newton raphson Algorithm
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.
# Implementing Newton Raphson method in Python
# Author: Syed Haseeb Shah (github.com/QuantumNovice)
# The Newton-Raphson method (also known as Newton's method) is a way to
# quickly find a good approximation for the root of a real-valued function
from decimal import Decimal
from math import * # noqa: F401, F403
from sympy import diff
def newton_raphson(func: str, a: int, precision: int = 10 ** -10) -> float:
""" Finds root from the point 'a' onwards by Newton-Raphson method
>>> newton_raphson("sin(x)", 2)
3.1415926536808043
>>> newton_raphson("x**2 - 5*x +2", 0.4)
0.4384471871911695
>>> newton_raphson("x**2 - 5", 0.1)
2.23606797749979
>>> newton_raphson("log(x)- 1", 2)
2.718281828458938
"""
x = a
while True:
x = Decimal(x) - (Decimal(eval(func)) / Decimal(eval(str(diff(func)))))
# This number dictates the accuracy of the answer
if abs(eval(func)) < precision:
return float(x)
# Let's Execute
if __name__ == "__main__":
# Find root of trigonometric function
# Find value of pi
print(f"The root of sin(x) = 0 is {newton_raphson('sin(x)', 2)}")
# Find root of polynomial
print(f"The root of x**2 - 5*x + 2 = 0 is {newton_raphson('x**2 - 5*x + 2', 0.4)}")
# Find Square Root of 5
print(f"The root of log(x) - 1 = 0 is {newton_raphson('log(x) - 1', 2)}")
# Exponential Roots
print(f"The root of exp(x) - 1 = 0 is {newton_raphson('exp(x) - 1', 0)}")
