import numpy as np
import matplotlib.pyplot as plt

def question_4():
    # x = 1.920 -> 2.080 (included) step = 0.001
    x = np.arange(1.920, 2.081, 0.001)
    # p = [1, -18, 144, -672, 2016, -4032, 5376, -4608, 2304, -512]
    # from x^9 to x^0
    coeffs = [1, -18, 144, -672, 2016, -4032, 5376, -4608, 2304, -512]
    p_coeff = np.polyval(coeffs, x)
    # p(x) = (x-2)^9
    p_fact = (x - 2.0) ** 9
    plt.plot(x, p_coeff, label='Coefficient Form', lw=2)
    plt.plot(x, p_fact, label='Factored Form', lw=2, ls='--')
    plt.yscale('symlog', linthresh=1e-10)
    plt.xlabel('x', fontsize=14)
    plt.ylabel('p(x)', fontsize=14)
    plt.title('Comparison of Polynomial Forms', fontsize=16)
    # save the figure
    plt.legend(fontsize=12)
    plt.grid(True, which='both', ls='--', lw=0.5)
    plt.savefig('question4.png')
    plt.show()
    abs_diff = np.abs(p_coeff - p_fact)
    rel_diff = abs_diff / (np.abs(p_fact) + 1e-30)
    print("Maximum Absolute Difference:", np.max(abs_diff))
    print("Maximum Relative Difference:", (abs_diff[np.argmax(np.abs(p_fact))] /
                               (np.abs(p_fact).max() + 1e-30)))

if __name__ == "__main__":
    question_4()