In [1]:

import numpy
import matplotlib.pyplot as plt

In [2]:

%matplotlib inline

Infinity norm¶

In [3]:

def norm(v, order):
    return sum([vi**order for vi in v])**(1/float(order))

In [4]:

v = [1, 2, 3, 2, 1]
norm(v, 10)

Out[4]:

3.010255858106243

In [5]:

plt.plot(range(1, 10), [norm(v, order) for order in range(1, 10)])

Out[5]:

[<matplotlib.lines.Line2D at 0x11b1295c0>]

../_images/notebooks_Infinity_norm_5_1.png

In [6]:

G = numpy.matrix([[5, 4],
                 [3, 2]])

In [7]:

d = numpy.array([[1], [1]])

In [8]:

G*d

Out[8]:

matrix([[9],
        [5]])

In [9]:

def gain(theta):
    d = numpy.matrix([[numpy.cos(theta[0])], [numpy.sin(theta[0])]])
    return numpy.linalg.norm(G*d)/numpy.linalg.norm(d)

In [10]:

import scipy.optimize

In [11]:

scipy.optimize.minimize(lambda theta: -gain(theta), [1])

Out[11]:

fun: -7.343420458864692
 hess_inv: array([[ 0.13636409]])
      jac: array([  5.96046448e-08])
  message: 'Optimization terminated successfully.'
     nfev: 18
      nit: 4
     njev: 6
   status: 0
  success: True
        x: array([ 0.65390079])