Everything seems to work ok

diff --git a/main.py b/main.py
index db9d52244ace8e26872ae3721be9bdfd305ae60f..96e8a16d5edde1610210b051c77027d2e479b281 100644 (file)
--- a/main.py
+++ b/main.py
@@ -4,22 +4,28 @@ import numpy as np
 import matplotlib.pyplot as plt
 from steepest import steepest
 from function import function_g
+from newton import newton
+from newtsteep import newtsteep
 
 def main():
     H = np.array([ ( 1.8580682, -1.01197101), (-1.01197101, 0.63456543)] )
     b = np.array([-0.3119187,  0.14508516])
     c = np.array([ 0.47736331, 1.48834978])
 
-    x0 = np.array([-0.1, 0.8])
+    x0 = np.array([ 3.1, 3.8])
     
     function_values = []
     
-    alpha = 0.3
-    optimal_points, optimal_residuals = steepest(x0, H, b, c, 0.01, 100, 0)
-    constant_points, constant_residuals = steepest(x0, H, b, c, 0.01, 100, alpha)
+    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)
 
     f_optimal = []
     f_constant = []
+    f_newton = []
+    f_nwst = []
 
     mlab.figure(figure="Surface", bgcolor=(1,1,1), fgcolor=(0,0,0))
 
@@ -32,7 +38,15 @@ def main():
         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:
+        z = function_g(point, H, b, c)
+        f_newton.append(z)
+
+    for point in nwst_points:
+        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)
 
@@ -50,7 +64,9 @@ def main():
     plt.ylabel("Relative residual")
     plt.plot(optimal_residuals)
     plt.plot(constant_residuals)
-    plt.legend([r'$\alpha$ = %.1f' % alpha, r'Optimal $\alpha$'])
+    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'])
     ax.set_yscale('log')
     
     ax = plt.subplot(212)
@@ -59,6 +75,8 @@ def main():
     plt.ylabel("g(x)")
     plt.plot(f_optimal)
     plt.plot(f_constant)
+    plt.plot(f_newton)
+    plt.plot(f_nwst)
     
     plt.show()
     fig.savefig("steepest.eps", format="eps")

diff --git a/newton.py b/newton.py
index 13c8f746500ed3d37877aa77cc0d4631e7ef2e2a..0be139a65ef87e73257866bfc7c78262f6714f16 100644 (file)
--- a/newton.py
+++ b/newton.py
@@ -9,8 +9,9 @@ from numpy.linalg import norm, inv
 def newton_iter(x, grad_g, hess_g):
     return x - inv(hess_g).dot(grad_g)
 
-def newton(xk, H, b, c, tolerance, maxiter):
-    x0_norm = norm(gradient_g(xk, H, b, c))
+def newton(xk, H, b, c, tolerance, maxiter, x0_norm=0):
+    if x0_norm == 0:
+        x0_norm = norm(gradient_g(xk, H, b, c))
     
     relative_residual = norm(gradient_g(xk, H, b, c)) / x0_norm
     relative_residuals = []

diff --git a/rapport.tex b/rapport.tex
index 6d514af77ee87b457d1a9acae8fc70e45de989a3..29a90b1b21a28db486890ad544c9154f52cb2284 100644 (file)
--- a/rapport.tex
+++ b/rapport.tex
@@ -133,7 +133,14 @@ Hence $\mathbf{v}^{T}(H+C(\mathbf{x}))\mathbf{v} > 0$, and we see by definition
 \label{fig:surface}\r
 \end{figure}\r
 \r
-% HER KOMMER EN UTLEDNING AV ALPHA OG SÅNT\r
+To find the optimal step length, $\alpha^*$, we solve the equation\r
+\begin{eqnarray}\r
+\frac{\partial}{\partial \mathbf{\alpha}}g\left(\mathbf{x}^{(k)}+\alpha\mathbf{u}^{(k)}\right) &=& 0, \text{where  } \mathbf{u}^{(k)} = -\nabla g(\mathbf{x}^{(k)} )  \nonumber \\\r
+&\Updownarrow& \nonumber\\\r
+%HER ER DET FEIL! \r
+\frac{1}{3}(\mathbf{u}^{(k)})^TC(\mathbf{u}^{(k)})\mathbf{u}^{(k)}\alpha^3+ (\mathbf{x}^{(k)})^TC(\mathbf{u}^{(k)}) \mathbf{u}^{(k)}\alpha^2+(\mathbf{u}^{(k)})^TH\mathbf{u}^{(k)}+(\mathbf{u}^{(k)})^TC(\mathbf{x}^{(k)})\mathbf{u}^{(k)})\alpha + \frac{1}{3}(\mathbf{u}^{(k)})^TC(\mathbf{x}^{(k)})\mathbf{x}^{(k)}+(\mathbf{u}^{(k)})^TH\mathbf{x}^{(k)}-\mathbf{b}^T\mathbf{u}^{(k)} = 0\r
+\end{eqnarray}\r
+\r
 % VI MÅ OGSÅ ha utledning av alt det andre, forklaring \r
 % av figurer, og en slags tabell med sammenlikning av metodene\r
 \r

diff --git a/steepest.eps b/steepest.eps
index 1b5770299ae42d9b6edd35333cb842c308fb6d30..7c543e697a367cae3f46e090cdf3840f4a64644d 100644 (file)
--- a/steepest.eps
+++ b/steepest.eps
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diff --git a/steepest.py b/steepest.py
index ecbf182d1ca864bc6d7dc3739dc85c1e8ffa7b8f..49147c48615ef9fa1d02e1b5913a0a21f7448640 100644 (file)
--- a/steepest.py
+++ b/steepest.py
@@ -30,9 +30,10 @@ def optimal_step(x, H, b, c, grad_f):
 
     return 0.1
 
-def steepest(x, H, b, c, tolerance, maxiter, step):
+def steepest(x, H, b, c, tolerance, maxiter, step, x0_norm=0):
     xk = np.array(x)
-    x0_norm = norm(gradient_g(xk, H, b, c))
+    if x0_norm == 0:
+        x0_norm = norm(gradient_g(xk, H, b, c))
     
     relative_residual = norm(gradient_g(xk, H, b, c)) / x0_norm
     relative_residuals = []