#### Newton Raphson

```"""
Author: P Shreyas Shetty
Implementation of Newton-Raphson method for solving equations of kind
f(x) = 0. It is an iterative method where solution is found by the expression
x[n+1] = x[n] + f(x[n])/f'(x[n])
If no solution exists, then either the solution will not be found when iteration
limit is reached or the gradient f'(x[n]) approaches zero. In both cases, exception
is raised. If iteration limit is reached, try increasing maxiter.
"""
import math as m

def calc_derivative(f, a, h=0.001):
"""
Calculates derivative at point a for function f using finite difference
method
"""
return (f(a + h) - f(a - h)) / (2 * h)

def newton_raphson(f, x0=0, maxiter=100, step=0.0001, maxerror=1e-6, logsteps=False):

a = x0  # set the initial guess
steps = [a]
error = abs(f(a))
f1 = lambda x: calc_derivative(f, x, h=step)  # noqa: E731  Derivative of f(x)
for _ in range(maxiter):
if f1(a) == 0:
raise ValueError("No converging solution found")
a = a - f(a) / f1(a)  # Calculate the next estimate
if logsteps:
steps.append(a)
if error < maxerror:
break
else:
raise ValueError("Iteration limit reached, no converging solution found")
if logsteps:
return a, error, steps
return a, error

if __name__ == "__main__":
from matplotlib import pyplot as plt

f = lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
solution, error, steps = newton_raphson(
f, x0=10, maxiter=1000, step=1e-6, logsteps=True
)
plt.plot([abs(f(x)) for x in steps])
plt.xlabel("step")
plt.ylabel("error")
plt.show()
print(f"solution = {{{solution:f}}}, error = {{{error:f}}}")
```  