Python/maths/newton_raphson.py
Caeden Perelli-Harris 490e645ed3
Fix minor typing errors in maths/ ()
* updating DIRECTORY.md

* types(maths): Fix pylance issues in maths

* reset(vsc): Reset settings changes

* Update maths/jaccard_similarity.py

Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

* revert(erosion_operation): Revert erosion_operation

* test(jaccard_similarity): Add doctest to test alternative_union

* types(newton_raphson): Add typehints to func bodies

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Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>
2023-08-15 14:27:41 -07:00

65 lines
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Python

"""
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
from collections.abc import Callable
DerivativeFunc = Callable[[float], float]
def calc_derivative(f: DerivativeFunc, a: float, h: float = 0.001) -> float:
"""
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: DerivativeFunc,
x0: float = 0,
maxiter: int = 100,
step: float = 0.0001,
maxerror: float = 1e-6,
logsteps: bool = False,
) -> tuple[float, float, list[float]]:
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__":
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}}}")