Function newton_raphson

maths/numerical_analysis/newton_raphson.py:35–98

Find a root of the given function f using the Newton-Raphson method. :param f: A real-valued single-variable function :param x0: Initial guess :param max_iter: Maximum number of iterations :param step: Step size of x, used to approximate f'(x) :param max_error: Maximum appr

(
    f: RealFunc,
    x0: float = 0,
    max_iter: int = 100,
    step: float = 1e-6,
    max_error: float = 1e-6,
    log_steps: bool = False,
)

def newton_raphson(
    f: RealFunc,
    x0: float = 0,
    max_iter: int = 100,
    step: float = 1e-6,
    max_error: float = 1e-6,
    log_steps: bool = False,
) -> tuple[float, float, list[float]]:
    """
    Find a root of the given function f using the Newton-Raphson method.

    :param f: A real-valued single-variable function
    :param x0: Initial guess
    :param max_iter: Maximum number of iterations
    :param step: Step size of x, used to approximate f'(x)
    :param max_error: Maximum approximation error
    :param log_steps: bool denoting whether to log intermediate steps

    :return: 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.
    :raises 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
    """

    def f_derivative(x: float) -> float:
        return calc_derivative(f, x, step)

    a = x0  # Set initial guess
    steps = []
    for _ in range(max_iter):
        if log_steps:  # Log intermediate steps
            steps.append(a)

        error = abs(f(a))
        if error < max_error: