Enable full 10.3min ionospere simulations

[Joshuaalbert]

Enable a simulate of full 10.3min observation.

Input:

• same as normal

How:

• Choose a gridding setup that can manage the simulation within 1TB ram.
• If we build row by row this means we need to store in total (Na*Nt*Nd)^2 values in matrix form.
• Cholesky only needs the lower triangular, so we can use compute just for that (for sampling that's all we need).
• We can fit this in memory with the 16x16 directions, 60 second time resolution (11 time grid points), 128 points in aperture plane.

[Joshuaalbert]

if __name__ == '__main__':
    import numpy as np
    from scipy.linalg import cholesky

    # Function to create a symmetric positive-definite matrix
    def create_spd_matrix(n):
        A = np.random.rand(n, n)
        A = np.dot(A, A.transpose())  # Make it symmetric
        A += n * np.eye(n)  # Make it positive-definite
        return A

    # Main demonstration
    n = 5  # Dimension of the matrix
    A = create_spd_matrix(n)  # Create SPD matrix
    A_lower = np.tril(A)  # Extract lower triangular part

    # Compute Cholesky decomposition using only the lower triangular part
    L = cholesky(A_lower, lower=True)

    # Verify the decomposition
    A_reconstructed = L @ L.T
    # Since we're working with floating point numbers, use np.allclose for comparison
    is_correct = np.allclose(A_lower, np.tril(A_reconstructed))

    print(f"Original Lower Triangular Part:\n{A_lower}")
    print(f"Reconstructed Lower Triangular Part from Cholesky Decomposition:\n{np.tril(A_reconstructed)}")
    print(f"Decomposition Correct: {is_correct}")

    import jax.numpy as jnp
    from jax import random
    from jax.scipy.linalg import cholesky

    # Function to create a symmetric positive-definite matrix using JAX
    def create_spd_matrix(n, key):
        A = random.normal(key, (n, n))
        A = jnp.dot(A, A.T)  # Make it symmetric
        A += n * jnp.eye(n)  # Make it positive-definite
        return A

    # Main demonstration
    n = 5  # Dimension of the matrix
    key = random.PRNGKey(0)  # Random key for JAX
    A = create_spd_matrix(n, key)  # Create SPD matrix
    A_lower = jnp.tril(A)  # Extract lower triangular part

    # Compute Cholesky decomposition using only the lower triangular part
    L = cholesky(A_lower, lower=True)

    # Verify the decomposition
    A_reconstructed = L @ L.T
    # Since we're working with floating point numbers, use jnp.allclose for comparison
    is_correct = jnp.allclose(A_lower, jnp.tril(A_reconstructed))

    print(f"Original Lower Triangular Part:\n{A_lower}")
    print(f"Reconstructed Lower Triangular Part from Cholesky Decomposition:\n{jnp.tril(A_reconstructed)}")
    print(f"Decomposition Correct: {is_correct}")