

In [1]:

    
# this line is needed for the web notebook only
%matplotlib inline

CIVL4250 Numerical Methods in Engineering

Dr Dorival Pedroso

The code for this notebook is available in https://github.com/cpmech/CIVL4250py

1 Sequence of real numbers



In [2]:

    
import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(0.0, 1.0, 11)
print x









    



[ 0.   0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1. ]

2 External library for plotting

Using Matplotlib [1] (aka PyPlot) for plotting and Numpy [2] for numerical computations.

[1] http://matplotlib.org/api/pyplot_summary.html

[2] http://www.scipy.org/getting-started.html



In [13]:

    
x = np.linspace(0.0, 2.0*np.pi, 101)
y = np.sin(x)

plt.plot(x, y, 'b-', lw=3, color='#ff9933')
plt.xlabel('x')
plt.ylabel('y')
plt.show()

3 Vectors and matrices

Vectors and matrices are created with the array structure from NumPy that takes one list as argument.

To create a matrix, array takes one list of lists.

The matrix multiplication is computed with the dot function from Numpy.



In [7]:

    
u = np.array([1.0, 2.0, 3.0]) # vector

M = np.array([[1.0, -1.0, 1.0],
              [-2.0, 2.0, 2.0],
              [-3.0, 3.0, 3.0]]) # matrix

v = np.dot(M, u) # matrix product: v = M times u
print v









    



[  2.   8.  12.]

To solve a linear system, the solve function from linalg can be used.

For example, given the following linear system of equations:

$$ \begin{bmatrix} 2 & \phantom{-}3 & 0 & 0 & 0 & \\ 3 & \phantom{-}0 & 4 & 0 & 6 & \\ 0 & -1 & -3 & 2 & 0 & \\ 0 & \phantom{-}0 & 1 & 0 & 0 & \\ 0 & \phantom{-}4 & 2 & 0 & 1 & \end{bmatrix} \begin{Bmatrix} x_0 \\ x_1 \\ x_2 \\ x_3 \\ x_4 \end{Bmatrix} = \begin{Bmatrix} 8 \\ 45 \\ -3 \\ 3 \\ 19 \end{Bmatrix} $$

The components $x_i$ can be found as follows



In [12]:

    
A = np.array([[2,  3,  0, 0, 0],
              [3,  0,  4, 0, 6],
              [0, -1, -3, 2, 0],
              [0,  0,  1, 0, 0],
              [0,  4,  2, 0, 1]])

b = np.array([8, 45, -3, 3, 19])

x = np.linalg.solve(A, b)
print x









    



[ 1.  2.  3.  4.  5.]