Untitled

import cmath

# Define the polynomial function
def f(x):
    return x**4 - 3*x**3 + x**2 + x + 1

# Muller's Method Implementation
def muller_method(x0, x1, x2, tol=1e-5, max_iter=100):
    for i in range(max_iter):
        # Compute function values
        f0, f1, f2 = f(x0), f(x1), f(x2)

        # Compute differences
        h1, h2 = x1 - x0, x2 - x1
        δ1, δ2 = (f1 - f0) / h1, (f2 - f1) / h2
        a = (δ2 - δ1) / (h2 + h1)
        b = a * h2 + δ2
        c = f2

        # Compute the quadratic formula roots
        disc = cmath.sqrt(b**2 - 4 * a * c)
        denom = b + disc if abs(b + disc) > abs(b - disc) else b - disc
        x3 = x2 - (2 * c) / denom  # Next approximation

        # Check convergence
        if abs(x3 - x2) < tol:
            return x3, i + 1  # Return root and iteration count

        # Shift values for next iteration
        x0, x1, x2 = x1, x2, x3

    return x3, max_iter  # Return last computed root if max iterations reached

# Define the intervals
intervals = [
    (-1, 0, 1),     # (a)
    (-0.5, 0, 0.5),  # (b)
    (0.5, 1, 1.5),   # (c)
    (1.5, 2, 2.5)    # (d)
]

# Compute roots for each interval
for idx, (x0, x1, x2) in enumerate(intervals, start=1):
    root, iterations = muller_method(x0, x1, x2)
    print(f"Interval {idx}: [{x0}, {x2}]")
    print(f"  Approximate Root: {root:.6f}")
    print(f"  Iterations Required: {iterations}\n")