In numerical analysis, Newton's method, also known as the Newton – Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the beginnings (or zeroes) of a real-valued function. The most basic version begins with a individual-variable function f specify for a real variable X, the function's derivative F ′, and an initial guess x0 for a root of f. Raphson again viewed Newton's method purely as an algebraic method and restricted its purpose to polynomials, but he describes the method in terms of the successive approximations xn instead of the more complicated sequence of polynomials used by Newton. The name" Newton's method" is derived from Isaac Newton's description of a special case of the method in De analysi per aequationesFinally, in 1740, Thomas Simpson described Newton's method as an iterative method for solve general nonlinear equations use calculus, essentially give the description above.

COMING SOON!

```
"""
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:
# If logstep is true, then log intermediate steps
return a, error, steps
return a, error
if __name__ == "__main__":
import matplotlib.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}}}")
```