Demo of Python in Jupyter Notebook

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<iframe width="560" height="315" src="https://www.youtube.com/embed/HW29067qVWk" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>

%load_ext version_information

%version_information numpy, scipy, sympy, matplotlib, version_information

Software	Version
Python	3.9.2 64bit [GCC 9.3.0]
IPython	7.21.0
OS	Linux 4.19.0 14 amd64 x86_64 with glibc2.28
numpy	1.20.1
scipy	1.6.0
sympy	1.7.1
matplotlib	3.3.4
version_information	1.0.3 (fixed)
Tue Mar 16 23:24:55 2021 CET

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

N = 50
x = np.random.rand(N)
y = np.random.rand(N)
colors = np.random.rand(N)
area = np.pi * (15 * np.random.rand(N))**2

plt.scatter(x, y, s=area, c=colors, alpha=0.5)
plt.show()

import pandas as pd
import numpy as np

df = pd.DataFrame(np.random.randn(10,5))
df

	0	1	2	3	4
0	1.244373	-2.375276	-1.249719	0.666848	0.917400
1	-0.299879	-0.886450	0.460110	-0.546190	0.340645
2	-1.110623	-0.544608	0.977602	-2.100853	0.234692
3	1.329575	0.341214	0.360423	-1.110605	0.690054
4	1.037706	-1.841255	-2.410571	0.249865	0.705019
5	-2.255515	0.092135	1.094941	0.150178	-1.117143
6	0.962582	1.498060	1.738207	-0.526848	0.337179
7	0.069187	-1.550710	0.096162	-0.250829	1.040123
8	0.533513	-0.818389	3.267903	0.171266	-0.786601
9	0.837141	1.124104	-0.587130	-0.399321	-1.489129

SymPy: Open Source Symbolic Mathematics

This section uses the SymPy package to perform symbolic manipulations, and combined with numpy and matplotlib, also displays numerical visualizations of symbolically constructed expressions.

We first load sympy printing extensions, as well as all of sympy:

from __future__ import division
from IPython.display import display

from sympy.interactive import printing
printing.init_printing(use_latex='mathjax')

import sympy as sym
from sympy import *
x, y, z = symbols("x y z")
k, m, n = symbols("k m n", integer=True)
f, g, h = map(Function, 'fgh')

Elementary operations

Rational(3,2)*pi + exp(I*x) / (x**2 + y)

$\displaystyle \frac{3 \pi}{2} + \frac{e^{i x}}{x^{2} + y}$

exp(I*x).subs(x,pi).evalf()

$\displaystyle -1.0$

e = x + 2*y

srepr(e)

"Add(Symbol('x'), Mul(Integer(2), Symbol('y')))"

exp(pi * sqrt(163)).evalf(50)

$\displaystyle 262537412640768743.99999999999925007259719818568888$

Algebra

eq = ((x+y)**2 * (x+1))
eq

$\displaystyle \left(x + 1\right) \left(x + y\right)^{2}$

expand(eq)

$\displaystyle x^{3} + 2 x^{2} y + x^{2} + x y^{2} + 2 x y + y^{2}$

a = 1/x + (x*sin(x) - 1)/x
a

$\displaystyle \frac{x \sin{\left(x \right)} - 1}{x} + \frac{1}{x}$

simplify(a)

$\displaystyle \sin{\left(x \right)}$

eq = Eq(x**3 + 2*x**2 + 4*x + 8, 0)
eq

$\displaystyle x^{3} + 2 x^{2} + 4 x + 8 = 0$

solve(eq, x)

$\displaystyle \left[ -2, \ - 2 i, \ 2 i\right]$

a, b = symbols('a b')
Sum(6*n**2 + 2**n, (n, a, b))

$\displaystyle \sum_{n=a}^{b} \left(2^{n} + 6 n^{2}\right)$

Calculus

limit((sin(x)-x)/x**3, x, 0)

$\displaystyle - \frac{1}{6}$

(1/cos(x)).series(x, 0, 6)

$\displaystyle 1 + \frac{x^{2}}{2} + \frac{5 x^{4}}{24} + O\left(x^{6}\right)$

diff(cos(x**2)**2 / (1+x), x)

$\displaystyle - \frac{4 x \sin{\left(x^{2} \right)} \cos{\left(x^{2} \right)}}{x + 1} - \frac{\cos^{2}{\left(x^{2} \right)}}{\left(x + 1\right)^{2}}$

integrate(x**2 * cos(x), (x, 0, pi/2))

$\displaystyle -2 + \frac{\pi^{2}}{4}$

eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)
display(eqn)
dsolve(eqn, f(x))

$\displaystyle 9 f{\left(x \right)} + \frac{d^{2}}{d x^{2}} f{\left(x \right)} = 1$

$\displaystyle f{\left(x \right)} = C_{1} \sin{\left(3 x \right)} + C_{2} \cos{\left(3 x \right)} + \frac{1}{9}$

Illustrating Taylor series

We will define a function to compute the Taylor series expansions of a symbolically defined expression at various orders and visualize all the approximations together with the original function

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

# You can change the default figure size to be a bit larger if you want,
# uncomment the next line for that:
#plt.rc('figure', figsize=(10, 6))

def plot_taylor_approximations(func, x0=None, orders=(2, 4), xrange=(0,1), yrange=None, npts=200):
    """Plot the Taylor series approximations to a function at various orders.

