Mathematics Question

Choose a Numerical Method:

• For this task, the Newton-Raphson method or the bisection method are common choices.
• The Newton-Raphson method converges faster but requires a derivative and may encounter issues near singularities or for poor initial guesses.
• The bisection method is robust but converges slower.

3. Implement the Method in Python:

Example: Newton-Raphson Method for Case (i)

def f(x):
    return x**3 - 5*x**2 + 8*x - 2  # Replace with your equation

def df(x):
    return 3*x**2 - 10*x + 8  # Replace with the derivative

def newton_raphson(x0, tol=1e-15):
    x = x0
    while abs(f(x)) > tol:
        x -= f(x) / df(x)
    return x

result = newton_raphson(0.5, tol=1e-15)
print(f"Root for case (i): {result:.15f}")

Adaptations:

• Replace f(x) and df(x) with the actual functions for each case.
• Adjust the initial guess (x0) and tolerance (tol) as needed.
• Remember to import numpy if using vectorized calculations.

4. Test and Verify:

• Run the program for both cases with different initial guesses to ensure robustness.
• Compare the results with analytical solutions (if available) or reference values to verify accuracy.
• Consider using visualization techniques (e.g., plotting) to analyze the convergence behavior.

Remember:

• Provide your specific equations and any additional requirements/constraints for tailored guidance.
• Include clear explanations and comments in your code for better understanding.
• Test and refine your program thoroughly before submission.

I hope this comprehensive explanation helps you create the program and understand the numerical methods involved. Feel free to ask if you have further questions!