




from typing import Any

import numpy as np




def        __SCREAMING_SNAKE_CASE (	lowercase_   )							->						bool:

    """simple docstring"""


    assert np.shape(lowercase_   )[0] == np.shape(lowercase_   )[1]

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

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

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




def        __SCREAMING_SNAKE_CASE (	lowercase_   )							->						Any:

    """simple docstring"""


    __UpperCamelCase			=    np.random.randn(lowercase_       ,  lowercase_   )
    __UpperCamelCase			=    np.dot(lowercase_       ,  random_matrix.T   )
    assert _is_matrix_spd(lowercase_   )
    return spd_matrix




def        __SCREAMING_SNAKE_CASE (	lowercase_       ,  lowercase_       ,  lowercase_ = 10_00       ,  lowercase_ = 1E-8       ,  )							->						Any:

    """simple docstring"""


    assert np.shape(lowercase_   )[0] == np.shape(lowercase_   )[1]
    assert np.shape(lowercase_   )[0] == np.shape(lowercase_   )[0]
    assert _is_matrix_spd(lowercase_   )

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

    # 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(lowercase_       ,  lowercase_   )

        # The main algorithm.

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

        # Update variables.
        __UpperCamelCase			=    np.copy(lowercase_   )
        __UpperCamelCase			=    np.copy(lowercase_   )
        __UpperCamelCase			=    np.copy(lowercase_   )

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

    return x




def        __SCREAMING_SNAKE_CASE (	)							->						None:

    """simple docstring"""


    __UpperCamelCase			=    3
    __UpperCamelCase			=    _create_spd_matrix(lowercase_   )
    __UpperCamelCase			=    np.random.randn(lowercase_       ,  1   )
    __UpperCamelCase			=    np.dot(lowercase_       ,  lowercase_   )

    # Numpy solution.
    __UpperCamelCase			=    np.linalg.solve(lowercase_       ,  lowercase_   )

    # Our implementation.
    __UpperCamelCase			=    conjugate_gradient(lowercase_       ,  lowercase_   )

    # 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'''

from __future__ import annotations

import sys







def     __UpperCamelCase					(				_lowercase       )			->      int:
	if n < 0:
		raise ValueError('Negative arguments are not supported'       )
	return _fib(_lowercase       )[0]







def     __UpperCamelCase					(				_lowercase       )			->      tuple[int, int]:
	if n == 0:  # (F(0), F(1))
		return (0, 1)

	# F(2n) = F(n)[2F(n+1) − F(n)]
	# F(2n+1) = F(n+1)^2+F(n)^2
	_lowercase	,     _lowercase       :							List[str]  	=		_fib(n // 2       )
	_lowercase       :							Dict  	=		a * (b * 2 - a)
	_lowercase       :							Union[str, Any]  	=		a * a + b * b
	return (d, c + d) if n % 2 else (c, d)


if __name__ == "__main__":
		_A					:     Optional[Any]      =int(sys.argv[1])
		print(F'''fibonacci({n}) is {fibonacci(n)}''')
