From: Bjørn Rustad <bjorn@rustad.me>
Date: Fri, 30 Sep 2011 14:57:53 +0000 (+0200)
Subject: Add timing and write more on the report
X-Git-Url: http://git.rustad.me/?a=commitdiff_plain;h=08482aebf52ee84c633f65e7289b965ea5aeecce;p=nummat1

Add timing and write more on the report
---

diff --git a/main.py b/main.py
index 981ee9e..df0e882 100644
--- a/main.py
+++ b/main.py
@@ -1,7 +1,8 @@
 from enthought.mayavi import mlab
-import surf
+import time
 import numpy as np
 import matplotlib.pyplot as plt
+import surf
 from steepest import steepest
 from function import function_g
 from newton import newton
@@ -12,15 +13,31 @@ def main():
     b = np.array([-0.3119187,  0.14508516])
     c = np.array([ 0.47736331, 1.48834978])
 
-    x0 = np.array([-2.1, 3.2])
+    x0 = np.array([-2.1, 5.2])
     
     function_values = []
     
-    alpha = 0.1
-    optimal_points, optimal_residuals = steepest(x0, H, b, c, 0.001, 100, 0)
-    constant_points, constant_residuals = steepest(x0, H, b, c, 0.001, 30, alpha)
-    newton_points, newton_residuals = newton(x0, H, b, c, 0.001, 100)
-    nwst_points, nwst_residuals = newtsteep(x0, H, b, c, 0.1, 0.001)
+    alpha = 0.05
+
+    start = time.clock()
+    optimal_points, optimal_residuals = steepest(x0, H, b, c, 0.000000001, 100000, 0)
+    stop = time.clock()
+    print "optimal steepest brukte %.4f" % (stop - start)
+
+    start = time.clock()
+    constant_points, constant_residuals = steepest(x0, H, b, c, 0.000000001, 100000, alpha)
+    stop = time.clock()
+    print "constant steepest brukte %.4f" % (stop - start)
+
+    start = time.clock()
+    newton_points, newton_residuals = newton(x0, H, b, c, 0.000000001, 100000)
+    stop = time.clock()
+    print "newton brukte %.4f" % (stop - start)
+
+    start = time.clock()
+    nwst_points, nwst_residuals = newtsteep(x0, H, b, c, 0.01, 0.000000001)
+    stop = time.clock()
+    print "newton + steepest brukte %.4f" % (stop - start)
 
     f_optimal = []
     f_constant = []
@@ -29,21 +46,22 @@ def main():
 
     mlab.figure(figure="Surface", bgcolor=(1,1,1), fgcolor=(0,0,0))
 
-    for point in optimal_points:
+    plen = 15
+    for point in optimal_points[:plen]:
         z = function_g(point, H, b, c)
         f_optimal.append(z)
         p = mlab.points3d([point[0]], [point[1]], [z], scale_factor=0.03, color=(0.7,0.2,0.2))
 
-    for point in constant_points:
+    for point in constant_points[:plen]:
         z = function_g(point, H, b, c)
         f_constant.append(z)
         p = mlab.points3d([point[0]], [point[1]], [z], scale_factor=0.02, color=(1,1,1))
 
-    for point in newton_points:
+    for point in newton_points[:plen]:
         z = function_g(point, H, b, c)
         f_newton.append(z)
 
-    for point in nwst_points:
+    for point in nwst_points[:plen]:
         z = function_g(point, H, b, c)
         f_nwst.append(z)
 
@@ -62,21 +80,21 @@ def main():
     plt.title("Relative residual")
     plt.xlabel("Iteration")
     plt.ylabel("Relative residual")
-    plt.plot(optimal_residuals)
-    plt.plot(constant_residuals)
-    plt.plot(newton_residuals)
-    plt.plot(nwst_residuals)
-    plt.legend([r'Optimal $\alpha$', r'$\alpha$ = %.1f' % alpha, r'Newtons method', r'Steepest first, then Newton'])
+    plt.plot(optimal_residuals[:plen])
+    plt.plot(constant_residuals[:plen])
+    plt.plot(newton_residuals[:plen])
+    plt.plot(nwst_residuals[:plen])
+    plt.legend([r'Optimal $\alpha$', r'$\alpha$ = %.2f' % alpha, r'Newtons method', r'Steepest first, then Newton'])
     ax.set_yscale('log')
     
