Types of roots

import numpy as np
import plotly.graph_objects as go
from scipy.optimize import root

try:
    import executable_engineering as exe
except ImportError:
    %pip install -q executable_engineering
    import executable_engineering as exe

from executable_engineering import weierstrass, plot_and_find_roots, complex_plot_with_shadow

Roots of some nonlinear functions¶

Let’s build some intuition by exploring some type of roots in 1D functions using the graphical method: Plot the function and examine where it crosses the x-axis.

NB: Note the structure of the code below - Since we don’t know a priori where the roots will be, we have to take a series of initial guesses and cross our finger.... and even then we may fail to find them all!

Example 1: Real roots - $x^2-4$ ¶

plot_and_find_roots(lambda x: x**2-4)

Example 2: No roots - $1/x$ ¶

(Noting that the vertical line is a plotting artifact).

plot_and_find_roots(lambda x: 1/x)

Example 3: Infinite roots $\sin(x^2)$ ¶

Only the roots closest to the initial guesses are found!

plot_and_find_roots(lambda x: np.sin(x**2))

Complex roots - $x^2+1$ ¶

Even the graphical method is not completely reliable due to the existence of complex roots.

But this is wrong! The quadratic has 2 roots but we need to use a different method.

def fun(x):
    return x**2 + 1

# Wrong!
plot_and_find_roots(fun)

# Find roots using krylov
root(lambda x: x**2+1, x0 = [1+1j, 1-1j], method = "krylov")

Visualizing complex roots¶

DON’T WORRY - We won’t be dealing with complex numbers in general in this course :-)

# Find roots and plot the complex surface with a 2D shadow
complex_plot_with_shadow(lambda z: z**2 + 1, roots=[1j, -1j])

Pathological / well-behaved functions¶

Some functions can be very tricky indeed, such as the Weierstrass Function,

\begin{aligned} \sum_{n=0}^\infty a^n \cos (b^n \pi x) \\ 0 < a < 1 \\ a b > 1+\frac{3 \pi}{2} \end{aligned}

(1)

which looks like:

This is a classic example of a continuous but nowhere differentiable function! It is a fractal, and yet roots do exist and may be on interest. It is a classic example of a pathological function.

Different applications will have different criteria for pathological vs well behaved functions, and no general definition exists although there is seldom disagreement among non-mathematicians about which is which.

Generally, well-behaved functions don’t have sharp corners, singularities, discontinuities, etc.

import plotly.graph_objects as go
import numpy as np

x_vals = np.linspace(-2, 2, 2000)
y_vals = weierstrass(x_vals, a=0.5, b=13)

fig = go.Figure()
fig.add_trace(go.Scatter(x=x_vals, y=y_vals, mode='lines', name='Weierstrass'))
fig.update_layout(title='Weierstrass Function (a=0.5, b=13)', xaxis_title='x', yaxis_title='f(x)')
fig.show()

Roots of some nonlinear functions¶

Example 1: Real roots - x2−4x^2-4x2−4¶

Example 2: No roots - 1/x1/x1/x¶

Example 3: Infinite roots sin⁡(x2)\sin(x^2)sin(x2)¶

Complex roots - x2+1x^2+1x2+1¶