1.5.12.15. Minima and roots of a function

Demos finding minima and roots of a function.

Define the function

import numpy as np

x = np.arange(-10, 10, 0.1)
def f(x):
    return x**2 + 10*np.sin(x)

Find minima

import scipy as sp

# Global optimization
grid = (-10, 10, 0.1)
xmin_global = sp.optimize.brute(f, (grid, ))
print(f"Global minima found {xmin_global}")

# Constrain optimization
xmin_local = sp.optimize.fminbound(f, 0, 10)
print(f"Local minimum found {xmin_local}")

Global minima found [-1.30641113]
Local minimum found 3.837467119498474

Root finding

root = sp.optimize.root(f, 1)  # our initial guess is 1
print(f"First root found {root.x}")
root2 = sp.optimize.root(f, -2.5)
print(f"Second root found {root2.x}")

First root found [0.]
Second root found [-2.47948183]

Plot function, minima, and roots

import matplotlib.pyplot as plt
fig = plt.figure(figsize=(6, 4))
ax = fig.add_subplot(111)

# Plot the function
ax.plot(x, f(x), 'b-', label="f(x)")

# Plot the minima
xmins = np.array([xmin_global[0], xmin_local])
ax.plot(xmins, f(xmins), 'go', label="Minima")

# Plot the roots
roots = np.array([root.x, root2.x])
ax.plot(roots, f(roots), 'kv', label="Roots")

# Decorate the figure
ax.legend(loc='best')
ax.set_xlabel('x')
ax.set_ylabel('f(x)')
ax.axhline(0, color='gray')
plt.show()