
'''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.
      snake_case_							,						snake_case_    							=				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      ):
      snake_case_    							=				np.random.randn(_a			,       _a      )
      snake_case_    							=				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_000			,       _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.
      snake_case_    							=				np.zeros((np.shape(_a      )[0], 1)      )
      snake_case_    							=				np.copy(_a      )
      snake_case_    							=				np.copy(_a      )

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

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

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

            # The main algorithm.

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

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

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

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

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

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

      # Our implementation.
      snake_case_    							=				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()







from __future__ import annotations

import sys
def 							__lowerCAmelCase			(UpperCAmelCase_		):
    if n < 0:
        raise ValueError("""Negative arguments are not supported"""		)
    return _fib(UpperCAmelCase_		)[0]
def 							__lowerCAmelCase			(UpperCAmelCase_		):
    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
    a__			,		a__  			=							_fib(n // 2		)
    a__  			=							a * (b * 2 - a)
    a__  			=							a * a + b * b
    return (d, c + d) if n % 2 else (c, d)


if __name__ == "__main__":
 SCREAMING_SNAKE_CASE_    :       Dict       =							int(sys.argv[1])
 print(F"fibonacci({n}) is {fibonacci(n)}")






