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])
 
     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 = []
     
     
     function_values = []
     

-    alpha = 0.001
+    alpha = 0.1

     tolerance = 0.000000001
 
     start = time.time()
     tolerance = 0.000000001
 
     start = time.time()


@@ -47,7 +47,7 @@ def main():
 
     mlab.figure(figure="Surface", bgcolor=(1,1,1), fgcolor=(0,0,0))
 
 
     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)
     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
 % Fag: Numerisk matematikk \r
 % Project 1\r

-% Utført av: Bjørn, Mats og Edvard\r

 \r
 \documentclass[12pt, a4paper]{article}\r
 \usepackage[latin1]{inputenc}\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
 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
   \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
   \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
   \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
   \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
 \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
 \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
   \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
 \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
 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
 \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
   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
 \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
     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
 \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
   \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
 \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
   \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
new file mode 100644 (file)
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diff --git a/snapshot.png b/snapshot.png
new file mode 100644 (file)
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diff --git a/steepest.eps b/steepest.eps
index 2cc61ff97f95648da5a8b0923ea68a5ff1bc1e10..f483d870ab33961a8adc8712032565f8a4101745 100644 (file)
--- a/steepest.eps
+++ b/steepest.eps
@@ -1,9 +1,9 @@
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 %%Title: steepest.eps
 %%Creator: matplotlib version 0.99.3, http://matplotlib.sourceforge.net/

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+%%CreationDate: Mon Oct  3 22:44:32 2011

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@@ -2814,7 +3088,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 166.856 o
+73.125 136.725 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2826,7 +3100,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 166.856 o
+526.5 136.725 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2838,7 +3112,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 174.351 o
+73.125 141.15 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2850,7 +3124,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 174.351 o
+526.5 141.15 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2862,7 +3136,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 178.736 o
+73.125 143.739 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2874,7 +3148,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 178.736 o
+526.5 143.739 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2886,7 +3160,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 181.847 o
+73.125 145.575 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2898,7 +3172,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 181.847 o
+526.5 145.575 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2910,7 +3184,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 191.756 o
+73.125 151.425 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2922,7 +3196,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 191.756 o
+526.5 151.425 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2934,7 +3208,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 199.251 o
+73.125 155.85 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2946,7 +3220,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 199.251 o
+526.5 155.85 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2958,7 +3232,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 203.636 o
+73.125 158.439 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2970,7 +3244,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 203.636 o
+526.5 158.439 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2982,7 +3256,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 206.747 o
+73.125 160.275 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -2994,7 +3268,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 206.747 o
+526.5 160.275 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3006,7 +3280,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 216.656 o
+73.125 166.125 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3018,7 +3292,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 216.656 o
+526.5 166.125 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3030,7 +3304,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 224.151 o
+73.125 170.55 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3042,7 +3316,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 224.151 o
+526.5 170.55 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3054,7 +3328,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 228.536 o
+73.125 173.139 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3066,7 +3340,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 228.536 o
+526.5 173.139 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3078,7 +3352,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 231.647 o
+73.125 174.975 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3090,7 +3364,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 231.647 o
+526.5 174.975 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3102,7 +3376,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 241.556 o
+73.125 180.825 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3114,7 +3388,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 241.556 o
+526.5 180.825 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3126,7 +3400,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 249.051 o
+73.125 185.25 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3138,7 +3412,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 249.051 o
+526.5 185.25 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3150,7 +3424,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 253.436 o
+73.125 187.839 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3162,7 +3436,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 253.436 o
+526.5 187.839 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3174,7 +3448,7 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-95.0625 256.547 o
+73.125 189.675 o

 grestore
 gsave
 /o {
 grestore
 gsave
 /o {


@@ -3186,9 +3460,9 @@ translate
 stroke
 grestore
 } bind def
 stroke
 grestore
 } bind def

-684.45 256.547 o
+526.5 189.675 o

 grestore
 grestore

-61.062 110.001 m
+33.125 68.241 m

 gsave
 90 rotate
 0 0.172 rmoveto
 gsave
 90 rotate
 0 0.172 rmoveto


@@ -3197,29 +3471,29 @@ grestore
 1.000 setlinewidth
 2 setlinecap
 gsave
 1.000 setlinewidth
 2 setlinecap
 gsave

-95.0625 258.96 m
-684.45 258.96 l
+73.125 191.1 m
+526.5 191.1 l

 stroke
 grestore
 gsave
 stroke
 grestore
 gsave

-684.45 59.76 m
-684.45 258.96 l
+526.5 44.1 m
+526.5 191.1 l

 stroke
 grestore
 gsave
 stroke
 grestore
 gsave

-95.0625 59.76 m
-684.45 59.76 l
+73.125 44.1 m
+526.5 44.1 l

 stroke
 grestore
 gsave
 stroke
 grestore
 gsave

-95.0625 59.76 m
-95.0625 258.96 l
+73.125 44.1 m
+73.125 191.1 l

 stroke
 grestore
 /DejaVuSans findfont
 14.400 scalefont
 setfont
 stroke
 grestore
 /DejaVuSans findfont
 14.400 scalefont
 setfont

-330.686 263.96 m
+240.742 196.1 m

 0 0.203 rmoveto
 (Relative residual) show
 
 0 0.203 rmoveto
 (Relative residual) show