Proving and stuffs

diff --git a/Makefile b/Makefile
index b2a7fb51441fd5f54d58e27800b706fca90011cb..13d90dda9eb65b82f1b0779816c72a28938cf2b6 100644 (file)
--- a/Makefile
+++ b/Makefile
@@ -1,3 +1,5 @@
 all:
        latex rapport.tex
        latex rapport.tex
+       dvips rapport.dvi
+       ps2pdf rapport.ps

diff --git a/main.py b/main.py
index 2477c2dd2cae1b9419b213c19bb4c962add5608d..633d01f09abd471905ed82e5a7fa90b1d29a6233 100644 (file)
--- a/main.py
+++ b/main.py
@@ -44,60 +44,61 @@ def main():
     f_newton = []
     f_nwst = []
 
-#    mlab.figure(figure="Surface", bgcolor=(1,1,1), fgcolor=(0,0,0))
-#
-#    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[: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[:plen]:
-#        z = function_g(point, H, b, c)
-#        f_newton.append(z)
-#
-#    for point in nwst_points[:plen]:
-#        z = function_g(point, H, b, c)
-#        f_nwst.append(z)
-#
-#    last = optimal_points[len(optimal_points)-1]
-#    X, Y, Z = surf.surf(H, b, c, last[0]-1, last[0]+1, last[1]-1, last[1]+1, -10, 3)
-#
-#    surface = mlab.mesh(X, Y, Z)
-#    axes = mlab.axes(color=(0,0,0))
-#    mlab.outline()
-#    mlab.show()
-#    
-#    fig = plt.figure(1)
-#    plt.rc('mathtext', default='regular')
-#    fig.subplots_adjust(hspace=0.4)
-#    ax = plt.subplot(211)
-#    plt.title("Relative residual")
-#    plt.xlabel("Iteration")
-#    plt.ylabel("Relative residual")
-#    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[:plen])
-#    plt.plot(f_constant[:plen])
-#    plt.plot(f_newton[:plen])
-#    plt.plot(f_nwst[:plen])
-#    
-#    plt.show()
-#    fig.savefig("steepest.eps", format="eps")
+    mlab.figure(figure="Surface", bgcolor=(1,1,1), fgcolor=(0,0,0))
+
+    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[: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[:plen]:
+        z = function_g(point, H, b, c)
+        f_newton.append(z)
+
+    for point in nwst_points[:plen]:
+        z = function_g(point, H, b, c)
+        f_nwst.append(z)
+
+    last = optimal_points[len(optimal_points)-1]
+    X, Y, Z = surf.surf(H, b, c, last[0]-1, last[0]+1, last[1]-1, last[1]+1, -10, 3)
+
+    surface = mlab.mesh(X, Y, Z)
+    axes = mlab.axes(color=(0,0,0))
+    mlab.outline()
+    mlab.show()
+    
+    fig = plt.figure(1)
+    plt.rc('mathtext', default='regular')
+    fig.subplots_adjust(hspace=0.4)
+
+    ax = plt.subplot(211)
+    plt.title("Function value")
+    plt.xlabel("Iteration")
+    plt.ylabel("g(x)")
+    plt.plot(f_optimal[:plen])
+    plt.plot(f_constant[:plen])
+    plt.plot(f_newton[:plen])
+    plt.plot(f_nwst[:plen])
+    plt.legend([r'Optimal $\alpha$', r'$\alpha$ = %.2f' % alpha, r'Newtons method', r'Steepest first, then Newton'])
+
+    ax = plt.subplot(212)
+    plt.title("Relative residual")
+    plt.xlabel("Iteration")
+    plt.ylabel("Relative residual")
+    plt.plot(optimal_residuals[:plen])
+    plt.plot(constant_residuals[:plen])
+    plt.plot(newton_residuals[:plen])
+    plt.plot(nwst_residuals[:plen])
+    ax.set_yscale('log')
+    
+    plt.show()
+    fig.savefig("steepest.eps", format="eps")
     
 if __name__ == "__main__":
     main()

