maths.numerical_analysis.newton_raphson

The Newton-Raphson method (aka the Newton method) is a root-finding algorithm that approximates a root of a given real-valued function f(x). It is an iterative method given by the formula

x_{n + 1} = x_n + f(x_n) / f’(x_n)

with the precision of the approximation increasing as the number of iterations increase.

Reference: https://en.wikipedia.org/wiki/Newton%27s_method

Attributes

RealFunc

Functions

calc_derivative(→ float)

Approximate the derivative of a function f(x) at a point x using the finite

func(→ float)

newton_raphson(→ tuple[float, float, list[float]])

Find a root of the given function f using the Newton-Raphson method.

Module Contents

maths.numerical_analysis.newton_raphson.calc_derivative(f: RealFunc, x: float, delta_x: float = 0.001) → float

Approximate the derivative of a function f(x) at a point x using the finite difference method

>>> import math
>>> tolerance = 1e-5
>>> derivative = calc_derivative(lambda x: x**2, 2)
>>> math.isclose(derivative, 4, abs_tol=tolerance)
True
>>> derivative = calc_derivative(math.sin, 0)
>>> math.isclose(derivative, 1, abs_tol=tolerance)
True
maths.numerical_analysis.newton_raphson.func(x: float) → float
maths.numerical_analysis.newton_raphson.newton_raphson(f: RealFunc, x0: float = 0, max_iter: int = 100, step: float = 1e-06, max_error: float = 1e-06, log_steps: bool = False) → tuple[float, float, list[float]]

Find a root of the given function f using the Newton-Raphson method.

Parameters:

  • f – A real-valued single-variable function

  • x0 – Initial guess

  • max_iter – Maximum number of iterations

  • step – Step size of x, used to approximate f’(x)

  • max_error – Maximum approximation error

  • log_steps – bool denoting whether to log intermediate steps

Returns:

A tuple containing the approximation, the error, and the intermediate steps. If log_steps is False, then an empty list is returned for the third element of the tuple.

Raises:

  • ZeroDivisionError – The derivative approaches 0.

  • ArithmeticError – No solution exists, or the solution isn’t found before the iteration limit is reached.

>>> import math
>>> tolerance = 1e-15
>>> root, *_ = newton_raphson(lambda x: x**2 - 5*x + 2, 0.4, max_error=tolerance)
>>> math.isclose(root, (5 - math.sqrt(17)) / 2, abs_tol=tolerance)
True
>>> root, *_ = newton_raphson(lambda x: math.log(x) - 1, 2, max_error=tolerance)
>>> math.isclose(root, math.e, abs_tol=tolerance)
True
>>> root, *_ = newton_raphson(math.sin, 1, max_error=tolerance)
>>> math.isclose(root, 0, abs_tol=tolerance)
True
>>> newton_raphson(math.cos, 0)
Traceback (most recent call last):
...
ZeroDivisionError: No converging solution found, zero derivative
>>> newton_raphson(lambda x: x**2 + 1, 2)
Traceback (most recent call last):
...
ArithmeticError: No converging solution found, iteration limit reached
maths.numerical_analysis.newton_raphson.RealFunc