ADDED BRENT'S METHOD

maths/numerical_analysis/brents_method.py

@@ -0,0 +1,146 @@
+"""
+Brent's Method for Root Finding
+
+Find a root of a function in a bracketing interval using Brent's method.
+
+Brent's method combines bisection, secant, and inverse quadratic interpolation to efficiently and robustly find a root of a continuous function. It is guaranteed to converge as long as the root is bracketed.
+
+See:
+    https://en.wikipedia.org/wiki/Brent%27s_method
+
+Author: Aryan Singh (2nd year LNMIIT)
+"""
+
+from collections.abc import Callable
+
+
+def brent_method(
+    function: Callable[[float], float],
+    lower_bound: float,
+    upper_bound: float,
+    tolerance: float = 1e-14,
+    max_iterations: int = 100,
+) -> float:
+    """
+    Find a root of a function in a bracketing interval using Brent's method.
+
+    Args:
+        function: A continuous function for which the root is sought.
+        lower_bound: One end of the bracketing interval.
+        upper_bound: The other end of the bracketing interval.
+        tolerance: The tolerance for convergence (default 1e-14).
+        max_iterations: Maximum number of iterations (default 100).
+
+    Returns:
+        A float value approximating the root.
+
+    Raises:
+        ValueError: If the root is not bracketed in [lower_bound, upper_bound].
+
+    Examples:
+        >>> brent_method(lambda x: x**3 - 1, -5, 5)
+        1.0
+        >>> brent_method(lambda x: x**2 - 4*x + 3, 0, 2)
+        1.0
+        >>> brent_method(lambda x: x**2 - 4*x + 3, 2, 4)
+        3.0
+        >>> brent_method(lambda x: x**2 - 4*x + 3, 4, 1000)
+        Traceback (most recent call last):
+            ...
+        ValueError: Root is not bracketed in the interval [4, 1000].
+    """
+    function_lower = function(lower_bound)
+    function_upper = function(upper_bound)
+    if function_lower * function_upper >= 0:
+        error_message = (
+            f"Root is not bracketed in the interval [{lower_bound}, {upper_bound}]."
+        )
+        raise ValueError(error_message)
+
+    if abs(function_lower) < abs(function_upper):
+        lower_bound, upper_bound = upper_bound, lower_bound
+        function_lower, function_upper = function_upper, function_lower
+
+    previous_bound = lower_bound
+    function_previous = function_lower
+    previous_step = upper_bound - lower_bound
+    bisect_flag = True
+
+    for _ in range(max_iterations):
+        if function_upper == 0:
+            return upper_bound
+        if function_previous not in {function_lower, function_upper}:
+            # Inverse quadratic interpolation
+            s = (
+                lower_bound
+                * function_upper
+                * function_previous
+                / (
+                    (function_lower - function_upper)
+                    * (function_lower - function_previous)
+                )
+                + upper_bound
+                * function_lower
+                * function_previous
+                / (
+                    (function_upper - function_lower)
+                    * (function_upper - function_previous)
+                )
+                + previous_bound
+                * function_lower
+                * function_upper
+                / (
+                    (function_previous - function_lower)
+                    * (function_previous - function_upper)
+                )
+            )
+        else:
+            # Secant method
+            s = upper_bound - function_upper * (upper_bound - lower_bound) / (
+                function_upper - function_lower
+            )
+
+        conditions = [
+            not (
+                (3 * lower_bound + upper_bound) / 4 < s < upper_bound
+                if upper_bound > lower_bound
+                else upper_bound < s < (3 * lower_bound + upper_bound) / 4
+            ),
+            bisect_flag
+            and abs(s - upper_bound) >= abs(upper_bound - previous_bound) / 2,
+            not bisect_flag
+            and abs(s - upper_bound) >= abs(previous_bound - previous_step) / 2,
+            bisect_flag and abs(upper_bound - previous_bound) < tolerance,
+            not bisect_flag and abs(previous_bound - previous_step) < tolerance,
+        ]
+        if any(conditions):
+            s = (lower_bound + upper_bound) / 2
+            bisect_flag = True
+        else:
+            bisect_flag = False
+
+        function_s = function(s)
+        previous_step, previous_bound = previous_bound, upper_bound
+        function_previous = function_upper
+
+        if function_lower * function_s < 0:
+            upper_bound = s
+            function_upper = function_s
+        else:
+            lower_bound = s
+            function_lower = function_s
+
+        if abs(function_lower) < abs(function_upper):
+            lower_bound, upper_bound = upper_bound, lower_bound
+            function_lower, function_upper = function_upper, function_lower
+
+        if abs(upper_bound - lower_bound) < tolerance or function_upper == 0:
+            return upper_bound
+
+    return upper_bound
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()