diff --git a/homework11/Homework11_Q5.py b/homework11/Homework11_Q5.py
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+++ b/homework11/Homework11_Q5.py
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+import numpy as np
+import matplotlib.pyplot as plt
+
+# --- 1. Define Function, Gradient, and Hessian ---
+# Objective function: Rosenbrock function
+def objective_function(x):
+    x1, x2 = x[0], x[1]
+    return 100 * (x2 - x1**2)**2 + (1 - x1)**2
+
+# Gradient vector calculation
+def gradient(x):
+    x1, x2 = x[0], x[1]
+    df_dx1 = -400 * x1 * (x2 - x1**2) - 2 * (1 - x1)
+    df_dx2 = 200 * (x2 - x1**2)
+    return np.array([df_dx1, df_dx2])
+
+# Hessian matrix calculation
+def hessian(x):
+    x1, x2 = x[0], x[1]
+    d2f_dx1dx1 = -400 * (x2 - 3 * x1**2) + 2
+    d2f_dx1dx2 = -400 * x1
+    d2f_dx2dx1 = -400 * x1
+    d2f_dx2dx2 = 200
+    return np.array([[d2f_dx1dx1, d2f_dx1dx2], [d2f_dx2dx1, d2f_dx2dx2]])
+
+# --- 2. Backtracking Line Search ---
+def backtracking_line_search(x, p, grad, alpha_bar=1.0, rho=0.5, c1=1e-3):
+    """
+    Finds a step size alpha that satisfies the 1st Wolfe condition (Armijo condition).
+    Condition: f(x + alpha*p) <= f(x) + c1 * alpha * grad^T * p
+    """
+    alpha = alpha_bar
+    f_val = objective_function(x)
+    grad_dot_p = np.dot(grad, p)
+    
+    # Loop until condition is met
+    while objective_function(x + alpha * p) > f_val + c1 * alpha * grad_dot_p:
+        alpha *= rho
+        # Safety break to prevent infinite loops if alpha becomes too small
+        if alpha < 1e-16: 
+            break
+    return alpha
+
+# --- 3. Optimization Algorithms ---
+def run_optimization(method, x0, tol_ratio=1e-12, max_iter=20000):
+    """
+    Runs the specified optimization method.
+    Returns: list of alphas used, total iterations
+    """
+    x = x0.copy()
+    grad_0_norm = np.linalg.norm(gradient(x0))
+    
+    alphas = []
+    
+    print(f"[{method}] Starting from {x0}...")
+    
+    for i in range(max_iter):
+        grad = gradient(x)
+        grad_norm = np.linalg.norm(grad)
+        
+        # Convergence Check: ||grad f(xk)|| < 10^-12 * ||grad f(x0)||
+        if grad_norm < tol_ratio * grad_0_norm:
+            print(f"   -> Converged at iteration {i}. Final Loss: {objective_function(x):.6e}")
+            return alphas, i
+            
+        # Determine Search Direction
+        if method == "Steepest_Descent":
+            direction = -grad
+        elif method == "Newton":
+            hess = hessian(x)
+            try:
+                # Solve H * d = -g for d
+                direction = np.linalg.solve(hess, -grad) 
+            except np.linalg.LinAlgError:
+                # Fallback if Hessian is singular
+                direction = -grad
+
+        # Perform Backtracking Line Search
+        alpha = backtracking_line_search(x, direction, grad, alpha_bar=1.0, rho=0.5, c1=1e-3)
+        
+        # Update Position
+        x = x + alpha * direction
+        
+        alphas.append(alpha)
+        
+    print(f"   -> Did not converge within {max_iter} iterations.")
+    print(f"   -> Final Gradient Norm: {grad_norm:.6e}")
+    return alphas, max_iter
+
+# --- 4. Main Execution and Plotting ---
+
+# Configuration
+initial_points = [np.array([1.2, 1.2]), np.array([-1.2, 1.0])]
+methods = ["Steepest_Descent", "Newton"]
+
+# Dictionary to hold simulation results
+results = {}
+
+# Run simulations
+for x0 in initial_points:
+    for method in methods:
+        # Construct a unique key for storage
+        key = f"{method}_x0_{x0}"
+        alphas, num_iters = run_optimization(method, x0)
+        results[key] = {'alphas': alphas, 'iters': num_iters, 'x0': x0}
+
+# Plot configuration
+fig, axes = plt.subplots(2, 2, figsize=(14, 10))
+plt.subplots_adjust(hspace=0.4, wspace=0.3)
+
+# Mapping results to subplots
+plot_mapping = {
+    ("Steepest_Descent", str(initial_points[0])): (0, 0),
+    ("Newton", str(initial_points[0])): (0, 1),
+    ("Steepest_Descent", str(initial_points[1])): (1, 0),
+    ("Newton", str(initial_points[1])): (1, 1),
+}
+
+for key, data in results.items():
+    method_name = key.split("_x0_")[0]
+    x0_str = str(data['x0'])
+    
+    ax_idx = plot_mapping[(method_name, x0_str)]
+    ax = axes[ax_idx]
+    
+    y_data = data['alphas']
+    x_data = range(len(y_data))
+    
+    # --- PLOTTING LOGIC MODIFICATION ---
+    # For Steepest Descent, show only first {limit} steps
+    if method_name == "Steepest_Descent":
+        limit = 30
+        if len(y_data) > limit:
+            y_data = y_data[:limit]
+            x_data = range(limit)
+            title_suffix = f" (First {limit} Steps Shown)"
+        else:
+            title_suffix = ""
+    else:
+        title_suffix = ""
+        
+    ax.plot(x_data, y_data, marker='o', linestyle='-', markersize=4, linewidth=1.5)
+    
+    # Titles and Labels
+    ax.set_title(f"{method_name}\nStart: {data['x0']}\nTotal Iters: {data['iters']}{title_suffix}")
+    ax.set_xlabel("Iteration")
+    ax.set_ylabel("Step Size (Alpha)")
+    ax.grid(True, which='both', linestyle='--', alpha=0.7)
+    
+    # Add annotation for total iterations
+    ax.text(0.95, 0.95, f"Total Iters: {data['iters']}", 
+            transform=ax.transAxes, 
+            verticalalignment='top', horizontalalignment='right',
+            bbox=dict(boxstyle='round', facecolor='white', alpha=0.8))
+
+# Save and Show
+#plt.suptitle("Step Size vs Iteration (Rosenbrock Function)", fontsize=16)
+plt.savefig("optimization_results.png")
+#plt.show()