




'''simple docstring'''







from typing import Any

import numpy as np





def 						snake_case_							(_a     :							np.ndarray					):
    assert np.shape(_a					)[0] == np.shape(_a					)[1]

    # If matrix not symmetric, exit right away.
    if np.allclose(_a   ,       matrix.T					) is False:
        return False

    # Get eigenvalues and eignevectors for a symmetric matrix.
    UpperCAmelCase	,					UpperCAmelCase      	=		np.linalg.eigh(_a					)

    # Check sign of all eigenvalues.
    # np.all returns a value of type np.bool_
    return bool(np.all(eigen_values > 0					)					)





def 						snake_case_							(_a     :							int					):
    UpperCAmelCase      	=		np.random.randn(_a   ,       _a					)
    UpperCAmelCase      	=		np.dot(_a   ,       random_matrix.T					)
    assert _is_matrix_spd(_a					)
    return spd_matrix





def 						snake_case_							(_a     :							np.ndarray   ,       _a     :							np.ndarray   ,       _a     :							int = 1_0_0_0   ,       _a     :							float = 1E-8   ,       ):
    assert np.shape(_a					)[0] == np.shape(_a					)[1]
    assert np.shape(_a					)[0] == np.shape(_a					)[0]
    assert _is_matrix_spd(_a					)

    # Initialize solution guess, residual, search direction.
    UpperCAmelCase      	=		np.zeros((np.shape(_a					)[0], 1)					)
    UpperCAmelCase      	=		np.copy(_a					)
    UpperCAmelCase      	=		np.copy(_a					)

    # Set initial errors in solution guess and residual.
    UpperCAmelCase      	=		1E9
    UpperCAmelCase      	=		1E9
    UpperCAmelCase      	=		1E9

    # Set iteration counter to threshold number of iterations.
    UpperCAmelCase      	=		0

    while error > tol:
        # Save this value so we only calculate the matrix-vector product once.
        UpperCAmelCase      	=		np.dot(_a   ,       _a					)

        # The main algorithm.

        # Update search direction magnitude.
        UpperCAmelCase      	=		np.dot(ra.T   ,       _a					) / np.dot(pa.T   ,       _a					)
        # Update solution guess.
        UpperCAmelCase      	=		xa + alpha * pa
        # Calculate new residual.
        UpperCAmelCase      	=		ra - alpha * w
        # Calculate new Krylov subspace scale.
        UpperCAmelCase      	=		np.dot(r.T   ,       _a					) / np.dot(ra.T   ,       _a					)
        # Calculate new A conjuage search direction.
        UpperCAmelCase      	=		r + beta * pa

        # Calculate errors.
        UpperCAmelCase      	=		np.linalg.norm(r - ra					)
        UpperCAmelCase      	=		np.linalg.norm(x - xa					)
        UpperCAmelCase      	=		np.maximum(_a   ,       _a					)

        # Update variables.
        UpperCAmelCase      	=		np.copy(_a					)
        UpperCAmelCase      	=		np.copy(_a					)
        UpperCAmelCase      	=		np.copy(_a					)

        # Update number of iterations.
        iterations += 1
        if iterations > max_iterations:
            break

    return x





def 						snake_case_							():
    UpperCAmelCase      	=		3
    UpperCAmelCase      	=		_create_spd_matrix(_a					)
    UpperCAmelCase      	=		np.random.randn(_a   ,       1					)
    UpperCAmelCase      	=		np.dot(_a   ,       _a					)

    # Numpy solution.
    UpperCAmelCase      	=		np.linalg.solve(_a   ,       _a					)

    # Our implementation.
    UpperCAmelCase      	=		conjugate_gradient(_a   ,       _a					)

    # Ensure both solutions are close to x_true (and therefore one another).
    assert np.linalg.norm(x_numpy - x_true					) <= 1E-6
    assert np.linalg.norm(x_conjugate_gradient - x_true					) <= 1E-6


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     test_conjugate_gradient()










'''simple docstring'''







def 						snake_case_							(_a     :							int = 9_9_8_0_0_1					):
    UpperCAmelCase      	=		0
    for i in range(9_9_9   ,       9_9   ,       -1					):  # 3 digit numbers range from 999 down to 100
        for j in range(9_9_9   ,       9_9   ,       -1					):
            UpperCAmelCase      	=		str(i * j					)
            if product_string == product_string[::-1] and i * j < n:
                UpperCAmelCase      	=		max(_a   ,       i * j					)
    return answer


if __name__ == "__main__":
   print(f"""{solution() = }""")




