Question

Brent’s method is a combination of bisection, secant and inverse quadratic interpolation. Like bisection, it is a ‘bracketed’ method (starts with points \((a,b)\) such that \(f(a)f(b)<0\).

Roughly speaking, the method begins by using the secant method to obtain a third point \(c\), then uses inverse quadratic interpolation to generate the next possible root. Without going into too much detail, the algorithm attempts to assess when interpolation will go awry, and if so, performs a bisection step. Also, it has certain criteria to reject an iterate. If that happens, the next step will be linear interpolation (secant method).

The Brent method is the default method that scypy uses to minimize a univariate function:

from scipy.optimize import minimize_scalar

def f(x):
    return (x - 2) * x * (x + 2)**2

res = minimize_scalar(f)
res.x

1.2807764040333458

x = np.arange(-5,5, 0.1);
p1=plt.plot(x, f(x))
plt.xlim(-4, 4)
plt.ylim(-10, 20)
plt.xlabel('x')
plt.axhline(0)

<matplotlib.lines.Line2D at 0x7f9c9b232bd0>

To find zeroes, use

scipy.optimize.brentq(f,-1,.5)

-7.864845203343107e-19

scipy.optimize.brentq(f,.5,3)

2.0

scipy.optimize.newton(f,-3)

-2.0000000172499592