Python matrix using matplotlib and mplot3d

Question

Hello,

The code below is with the following error:

ValueError: shapes (1,2) and (3,3) not aligned: 2 (dim 1) != 3 (dim 0)

Someone knows a solution?

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
import itertools
A = np.matrix([[100.0,100.0,100.0],[0.0,0.0 ,64.3],[0.0 ,0.0,57.1]])
b = np.matrix([[100.0], [100.0],[100]])  # we will use the convention that a vector is a column vector
c = 0.0

def f(x, A, b, c):
    return float(0.5 * x.T * A * x - b.T * x + c)
def bowl(A, b, c):
    fig = plt.figure(figsize=(10,8))
    qf = fig.gca(projection='3d')
    size = 20
    x1 = list(np.linspace(-6, 6, size))
    x2 = list(np.linspace(-6, 6, size))
    x1, x2 = np.meshgrid(x1, x2)
    zs = np.zeros((size, size))
    for i in range(size):
        for j in range(size):
            x = np.matrix([[x1[i,j]], [x2[i,j]]])
            zs[i,j] = f(x, A, b, c)
    qf.plot_surface(x1, x2, zs, rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=0)
    fig.show()
    return x1, x2, zs
x1, x2, zs = bowl(A, b, c)

def contoursteps(x1, x2, zs, steps=None):
    fig = plt.figure(figsize=(6,6))
    cp = plt.contour(x1, x2, zs, 10)
    plt.clabel(cp, inline=1, fontsize=10)
    if steps is not None:
        steps = np.matrix(steps)
        plt.plot(steps[:,0], steps[:,1], '-o')
    fig.show()
contoursteps(x1, x2, zs)
x = np.matrix([[-2.0],[-2.0]])
steps = [(-2.0, -2.0)]
i = 0
imax = 10
eps = 0.01
r = b - A * x
delta = r.T * r
delta0 = delta
while i < imax and delta > eps**2 * delta0:
    alpha = float(delta / (r.T * (A * r)))
    x = x + alpha * r
    steps.append((x[0,0], x[1,0]))  # store steps for future drawing
    r = b - A * x
    delta = r.T * r
    i += 1
contoursteps(x1, x2, zs, steps)
x = np.matrix([[-2.0],[-2.0]])
steps = [(-2.0, -2.0)]
i = 0
imax = 10
eps = 0.01
print ('b:\n', b)
print ('A:\n', A)
print ('x:\n', x)
r = b - A * x
print ('r:\n', r)
contoursteps(x1, x2, zs, None)
plt.plot([0, r[0, 0] * 0.5], [0, r[1, 0] * 0.5], 'g')
plt.show()
print ('||r||^2 =', np.linalg.norm(r)**2)
delta = r.T * r
print ('r.T * r = ',  delta)
delta0 = delta

ValueError: shapes (1,2) and (3,3) not aligned: 2 (dim 1) != 3 (dim 0)

Answer 1

In 3 points, the script declares the matrix x with Shape (2, 1), however, uses this matrix in multiplication operations with matrices of Shape different (incompatible, for matrix multiplication operation).

An example of where this error occurs in the function f():

def f(x, A, b, c):
    return float(0.5 * x.T * A * x - b.T * x + c)
                           ^--aqui

Just as an example, if you change the 3 points of the script and add another element to the matrix x, the script will work:

...
x = np.matrix([ [x1[i,j]], [x2[i,j]], [0.0] ])
...
x = np.matrix([[-2.0],[-2.0], [0.0]])
...
x = np.matrix([[-2.0],[-2.0], [0.0]])
...

Important: I put the value 0.0 just to work, but you need to check in the description of the algorithm, which must be the correct value to initialize the matrix x.

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The script with the 0.0s above produces the following images:

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_{tested with: Python 3.5.2 |Anaconda custom (64-bit)}