diff --git a/rapport.tex b/rapport.tex
index 015b48108909e9ae44487db25800c6f33971a5e1..58a1bc245ba641ff29be02bc0def2bcf56de4c4a 100644 (file)
--- a/rapport.tex
+++ b/rapport.tex
@@ -93,14 +93,14 @@ The function analyzed in this project was
   \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
-From equation \ref{eq:func_g} we get\r
+From equation \eqref{eq:func_g} we get\r
 \begin{eqnarray}\r
   \label{eq:grad_g}\r
   \nabla g(\mathbf{x}) &=& -\mathbf{b}+H\mathbf{x}+\frac{1}{3}C(\mathbf{x})\mathbf{x} \\\r
   \label{eq:hess_g}\r
   \nabla^{2}g(\mathbf{x}) &=& H+C(\mathbf{x}).\r
 \end{eqnarray}\r
-The three functions, \ref{eq:func_g}, \ref{eq:grad_g} and \ref{eq:hess_g}, are\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
 \r
 \lstinputlisting[language=Python]{bj_function.py}\r
@@ -147,10 +147,12 @@ vector equation, we get the following iteration:
   \label{eq:forward_iter}\r
   \mathbf{r}^{(k+1)} = \mathbf{r}^{(k)} - \alpha \nabla g\left(\mathbf{r}^{(k)}\right)\r
 \end{equation}\r
-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
+where $t_{n+1} = t_n + \alpha$. Looking at this result, we see that it is\r
+equivalent to the steepest descent iteration with constant step size $\alpha$. \r
 \r
 To find the optimal step length, $\alpha^*$, we solve the equation\r
 \begin{eqnarray}\r
+  \label{eq:poly}\r
   \frac{\partial}{\partial\r
     \mathbf{\alpha}}g\left(\mathbf{x}+\alpha\mathbf{u}\right) &=& 0, \text{where  }\r
     \mathbf{u} = -\nabla g(\mathbf{x} )  \nonumber \\\r
@@ -164,6 +166,20 @@ 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
+Deriving the polynomial results in:\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
+    \left[C\left(\alpha\mathbf{u}\right) +\r
+    C\left(\mathbf{x}\right)\right]\mathbf{u} + 2\mathbf{x}^T\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) \nonumber \\\r
+  &=& \mathbf{u}^T\nabla^2 g\left(\mathbf{x} + \alpha\mathbf{u}\right)\mathbf{u}.\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
+optimal step length.\r
 \r
 \lstinputlisting[language=Python]{bj_optimal.py}\r
 \r

diff --git a/steepest.eps b/steepest.eps
index b44133282aedc8edb0da418855176cdde3864014..62abd42c426ab88c72230f7ea7737081f6bd1cc4 100644 (file)
--- a/steepest.eps
+++ b/steepest.eps
@@ -1,9 +1,9 @@
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diff --git a/steepest.py b/steepest.py
index 976388c886c5db64e78626b3b5d65832743af259..581627c8d226c25679462ded8539b98df7f96eb1 100644 (file)
--- a/steepest.py
+++ b/steepest.py
@@ -58,7 +58,7 @@ def steepest(x, H, b, c, tolerance, maxiter, step, x0_norm=0):
         relative_residuals.append(relative_residual)
         points.append(xk)
     
-        xk = steepest_iter(xk, alpha, gradient_g(xk))
+        xk = steepest_iter(xk, alpha, gradient_g(xk, H, b, c))
     
     return (points, relative_residuals)
     

diff --git a/steepestnewton.py b/steepestnewton.py
index 2498575515605b64713ad1bb5924d2289984291b..ee9cfd725b5295654c54612ef06789d507c97054 100644 (file)
--- a/steepestnewton.py
+++ b/steepestnewton.py
@@ -16,7 +16,7 @@ def steepestnewton(x0, H, b, c, steep_tol, newt_tol):
     r.extend(residuals)
 
     # Newton starts in the same point as steepest ends in
-    points, residuals = newton.newton(x[len(x)-1], H, b, c, newt_tol, 100, x0_norm)
+    points, residuals = newton(x[len(x)-1], H, b, c, newt_tol, 100, x0_norm)
     x.extend(points[1:])
     r.extend(residuals[1:])