Importing



In [5]:

    
import numpy as np

Numpy arrays



In [6]:

    
a = np.array( [1, 2, 3, 4, 5], float)
a









    Out[6]:





array([ 1.,  2.,  3.,  4.,  5.])



In [7]:

    
b = np.array( [9,8,9], int)
print(b.dtype)









    



int64

Numpy arrays behave similar to Python arrays



In [8]:

    
print(a[2:]) # all elements after the 2nd
print(a[-1]) # last element
print(a[2:4]) # slice 3rd to 4th
print(a[::2]) # every second element
a[0] = 5 
print(a)









    



[ 3.  4.  5.]
5.0
[ 3.  4.]
[ 1.  3.  5.]
[ 5.  2.  3.  4.  5.]



In [9]:

    
# iteration
for elem in a:
    print(elem)

Multi-dimensional arrays



In [10]:

    
A = np.array( [[1,2,3], [4,5,6], [7,8,9]])
A









    Out[10]:





array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]])



In [11]:

    
print(A[0,0]) # access single element
print(A[2,:]) # access 3rd row
print(A[:,0]) # access first column
print(A[::2,:]) # get every second row









    



1
[7 8 9]
[1 4 7]
[[1 2 3]
 [7 8 9]]

Slicing



In [12]:

    
a = np.array( range(15) )
print(a[1:10:2]) # every second elemtn from second to 10th
a[1::2] = 0 # set every second element to zero
print(a)









    



[1 3 5 7 9]
[ 0  0  2  0  4  0  6  0  8  0 10  0 12  0 14]

Array dimensions



In [13]:

    
print(len(A))     # length of 1st dimension
print(A.shape)    # shape of the arrays

Check occurrence



In [14]:

    
print(2 in A)
print(13 in A)









    



True
False

Changing the shape of an array



In [15]:

    
c = np.array([1,2,3,4,5,6])
print(c)
c.reshape( (3,2) ) # reshape takes a dimension tuple as input (#rows,#cols)









    



[1 2 3 4 5 6]






    Out[15]:





array([[1, 2],
       [3, 4],
       [5, 6]])

Attention: deep-copy vs. shallow-copy



In [16]:

    
a = np.array([1,2,3])
b = a
c = a.copy()
a[0]=6
print(a)
print(b)
print(c)

Further array functions



In [17]:

    
# convert to python list
a.tolist()

# fill up with new value
a.fill(12)

print(A)
print(A.transpose()) # transpose matrix
print(A.T)           # same as A.transpose()









    



[[1 2 3]
 [4 5 6]
 [7 8 9]]
[[1 4 7]
 [2 5 8]
 [3 6 9]]
[[1 4 7]
 [2 5 8]
 [3 6 9]]



In [18]:

    
# concatenate arrays
B = A.transpose()

print(np.concatenate((A,B))) # concatenate vertically
print(np.concatenate((A,B), axis=1)) # concatenate horizontally









    



[[1 2 3]
 [4 5 6]
 [7 8 9]
 [1 4 7]
 [2 5 8]
 [3 6 9]]
[[1 2 3 1 4 7]
 [4 5 6 2 5 8]
 [7 8 9 3 6 9]]

Convenient functions to construct arrays



In [19]:

    
# equally spaced values within an interval: arange( start, stop, step )
np.arange(0, 5, 0.1)









    Out[19]:





array([ 0. ,  0.1,  0.2,  0.3,  0.4,  0.5,  0.6,  0.7,  0.8,  0.9,  1. ,
        1.1,  1.2,  1.3,  1.4,  1.5,  1.6,  1.7,  1.8,  1.9,  2. ,  2.1,
        2.2,  2.3,  2.4,  2.5,  2.6,  2.7,  2.8,  2.9,  3. ,  3.1,  3.2,
        3.3,  3.4,  3.5,  3.6,  3.7,  3.8,  3.9,  4. ,  4.1,  4.2,  4.3,
        4.4,  4.5,  4.6,  4.7,  4.8,  4.9])



In [42]:

    
# alternative version: linspace(start, stop, no_points)
np.linspace(0,5,50)









    Out[42]:





array([ 0.        ,  0.10204082,  0.20408163,  0.30612245,  0.40816327,
        0.51020408,  0.6122449 ,  0.71428571,  0.81632653,  0.91836735,
        1.02040816,  1.12244898,  1.2244898 ,  1.32653061,  1.42857143,
        1.53061224,  1.63265306,  1.73469388,  1.83673469,  1.93877551,
        2.04081633,  2.14285714,  2.24489796,  2.34693878,  2.44897959,
        2.55102041,  2.65306122,  2.75510204,  2.85714286,  2.95918367,
        3.06122449,  3.16326531,  3.26530612,  3.36734694,  3.46938776,
        3.57142857,  3.67346939,  3.7755102 ,  3.87755102,  3.97959184,
        4.08163265,  4.18367347,  4.28571429,  4.3877551 ,  4.48979592,
        4.59183673,  4.69387755,  4.79591837,  4.89795918,  5.        ])



In [27]:

    
# generate 1 or 0-arrays
print(np.ones( (2,3) ))
print(np.zeros( (3,4) ))









    



[[ 1.  1.  1.]
 [ 1.  1.  1.]]
[[ 0.  0.  0.  0.]
 [ 0.  0.  0.  0.]
 [ 0.  0.  0.  0.]]



