ADDED BRENT'S METHOD

maths/numerical_analysis/brents_method.py

@@ -0,0 +1,114 @@
+"""
+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: float,
+    upper: 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: One end of the bracketing interval.
+        upper: 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, upper].
+
+    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].
+    """
+    fa = function(lower)
+    fb = function(upper)
+    if fa * fb >= 0:
+        raise ValueError(f"Root is not bracketed in the interval [{lower}, {upper}].")
+
+    if abs(fa) < abs(fb):
+        lower, upper = upper, lower
+        fa, fb = fb, fa
+
+    c = lower
+    fc = fa
+    d = upper - lower
+    mflag = True
+
+    for _ in range(max_iterations):
+        if fb == 0:
+            return upper
+        if fc not in {fa, fb}:
+            # Inverse quadratic interpolation
+            s = (
+                lower * fb * fc / ((fa - fb) * (fa - fc))
+                + upper * fa * fc / ((fb - fa) * (fb - fc))
+                + c * fa * fb / ((fc - fa) * (fc - fb))
+            )
+        else:
+            # Secant method
+            s = upper - fb * (upper - lower) / (fb - fa)
+
+        conditions = [
+            not ((3 * lower + upper) / 4 < s < upper if upper > lower else upper < s < (3 * lower + upper) / 4),
+            mflag and abs(s - upper) >= abs(upper - c) / 2,
+            not mflag and abs(s - upper) >= abs(c - d) / 2,
+            mflag and abs(upper - c) < tolerance,
+            not mflag and abs(c - d) < tolerance,
+        ]
+        if any(conditions):
+            s = (lower + upper) / 2
+            mflag = True
+        else:
+            mflag = False
+
+        fs = function(s)
+        d, c = c, upper
+        fc = fb
+
+        if fa * fs < 0:
+            upper = s
+            fb = fs
+        else:
+            lower = s
+            fa = fs
+
+        if abs(fa) < abs(fb):
+            lower, upper = upper, lower
+            fa, fb = fb, fa
+
+        if abs(upper - lower) < tolerance or fb == 0:
+            return upper
+
+    return upper
+
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()