Add Brent's Method for root finding (numerical analysis)

maths/numerical_analysis/brent_method.py

@@ -0,0 +1,121 @@
+from collections.abc import Callable
+
+
+def brent_method(
+    func: Callable[[float], float],
+    left: float,
+    right: float,
+    tol: float = 1e-8,
+    max_iter: int = 100,
+) -> float:
+    """
+    Find the root of function func in the interval [left, right] using Brent's Method.
+
+    Brent's Method combines bisection, secant, and inverse quadratic interpolation.
+
+
+    Parameters
+    ----------
+    func : Callable[[float], float]
+        Function for which to find the root.
+    left : float
+        Left endpoint of interval.
+    right : float
+        Right endpoint of interval.
+    tol : float
+        Tolerance for convergence (default 1e-8).
+    max_iter : int
+        Maximum number of iterations (default 100).
+
+
+    Returns
+    -------
+    float
+        Approximate root of func in [left, right].
+
+    Raises
+    ------
+    ValueError
+        If func(left) and func(right) do not have opposite signs.
+
+    Examples
+    --------
+    >>> def f(x): return x**3 - x - 2
+    >>> round(brent_method(f, 1, 2), 5)
+    1.52138
+
+    >>> def f2(x): return x**2 + 1
+    >>> brent_method(f2, 0, 1)
+    Traceback (most recent call last):
+    ...
+    ValueError: func(left) and func(right) must have opposite signs
+    """
+    fl = func(left)
+    fr = func(right)
+
+    if fl * fr >= 0:
+        raise ValueError("func(left) and func(right) must have opposite signs")
+
+    if abs(fl) < abs(fr):
+        left, right = right, left
+        fl, fr = fr, fl
+
+    c = left
+    fc = fl
+    d = right - left
+
+    for iteration in range(max_iter):
+        if fr == 0:
+            return right
+
+        if fc not in (fl, fr):
+            # Inverse quadratic interpolation
+            s = (
+                left * fr * fc / ((fl - fr) * (fl - fc))
+                + right * fl * fc / ((fr - fl) * (fr - fc))
+                + c * fl * fr / ((fc - fl) * (fc - fr))
+            )
+        else:
+            # Secant method
+            s = right - fr * (right - left) / (fr - fl)
+
+        conditions = [
+            not ((3 * left + right) / 4 < s < right)
+            if right > left
+            else not (right < s < (3 * left + right) / 4),
+            iteration > 1 and abs(s - right) >= abs(right - c) / 2,
+            iteration <= 1 and abs(s - right) >= abs(c - d) / 2,
+            iteration > 1 and abs(right - c) < tol,
+            iteration <= 1 and abs(c - d) < tol,
+        ]
+
+        if any(conditions):
+            # Bisection fallback
+            s = (left + right) / 2
+            d = right - left
+
+        fs = func(s)
+        d, c = c, right
+        fc = fr
+
+        if fl * fs < 0:
+            right = s
+            fr = fs
+        else:
+            left = s
+            fl = fs
+
+        if abs(fl) < abs(fr):
+            left, right = right, left
+            fl, fr = fr, fl
+
+        if abs(right - left) < tol:
+            return right
+
+    return right
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod(verbose=True)