In [28]:

    
# construct an array with a similar shape to another one
np.zeros_like(A)









    Out[28]:





array([[0, 0],
       [0, 0]])



In [29]:

    
# identity matrix
Id = np.identity(4)
Id









    Out[29]:





array([[ 1.,  0.,  0.,  0.],
       [ 0.,  1.,  0.,  0.],
       [ 0.,  0.,  1.,  0.],
       [ 0.,  0.,  0.,  1.]])

Linear algebra in numpy



In [30]:

    
A = np.array([[1,2],[6,7]])
np.linalg.det(A) # determinant
np.linalg.eig(A) # eigenvalues and eigenvectors
np.linalg.inv(A) # matrix inverse
np.linalg.svd(A) # singular value decomposition









    Out[30]:





(array([[-0.22975292, -0.97324899],
        [-0.97324899,  0.22975292]]),
 array([ 9.47213595,  0.52786405]),
 array([[-0.64074744, -0.76775173],
        [ 0.76775173, -0.64074744]]))

numpy mathematics

All operations on arrays are elementwise per default



In [31]:

    
A = np.array([[1,2],[3,4]])
B = np.array([[0,1],[1,0]])
A + B
A - B
A * B
B / A
A ** B









    Out[31]:





array([[1, 2],
       [3, 1]])

smaller arrays are broadcasted automatically



In [32]:

    
A = np.ones( (3,3) )
c = np.array([1,2,3])
# per default arrays are added row-wise
A + c
# like this col-wise
A + c[:,np.newaxis]









    Out[32]:





array([[ 2.,  2.,  2.],
       [ 3.,  3.,  3.],
       [ 4.,  4.,  4.]])

useful array functions



In [33]:

    
# element-wise functions
np.sqrt(a) # square root
np.sign(a) # sign
np.log(a) # natural logarithm
np.log10(a) # decadic logarithm
np.exp(a) #exponential
np.sin(a) # trigonometric (also cos, tan, arcsin, arccos, arctan)

# non-element-wise
a.sum() # sum of all elements 
a.prod() # product of all elements
a.mean() # mean
a.var() # variance
a.std() # standard deviation
a.max() # maximum
a.min() # minimum
a.sort()

np.unique( [1,1,3,3,5] ) # get unique elements









    Out[33]:





array([1, 3, 5])

Matrix multiplication



In [34]:

    
A = np.array( [[0,1], [1,0]] )
v = np.array( [6,7] )
np.dot( A,v ) # matrix product









    Out[34]:





array([7, 6])

Advaned array acessing



In [35]:

    
a = np.array( [1,6,3,4,9,6,7,3,2,4,5] )
a[ a>4 ] # get all elements larger than 4
a[ np.logical_and(a>4, a<12) ]

# acessing via index
indices = [1,3]
a[indices]









    Out[35]:





array([6, 4])

Scipy

Scipy is a collection of useful scientific algorithms

scipy.integrate
scipy.optimize
scipy.interpolate
etc...



In [36]:

    
import scipy as sp 
import scipy.optimize



In [37]:

    
from matplotlib import pyplot as plt
plt.style.use('ggplot')
%matplotlib inline

Example 1: numerical minimization of a function



In [38]:

    
# define a function
def myfunc( x ):
    return ( 3 + x ) * (3 - x)  * -12


# minimize using scipy
sp.optimize.minimize_scalar( myfunc )









    Out[38]:





  fun: -108.0
 nfev: 39
  nit: 38
    x: -3.195989243713537e-11



In [39]:

    
# plot the function 
x = np.arange( -100, 100, 1 )
plt.plot(x, myfunc(x) )









    Out[39]:





[<matplotlib.lines.Line2D at 0x7fa1458a0750>]

Example 2: numerical integration

Consider a growing bacterial population with a doubling time of 2 hours

this means that the growth rate is $$ k_g = \frac {ln( 2 )} {2} = 0.34657359027997264 $$

and the growth model: $$ \frac {dx} {dt} = k_g \cdot x $$

lets integrate this ODE using scipy.integrate.odeint



In [40]:

    
import scipy.integrate

k_g = np.log(2)/2
x0  = 1e-5
time = np.arange(0,10,1)


def dxdt( x, t ):
    return k_g * x


result = scipy.integrate.odeint(dxdt, x0, time )

plt.plot( time, result)









    Out[40]:





[<matplotlib.lines.Line2D at 0x7fa14435c310>]

Compare exact and numerical solutions



In [41]:

    
exact = x0 * np.exp( k_g*time )
exact - result.T

plt.plot( time, (exact - result.T).T )









    Out[41]:





[<matplotlib.lines.Line2D at 0x7fa1442a3350>]