     ax = plt.subplot(212)
     plt.title("Function value")
     plt.xlabel("Iteration")
     plt.ylabel("g(x)")
-    plt.plot(f_optimal)
-    plt.plot(f_constant)
-    plt.plot(f_newton)
-    plt.plot(f_nwst)
+    plt.plot(f_optimal[:plen])
+    plt.plot(f_constant[:plen])
+    plt.plot(f_newton[:plen])
+    plt.plot(f_nwst[:plen])
     
     plt.show()
     fig.savefig("steepest.eps", format="eps")
diff --git a/rapport.tex b/rapport.tex
index c7e1fd4..f7816dd 100644
--- a/rapport.tex
+++ b/rapport.tex
@@ -55,96 +55,113 @@ ABSTRACTABSTBSSTRACTABSTRACTABSTRACT
 %                                %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Resultater}
-We used the following data:
-
+This project explores how one can approximate the local minima of a function.
+The function considered takes in a symmetric positive definite matrix, and two
+vectors. For this purpose we used the following randomly generated data:
 \begin{equation}
-H = 
-\begin{bmatrix}
-1.8580682 & -1.01197101 \\
--1.01197101 & 0.63456543 
-\end{bmatrix}
-,  \mathbf{b} =
-\begin{bmatrix}
--0.3119187 \\
-0.14508516
-\end{bmatrix}
-,  \mathbf{c} =
-\begin{bmatrix}
-0.47736331 \\
-1.48834978
-\end{bmatrix}.
-\nonumber
+  H = 
+  \begin{bmatrix}
+    1.8580682 & -1.01197101 \\
+    -1.01197101 & 0.63456543 
+  \end{bmatrix}
+  ,  \mathbf{b} =
+  \begin{bmatrix}
+    -0.3119187 \\
+    0.14508516
+  \end{bmatrix}
+  ,  \mathbf{c} =
+  \begin{bmatrix}
+    0.47736331 \\
+    1.48834978
+  \end{bmatrix}.
+  \nonumber
 \end{equation}
+The elements of $\mathbf{b}$ are randomly generated numbers from a uniform
+distribution over $[-1, 1]$, and the elements of $\mathbf{c}$ are randomly
+generated positive numbers from a uniform distribution over $(0, 3]$
 
-Where the elements of $\mathbf{b}$ are randomly generated numbers from the interval INTERVAL, and the elements of $\mathbf{c}$ are randomly generated positive numbers.
-
-To generate the nxn matrix, $H$, we first generate a upper triangular nxn matrix $U$.
-The elements along the diagonal of $U$ are the square roots of randomly generated positive numbers, and the elements above the diagonal are randomly generated numbers from the interval INTERVAL.
-$H = U^{T}U$ is then a symmetric positive definite matrix.
+To generate the $n\times n$ matrix $H$, we first generate a upper triangular
+$n\times n$ matrix $U$. The elements above the diagonal are randomly generated
+in the uniform distribution over $[-3, 3]$, while the numbers on the diagonal
+are square roots of positive numbers from the same distribution. $H = U^{T}U$ is
+then a symmetric positive definite matrix.
 
 The function analyzed in this project was
-
 \begin{equation}
-\label{eq:func_g}
-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}
+  \label{eq:func_g}
+  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}
 From equation \ref{eq:func_g} we get
 \begin{eqnarray}
-\label{eq:grad_g}
-\nabla g(\mathbf{x}) &=& -\mathbf{b}+H\mathbf{x}+\frac{1}{3}C(\mathbf{x})\mathbf{x} \\
-\label{eq:hess_g}
-\nabla^{2}g(\mathbf{x}) &=& H+C(\mathbf{x}).
+  \label{eq:grad_g}
+  \nabla g(\mathbf{x}) &=& -\mathbf{b}+H\mathbf{x}+\frac{1}{3}C(\mathbf{x})\mathbf{x} \\
+  \label{eq:hess_g}
+  \nabla^{2}g(\mathbf{x}) &=& H+C(\mathbf{x}).
 \end{eqnarray}
-The three functions, \ref{eq:func_g}, \ref{eq:grad_g} and \ref{eq:hess_g}, are implemented in the Python code below.
+The three functions, \ref{eq:func_g}, \ref{eq:grad_g} and \ref{eq:hess_g}, are
+implemented in the Python code below.
 