    Parameters
    ----------
    func : a sympy function
    x0 : float
      Origin of the Taylor series expansion.  If not given, x0=xrange[0].
    orders : list
      List of integers with the orders of Taylor series to show.  Default is (2, 4).
    xrange : 2-tuple or array.
      Either an (xmin, xmax) tuple indicating the x range for the plot (default is (0, 1)),
      or the actual array of values to use.
    yrange : 2-tuple
      (ymin, ymax) tuple indicating the y range for the plot.  If not given,
      the full range of values will be automatically used. 
    npts : int
      Number of points to sample the x range with.  Default is 200.
    """
    if not callable(func):
        raise ValueError('func must be callable')
    if isinstance(xrange, (list, tuple)):
        x = np.linspace(float(xrange[0]), float(xrange[1]), npts)
    else:
        x = xrange
    if x0 is None: x0 = x[0]
    xs = sym.Symbol('x')
    # Make a numpy-callable form of the original function for plotting
    fx = func(xs)
    f = sym.lambdify(xs, fx, modules=['numpy'])
    # We could use latex(fx) instead of str(), but matploblib gets confused
    # with some of the (valid) latex constructs sympy emits.  So we play it safe.
    plt.plot(x, f(x), label=str(fx), lw=2)
    # Build the Taylor approximations, plotting as we go
    apps = {}
    for order in orders:
        app = fx.series(xs, x0, n=order).removeO()
        apps[order] = app
        # Must be careful here: if the approximation is a constant, we can't
        # blindly use lambdify as it won't do the right thing.  In that case, 
        # evaluate the number as a float and fill the y array with that value.
        if isinstance(app, sym.core.numbers.Number):
            y = np.zeros_like(x)
            y.fill(app.evalf())
        else:
            fa = sym.lambdify(xs, app, modules=['numpy'])
            y = fa(x)
        tex = sym.latex(app).replace('$', '')
        plt.plot(x, y, label=r'$n=%s:\, %s$' % (order, tex) )
        
    # Plot refinements
    if yrange is not None:
        plt.ylim(*yrange)
    plt.grid()
    plt.legend(loc='best').get_frame().set_alpha(0.8)

With this function defined, we can now use it for any sympy function or expression

plot_taylor_approximations(sin, 0, [2, 4, 6], (0, 2*pi), (-2,2))

plot_taylor_approximations(cos, 0, [2, 4, 6], (0, 2*pi), (-2,2))

This shows easily how a Taylor series is useless beyond its convergence radius, illustrated by a simple function that has singularities on the real axis:

# For an expression made from elementary functions, we must first make it into
# a callable function, the simplest way is to use the Python lambda construct.
plot_taylor_approximations(lambda x: 1/cos(x), 0, [2,4,6], (0, 2*pi), (-5,5))

Basic Numerical Integration: the Trapezoid Rule

A simple illustration of the trapezoid rule for definite integration:

$$ \int_{a}^{b} f(x)\, dx \approx \frac{1}{2} \sum_{k=1}^{N} \left( x_{k} - x_{k-1} \right) \left( f(x_{k}) + f(x_{k-1}) \right). $$

First, we define a simple function and sample it between 0 and 10 at 200 points

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

def f(x):
    return (x-3)*(x-5)*(x-7)+85

x = np.linspace(0, 10, 200)
y = f(x)

Choose a region to integrate over and take only a few points in that region

a, b = 1, 8 # the left and right boundaries
N = 5 # the number of points
xint = np.linspace(a, b, N)
yint = f(xint)

Plot both the function and the area below it in the trapezoid approximation

plt.plot(x, y, lw=2)
plt.axis([0, 9, 0, 140])
plt.fill_between(xint, 0, yint, facecolor='gray', alpha=0.4)
plt.text(0.5 * (a + b), 30,r"$\int_a^b f(x)dx$", horizontalalignment='center', fontsize=20);

Compute the integral both at high accuracy and with the trapezoid approximation

from __future__ import print_function
from scipy.integrate import quad
integral, error = quad(f, a, b)
integral_trapezoid = sum( (xint[1:] - xint[:-1]) * (yint[1:] + yint[:-1]) ) / 2
print("The integral is:", integral, "+/-", error)
print("The trapezoid approximation with", len(xint), "points is:", integral_trapezoid)

The integral is: 565.2499999999999 +/- 6.275535646693696e-12
The trapezoid approximation with 5 points is: 559.890625