Python is a full, object oriented, interpreted programming language. It has been around for a long time now and enjoys the attention of many good programmers who expanded it with packages. These packages often make use of the C API which lets native programs connect to the Python kernel. Thus, they are much faster and often state of the art. Python itself is more than ten times slower than native code.

Were are mainly going to use the following packages.

• Numpy. To get around the slow loops and array handling in Python, this package provides operations on a matrix level, just like Euler Math Toolbox or Matlab.
• Sympy. This is a package for symbolic computations. It provides symbolic variables and expressions.
• Matplotlib. To get beautiful plots in Python, you will have to use this package.
• Scilab. This is the high level package for mathematical computations, such as optimization or signal processing.

To install all this, the easiest and best way is Anaconda. It installs the recent Python version (3.10) and the packages mentioned above. Moreover, you get a user interface called IPython or Jupyter Notebooks.

Unfortunately, the Anaconda Navigator seems not to work on M1 processors. But you can open a command line and type

jupyter notebook

The default browser (Safari in my case) will open automatically and display the Jupyter notebook showing the files in your home directory. You can then switch to a subdirectory and create a new one. A notebook has input lines and output lines. You can just start typing in Python commands and execute with Ctrl-Return or Shift-Return, or start the Help to learn about this interface.

For users of Euler Math Toolbox (or Matlab), it is important to know, that Python does not handle operators on matrices element by element. In fact, using its list type like vectors, or lists of lists like matrices, yields surprising results.

2 * [4,5,6] 
  -> [4, 5, 6, 4, 5, 6]
[4,5,6] + [4,5,6]
  -> [4, 5, 6, 4, 5, 6]
[4,5,6] * [4,5,6]
------------------------------------------------------
TypeError
...
TypeError: can't multiply sequence by non-int of type 'list'

I have used an indented „->“ to indicate the output of the command. This convention will be used in the future on this blog.

Obviously, the Python folks have used math operators to manipulate lists in „intuitive“ ways.

This can be fixed with Numpy. You get a new data type, the „ndarray“. It is easily created, albeit requiring an additional function call. It provides the matrix functionality that we are used to from matrix languages like Matlab, R or Euler Math Toolbox.

import numpy
v = numpy.array([4,5,6])
2*v
  -> array([ 8, 10, 12])

v+v
  -> array([ 8, 10, 12])

v*v
  -> array([16, 25, 36])

Euler Math Toolbox often demonstrates the value of matrix language by creating tables of function values y=f(x). Here is a short demonstration and a plot.

import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt

x = np.arange(0,1,0.01)
y = x**3-x

print(f"Riemann-Summe: {np.sum(y)*0.01:0.4}")

fig, ax = plt.subplots(figsize=(10,6))
ax.grid(True)
ax.plot(x,y);

  -> Riemann-Summe: -0.25

As I said, there is some learning curve. Here are some differences to Euler Math Toolbox to start with.

• You need to import the packages. The „load“ command of EMT is similar. But EMT loads most packages at the start. You can import and use a package under a different, shorter name, as shown in the example.
• The „arange(a,b,delta)“ function works similar to the ranges in EMT. But it has the step size as a third parameter. There is also a „linspace(a,b,n)“ function like „equispace“ in EMT. But it generates exactly n points, not n intervals.
• In the example, an important difference is not visible: Array start with index 0 in Python and Numpy. This is a source of much confusion.
• For the plot, we cannot use a function, but must provide vectors of data.
• The plot will be inserted automatically into to Jupyter notebook. Of course, the graphics can also be saved to a file. That file will be in the directory of your notebook.

You will find numerous tutorials on Numpy and Matplotlib, explaining the steps above in much detail. So, I won’t give an introduction in this blog. Rather, my job will be to show you how EMT compares to the things you can do in Python, and to point out the main differences and problems that could occur.