 \lstinputlisting[language=Python]{bj_function.py}
 
-Consider the Hessian $\nabla^{2}g(\mathbf{x})$, and any vector $\mathbf{v} \in \mathbb{R}^{n}_{*}$.
-Then we have
+Consider the Hessian $\nabla^{2}g(\mathbf{x})$, and any vector $\mathbf{v} \in
+\mathbb{R}^{n}_{*}$.  Then we have
 
 \begin{eqnarray}
-\mathbf{v}^{T}(\nabla^{2}g(\mathbf{x}))\mathbf{v} &=& \mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} \nonumber \\
-&=& \mathbf{v}^{T}H(\mathbf{x}))\mathbf{v}+\mathbf{v}^{T}C(\mathbf{x})\mathbf{v}.
+  \mathbf{v}^{T}(\nabla^{2}g(\mathbf{x}))\mathbf{v} &=& \mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} \nonumber \\
+  &=& \mathbf{v}^{T}H(\mathbf{x}))\mathbf{v}+\mathbf{v}^{T}C(\mathbf{x})\mathbf{v}.
 \end{eqnarray}
 We already know H is positive definite, so $\mathbf{v}^{T}H(\mathbf{x}))\mathbf{v} > 0$. 
 Since $C(\mathbf{x})$ only has positive entries, we easily see that
-
 \begin{equation}
-\mathbf{v}^{T}C(\mathbf{x})\mathbf{v} = \sum\limits_{i=1}^n c_{i}(v_{i}x_{i})^2 \geq 0
+  \mathbf{v}^{T}C(\mathbf{x})\mathbf{v} = \sum\limits_{i=1}^n c_{i}(v_{i}x_{i})^2 \geq 0
 \end{equation}
-Hence $\mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} > 0$, and we see by definition that the Hessian, $\nabla^{2}g(\mathbf{x})$, is positive definite.
+Hence $\mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} > 0$, and we see by definition
+that the Hessian, $\nabla^{2}g(\mathbf{x})$, is positive definite.
 
 \begin{figure}[htb]
-\centering
-\includegraphics[width=0.8\textwidth]{steepest}
-\caption{Plot of relative residual and function values for iterations of the steepest descent method with optimal $\alpha$}
-\label{fig:fn_resid}
+  \centering
+  \includegraphics[width=0.9\textwidth]{steepest}
+  \caption{Plot of relative residual and function values for iterations of the steepest descent method with optimal $\alpha$}
+  \label{fig:fn_resid}
 \end{figure}
 
 \begin{figure}[htb]
-\centering
-\includegraphics[width=0.9\textwidth]{surface}
-\caption{Surface plot of function with points showing iteration of steepest descent method, both with optimal and constant $\alpha$}
-\label{fig:surface}
+  \centering
+  \includegraphics[width=0.9\textwidth]{surface}
+  \caption{Surface plot of function with points showing iteration of steepest descent method, both with optimal and constant $\alpha$}
+  \label{fig:surface}
 \end{figure}
 
+Consider the equation
+\begin{equation}
+  \label{eq:forward}
+  \mathbf{r}'(t) = -\nabla g\left(\mathbf{r}(t)\right)\text{,  } \mathbf{r}(0) = \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:
+\begin{equation}
+  \label{eq:forward_iter}
+  \mathbf{r}^{(k+1)} = \mathbf{r}^{(k)} - \alpha \nabla g\left(\mathbf{r}^{(k)}\right)
+\end{equation}
+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 solve the equation
 \begin{eqnarray}
-\frac{\partial}{\partial \mathbf{\alpha}}g\left(\mathbf{x}+\alpha\mathbf{u}\right) &=& 0, \text{where  } \mathbf{u} = -\nabla g(\mathbf{x} )  \nonumber \\
-&\Updownarrow& \nonumber\\
-%HER ER DET FEIL! 
-A\alpha^3 + B\alpha^2 + C\alpha + D &=& 0,
+  \frac{\partial}{\partial \mathbf{\alpha}}g\left(\mathbf{x}+\alpha\mathbf{u}\right) &=& 0, \text{where  } \mathbf{u} = -\nabla g(\mathbf{x} )  \nonumber \\
+  &\Updownarrow& \nonumber\\
+  A\alpha^3 + B\alpha^2 + C\alpha + D &=& 0,
+  \end{eqnarray}
+where $A$, $B$, $C$, and $D$ is 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}
 \end{eqnarray}
-der $A$, $B$, $C$ og $D$ er gitt ved
-\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}
-\end{eqnarray}
-
-% VI MÅ OGSÅ ha utledning av alt det andre, forklaring 
-% av figurer, og en slags tabell med sammenlikning av metodene
 
 \lstinputlisting[language=Python]{bj_optimal.py}
 
diff --git a/steepest.eps b/steepest.eps
index bae093b..b441332 100644
--- a/steepest.eps
+++ b/steepest.eps
@@ -1,9 +1,9 @@
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