add homework 8

homework8/Homework8_Q5.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+# --- FUNCTION DEFINITION CORRECTED HERE ---
+def g(t):
+    """
+    Calculates g(t) = t^4 - 3t^3 + 2t^2 in a vectorized way.
+    It handles both single float inputs and NumPy array inputs.
+    If an element's absolute value is too large, it returns infinity
+    to prevent overflow and handle plotting of diverging paths.
+    """
+    # Use np.where for a vectorized if-else statement.
+    # This correctly handles both scalar and array inputs.
+    return np.where(np.abs(t) > 1e10,  # Condition
+                    np.inf,             # Value if condition is True
+                    t**4 - 3*t**3 + 2*t**2) # Value if condition is False
+
+# Define the gradient (derivative) of g(t)
+def grad_g(t):
+    # This function is only called with scalars in our loop,
+    # but it's good practice for it to be vectorization-safe as well.
+    # Standard arithmetic operations are already vectorized in NumPy.
+    return 4*t**3 - 9*t**2 + 4*t
+
+# Implement the gradient descent algorithm
+def gradient_descent(x0, alpha, iterations=100):
+    history = [x0]
+    x = x0
+    for _ in range(iterations):
+        grad = grad_g(x)
+        
+        if np.isinf(x) or np.isnan(x) or np.abs(x) > 1e9:
+            print(f"Divergence detected for alpha = {alpha}. Stopping early.")
+            break
+
+        x = x - alpha * grad
+        history.append(x)
+    if not np.isinf(x) and not np.isnan(x) and np.abs(x) <= 1e9:
+        print(f"Alpha: {alpha:<.2f}, Current x: {x:<.2f}, g(x): {g(x):<.2f}")
+    return history
+
+# --- Main part of the experiment ---
+
+# Set an initial point
+initial_xs = [-1.0, 0.5, 0.65, 2.5]
+
+# Define a list of different alpha values to test
+alphas_to_test = [0.05, 0.15, 0.25, 0.4, 0.5, 0.6]
+
+# Run the experiment for each alpha value and plot the results
+for initial_x in initial_xs:
+    print(f"Running gradient descent from initial x = {initial_x}")
+    # Create a plot
+    plt.figure(figsize=(14, 8))
+
+    # Plot the function g(t) for context
+    t_plot = np.linspace(-1.5, 3.5, 400)
+    plt.plot(t_plot, g(t_plot), 'k-', label='g(t) = t^4 - 3t^3 + 2t^2', linewidth=2)
+    for alpha in alphas_to_test:
+        history = gradient_descent(initial_x, alpha)
+        history_np = np.array(history)
+        plt.plot(history_np, g(history_np), 'o-', label=f'alpha = {alpha}', markersize=4, alpha=0.8)
+
+    # Add stationary points
+    minima1 = 0.0
+    minima2 = (9 + np.sqrt(17)) / 8
+    maxima = (9 - np.sqrt(17)) / 8
+    plt.plot(minima1, g(minima1), 'g*', markersize=15, label=f'Local Minimum at t={minima1:.2f}')
+    plt.plot(minima2, g(minima2), 'g*', markersize=15, label=f'Local Minimum at t≈{minima2:.2f}')
+    plt.plot(maxima, g(maxima), 'rX', markersize=10, label=f'Local Maximum at t≈{maxima:.2f}')
+
+    # Final plot formatting
+    plt.title('Gradient Descent Convergence for Different Step Sizes (alpha, initial x={})'.format(initial_x))
+    plt.xlabel('t')
+    plt.ylabel('g(t)')
+    plt.legend()
+    plt.grid(True)
+    plt.ylim(-1, 5)
+    plt.xlim(-1.5, 3.5)
+    plt.savefig(f'Homework8_Q5_Gradient_Descent_Convergence_x={initial_x}.png')
+

homework8/Homework8_Q5.txt

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+Running gradient descent from initial x = -1.0
+Alpha: 0.05, Current x: -0.00, g(x): 0.00
+Alpha: 0.15, Current x: 1.64, g(x): -0.62
+Divergence detected for alpha = 0.25. Stopping early.
+Divergence detected for alpha = 0.4. Stopping early.
+Divergence detected for alpha = 0.5. Stopping early.
+Divergence detected for alpha = 0.6. Stopping early.
+Running gradient descent from initial x = 0.5
+Alpha: 0.05, Current x: 0.00, g(x): 0.00
+Alpha: 0.15, Current x: 0.00, g(x): 0.00
+Alpha: 0.25, Current x: 0.00, g(x): 0.00
+Alpha: 0.40, Current x: -0.00, g(x): 0.00
+Alpha: 0.50, Current x: 0.02, g(x): 0.00
+Alpha: 0.60, Current x: 0.18, g(x): 0.05
+Running gradient descent from initial x = 0.65
+Alpha: 0.05, Current x: 1.64, g(x): -0.62
+Alpha: 0.15, Current x: 1.64, g(x): -0.62
+Alpha: 0.25, Current x: 1.64, g(x): -0.62
+Alpha: 0.40, Current x: 1.51, g(x): -0.57
+Alpha: 0.50, Current x: 0.01, g(x): 0.00
+Divergence detected for alpha = 0.6. Stopping early.
+Running gradient descent from initial x = 2.5
+Alpha: 0.05, Current x: 1.64, g(x): -0.62
+Alpha: 0.15, Current x: 0.00, g(x): 0.00
+Divergence detected for alpha = 0.25. Stopping early.
+Divergence detected for alpha = 0.4. Stopping early.
+Divergence detected for alpha = 0.5. Stopping early.
+Divergence detected for alpha = 0.6. Stopping early.