Consolidate Newton-Raphson implementations (#10859)

* updating DIRECTORY.md

* updating DIRECTORY.md

* Consolidate Newton-Raphson duplicates

* Rename consolidated Newton-Raphson file

* updating DIRECTORY.md

* updating DIRECTORY.md

* Fix doctest precision

* Fix doctest precision again

---------

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

Author: tianyizheng02

DIRECTORY.md

@@ -642,10 +642,7 @@
     * [Intersection](maths/numerical_analysis/intersection.py)
     * [Nevilles Method](maths/numerical_analysis/nevilles_method.py)
     * [Newton Forward Interpolation](maths/numerical_analysis/newton_forward_interpolation.py)
-    * [Newton Method](maths/numerical_analysis/newton_method.py)
     * [Newton Raphson](maths/numerical_analysis/newton_raphson.py)
-    * [Newton Raphson 2](maths/numerical_analysis/newton_raphson_2.py)
-    * [Newton Raphson New](maths/numerical_analysis/newton_raphson_new.py)
     * [Numerical Integration](maths/numerical_analysis/numerical_integration.py)
     * [Runge Kutta](maths/numerical_analysis/runge_kutta.py)
     * [Runge Kutta Fehlberg 45](maths/numerical_analysis/runge_kutta_fehlberg_45.py)

maths/numerical_analysis/newton_method.py

@@ -1,45 +1,113 @@
-# 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 __future__ import annotations
-
-from decimal import Decimal
-
-from sympy import diff, lambdify, symbols
-
-
-def newton_raphson(func: str, a: float | Decimal, precision: float = 1e-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
+"""
+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
+"""
+from collections.abc import Callable
+
+RealFunc = Callable[[float], float]
+
+
+def calc_derivative(f: RealFunc, x: float, delta_x: float = 1e-3) -> 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
+    """
+    return (f(x + delta_x / 2) - f(x - delta_x / 2)) / delta_x
+
+
+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]]:
     """
-    x = symbols("x")
-    f = lambdify(x, func, "math")
-    f_derivative = lambdify(x, diff(func), "math")
-    x_curr = a
-    while True:
-        x_curr = Decimal(x_curr) - Decimal(f(x_curr)) / Decimal(f_derivative(x_curr))
-        if abs(f(x_curr)) < precision:
-            return float(x_curr)
+    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:
+            return a, error, steps
+
+        if f_derivative(a) == 0:
+            raise ZeroDivisionError("No converging solution found, zero derivative")
+        a -= f(a) / f_derivative(a)  # Calculate next estimate
+    raise ArithmeticError("No converging solution found, iteration limit reached")
 
 
 if __name__ == "__main__":
     import doctest
+    from math import exp, tanh
 
     doctest.testmod()
 
-    # 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 value of e
-    print(f"The root of log(x) - 1 = 0 is {newton_raphson('log(x) - 1', 2)}")
-    # Find root of exponential function
-    print(f"The root of exp(x) - 1 = 0 is {newton_raphson('exp(x) - 1', 0)}")
+    def func(x: float) -> float:
+        return tanh(x) ** 2 - exp(3 * x)
+
+    solution, err, steps = newton_raphson(
+        func, x0=10, max_iter=100, step=1e-6, log_steps=True
+    )
+    print(f"{solution=}, {err=}")
+    print("\n".join(str(x) for x in steps))