Root finding

In this section you will find 3 short code snippets to find roots with different methods. Try to understand how the code works and adapt it to solve your own problems.

Secant method

This code will find the roots of $f(x)=x^2-3x+1=0$ with the secant method.

def f(x):
    return x**2-3*x+1

def secant(x0,x1,e,N):
    step = 1
    condition = True
    while condition:
        if f(x0) == f(x1):
            print('Divide by zero error!')
            break
        
        x2 = x0 - (x1-x0)*f(x0)/( f(x1) - f(x0) ) 
        x0 = x1
        x1 = x2
        step = step + 1
        
        if step > N:
            print('Not Convergent!')
            break
        
        condition = abs(f(x2)) > e
    print('Root is: ', x2)

x0 = -2 # Guess 1
x1 = 3 # Guess 2
e = 0.0001 # Precision
N = 1000 # Max number of trials

secant(x0,x1,e,N)

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Newton-Raphson method

This code snippet will find the root of $f(x)=x^3-1$ with an initial guess of 1.5.

from scipy import optimize

def f(x):
  return (x**3 - 1) 
  
root = optimize.newton(f, 1.5)

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Bisection method

The following code will find the positive root of $f(x)=x^3-6x^2+11x-6=0$ using the bisection method from the interval [2.5, 4].

def samesign(a, b):
    return a * b > 0

def bisect(func, low, high):
    assert not samesign(func(low), func(high))

    for i in range(54):
        midpoint = (low + high) / 2.0
        if samesign(func(low), func(midpoint)):
            low = midpoint
        else:
            high = midpoint

    return midpoint

def f(x):
    return x**3 - 6*x**2 + 11*x - 6

x = bisect(f, 2.5, 4)
print(x, f(x))

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