def gradient_descent(A, b, x_init, n_iter, alpha=0.1):

x = x_init

f_values = []

for t in range(n_iter):

# Compute the gradient

grad_f = A.dot(x) - b

# Update the solution

x = x - alpha * grad_f

# Store the function value

f_values.append(0.5 * x.T.dot(A).dot(x) - b.T.dot(x))

return x, f_values

# Gradient descent for n = 20

A_20 = np.diag(np.arange(1, 21))

b_20 = np.ones(20)

x_init_20 = np.zeros(20)

x_gd_20, f_gd_20 = gradient_descent(A_20, b_20, x_init_20, 20)

# Gradient descent for n = 100

A_100 = np.diag(np.arange(1, 101))

b_100 = np.ones(100)

x_init_100 = np.zeros(100)

x_gd_100, f_gd_100 = gradient_descent(A_100, b_100, x_init_100, 100) in this can we use the function optimal solution

Yes, you can use the function's optimal solution in the context of gradient descent, though the concept of an optimal solution typically refers to finding the point where the gradient of a function equals zero. For your quadratic function \( f(x) = 0.5 x^T A x - b^T x \), the optimal solution is the point where the gradient is zero. In other words, it is the solution to the following system:

\[
\nabla f(x) = A x - b = 0
\]

This implies that the optimal solution \( x_{\text{opt}} \) is:

\[
x_{\text{opt}} = A^{-1} b
\]

For the given quadratic function, assuming \( A \) is invertible (which it is since it's a diagonal matrix with nonzero entries), you can calculate the optimal solution directly using the formula above. The gradient descent method will iteratively converge towards this optimal solution, but the actual result after convergence should ideally be very close to \( A^{-1} b \).

### Here's how you could calculate the optimal solution:

```python
import numpy as np

# Function to compute the optimal solution
def optimal_solution(A, b):
    return np.linalg.inv(A).dot(b)

# Gradient descent function as defined
def gradient_descent(A, b, x_init, n_iter, alpha=0.1):
    x = x_init
    f_values = []
    for t in range(n_iter):
        # Compute the gradient
        grad_f = A.dot(x) - b
        # Update the solution
        x = x - alpha * grad_f
        # Store the function value
        f_values.append(0.5 * x.T.dot(A).dot(x) - b.T.dot(x))
    return x, f_values

# Gradient descent for n = 20
A_20 = np.diag(np.arange(1, 21))
b_20 = np.ones(20)
x_init_20 = np.zeros(20)
x_gd_20, f_gd_20 = gradient_descent(A_20, b_20, x_init_20, 20)

# Calculate the optimal solution for n = 20
x_opt_20 = optimal_solution(A_20, b_20)

# Gradient descent for n = 100
A_100 = np.diag(np.arange(1, 101))
b_100 = np.ones(100)
x_init_100 = np.zeros(100)
x_gd_100, f_gd_100 = gradient_descent(A_100, b_100, x_init_100, 100)

# Calculate the optimal solution for n = 100
x_opt_100 = optimal_solution(A_100, b_100)

# Output optimal solutions
print("Optimal solution for n=20:", x_opt_20)
print("Optimal solution for n=100:", x_opt_100)
```

### Explanation:
1. **Optimal Solution Calculation**: The `optimal_solution` function computes the exact optimal solution by solving \( A x = b \) (i.e., \( x = A^{-1} b \)).
2. **Gradient Descent Convergence**: The gradient descent will iteratively approach the optimal solution \( x_{\text{opt}} \) as the number of iterations increases.

You can compare the results from the gradient descent method (`x_gd_20`, `x_gd_100`) with the optimal solution (`x_opt_20`, `x_opt_100`) to observe how closely the gradient descent solution matches the theoretical optimal solution.

### Would you like more information on gradient descent convergence or the impact of the learning rate? Here are a few related questions:
1. How does the learning rate \( \alpha \) affect the convergence speed in gradient descent?
2. What happens if the matrix \( A \) is not invertible? Can we still apply gradient descent?
3. How do you evaluate the performance of gradient descent (e.g., by monitoring the function value \( f(x) \))?
4. How can we implement a stopping criterion for gradient descent?
5. What is the difference between gradient descent and other optimization algorithms like Newton's method?

**Tip**: To ensure gradient descent converges efficiently, consider adjusting the learning rate based on the

Learn how to implement gradient descent for quadratic optimization, including calculating the optimal solution using linear algebra. This example explores convergence for matrices of size 20 and 100, key formulas, and step-by-step explanations. Suitable for undergraduate students studying optimization or linear algebra.

Math Problem Statement

Compute the gradient

Update the solution

Store the function value

Gradient descent for n = 20

Gradient descent for n = 100

Solution

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