Functions and plotting.ipynb

Kernel: Python 3 (Ubuntu Linux)

Functions and Plotting

In this notebook you will find some examples on working with functions and basic plotting using numpy and pyplot. Documentation can be found here and here.

# import the required modules
import numpy as np
import matplotlib.pyplot as plt

Elementary operations

We will frequently be computing with arrays of numbers, so let's see how various elementary computations are defined

# define array
x = np.array([1,2,3,4])

# inspect the array
print(x)

[1 2 3 4]

# elementary operations with two arrays
y = np.array([5,6,7,8])

print(x + y)

[ 6  8 10 12]

Exercise

Try various elementary operations (+,-,*,/). What do the operations x**2 and x**y produce? And how about x^2?

Univariate functions

To plot a function $f$ we first need to evaluate it at a number of points $x_i$ and subsequently produce a plot.

# uniform grid of 100 points on [-1,1]
x = np.linspace(-1,1,100)

# plot sine and cosine
plt.plot(x,np.sin(x),x,np.cos(x))

[<matplotlib.lines.Line2D at 0x7f3e41edacc0>,
 <matplotlib.lines.Line2D at 0x7f3e41edaeb8>]

We can add labels, a legend and change the linestyle of the plots:

# uniform grid of 100 points on [-1,1]
x = np.linspace(-1,1,100)

# plot sine and cosine
plt.plot(x,np.sin(x),'r-',x,np.cos(x),'k--')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.legend(('sin(x)','cos(x)'))

<matplotlib.legend.Legend at 0x7f3e18119f60>

Defining your own functions

We can define our own functions as follows

# define f(x) = x^2
f = lambda x : x**2

# define piecewise function (uses list comprehension and conditional expression)
g = lambda x : np.array([s if s > 0 else s**2 for s in x])
    
# plot
x = np.linspace(-1,1,100)
plt.plot(x,f(x),x,g(x))

[<matplotlib.lines.Line2D at 0x7f3e115b0908>,
 <matplotlib.lines.Line2D at 0x7f3e115b0a90>]

Exercise

Define the function ParseError: KaTeX parse error: Unknown column alignment: - at position 29: …{\begin{array} -̲-x & x < -1 \\ …

and plot it.

Multivariate functions

When plotting a function of two variables, we need to define a grid $(x_i, y_i)$ first:

# define grid with 100 x 100 points on [0,1]^2
x = np.linspace(0,1,100)
y = np.linspace(0,1,100)
xx,yy = np.meshgrid(x,y)

# define function
f = lambda x,y : np.exp(-(x - .5)**2-5*(y-.2)**2)

# various ways of plotting
plt.subplot(121)
plt.contour(xx,yy,f(xx,yy))
plt.axis('square')

plt.subplot(122)
plt.contourf(xx,yy,f(xx,yy))
plt.axis('square')

(0.0, 1.0, 0.0, 1.0)