     1.48834978\r
@@ -105,6 +104,7 @@ From equation \eqref{eq:func_g} we get
 \end{eqnarray}\r
 The three functions, \eqref{eq:func_g}, \eqref{eq:grad_g} and \eqref{eq:hess_g}, are\r
 implemented in the Python code below.\r
+\newpage\r
 \r
 \lstinputlisting[language=Python]{bj_function.py}\r
 \r
@@ -143,9 +143,9 @@ Consider the equation
   \mathbf{r}^{(0)}.\r
 \end{equation}\r
 One can view the function $\mathbf{r}(t)$ as a parameterisation of the curve\r
-going through the starting point to the minima, always going in the direction of\r
-the steepest descent. By applying the forward Euler method to this differential\r
-vector equation, we get the following iteration:\r
+going through the starting point and to the minima, always going in the\r
+direction of the steepest descent. By applying the forward Euler method to this\r
+differential vector equation, we get the following iteration:\r
 \begin{equation}\r
   \label{eq:forward_iter}\r
   \mathbf{r}^{(k+1)} = \mathbf{r}^{(k)} - \alpha \nabla g\left(\mathbf{r}^{(k)}\right)\r
@@ -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 $\alpha$. \r
 \r
 To find the optimal step length, $\alpha^*$, we walk along the line of steepest\r
-descent until we get to the lowest point. This point can be found analytically\r
-by solving the following equation for $\alpha$:\r
+descent until we get to the lowest point and it starts going upwards again.\r
+This point can be found analytically by solving the following equation for\r
+$\alpha$:\r
 \begin{eqnarray}\r
   \label{eq:poly}\r
   \frac{\partial}{\partial\r
@@ -171,7 +172,8 @@ Where $A$, $B$,
   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
 \end{eqnarray}\r
-Differentiating the polynomial results in:\r
+Differentiating the polynomial and using the fact that the Hessian is positive\r
+definite we get the following:\r
 \begin{eqnarray}\r
   \frac{\partial^2}{\partial \alpha^2}\left(g\left(\mathbf{x} +\r
     \alpha\mathbf{u}\right)\right) &=& \mathbf{u}^T H \mathbf{u} + \mathbf{u}^T\r
@@ -180,7 +182,7 @@ Differentiating the polynomial results in:
     C\left(\mathbf{u}\right) \mathbf{u}\alpha \nonumber \\\r
   &=& \mathbf{u}^T H \mathbf{u} + \mathbf{u}^T C\left(\mathbf{x} +\r
     \alpha\mathbf{u}\right)\mathbf{u} \nonumber \\\r
-  &=& \mathbf{u}^T\nabla^2 g\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u}.\r
+  &=& \mathbf{u}^T\nabla^2 g\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u} > 0.\r
 \end{eqnarray}\r
 From this we see that the polynomial \eqref{eq:poly} is monotonically increasing\r
 and therefore only has one real zero. This is the zero we want to use as our\r
@@ -207,15 +209,17 @@ The Newton method was also implemented to solve $\nabla g\left(\mathbf{x}^*\righ
   \end{center}\r
   \label{tab:time}\r
 \end{table}\r
-We see from table \ref{tab:time} that newtons method uses the least time to\r
-solve the problem, even though figure \ref{fig:fn_resid} shows that the\r
-combination method results in fewer iterations. This comes from the fact that\r
-computing the optimal step size requires solving a cubic equation and is\r
-computationally expensive.\r
+We see from table \ref{tab:time} that Newtons method uses the least time to\r
+solve an instance of the problem, even though we adjusted the combination method\r
+to use as few iterations as possible as shown in figure \ref{fig:fn_resid}.\r
+This comes from the fact that computing the optimal step size requires solving a\r
+cubic equation and is computationally expensive. A faster and more approximated\r
+solution for the optimal step size could make the steepest descent method even\r
+faster.\r
 \r
 \begin{figure}[hb]\r
   \centering\r
-  \includegraphics[width=0.9\textwidth]{surface}\r
+  \includegraphics[width=0.9\textwidth]{snapshot}\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

diff --git a/snapshot.eps b/snapshot.eps
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diff --git a/snapshot.png b/snapshot.png
new file mode 100644 (file)
index 0000000..bddca28
Binary files /dev/null and b/snapshot.png differ

diff --git a/steepest.eps b/steepest.eps
index 2cc61ff97f95648da5a8b0923ea68a5ff1bc1e10..f483d870ab33961a8adc8712032565f8a4101745 100644 (file)
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