


'''simple docstring'''







import timeit

class   _snake_case:






 def __init__(self					:     str						,		a					:     Tuple						)					->      Optional[Any]:






    """simple docstring"""


    if isinstance(a						,		a						):  # Initializes a matrix identical to the one provided.
       A__         =    arg
       A__         =    len(a						)
    else:  # Initializes a square matrix of the given size and set values to zero.
       A__         =    arg
       A__         =    [[0 for _ in range(self.n						)] for _ in range(self.n						)]






 def __mul__(self					:     Dict						,		a					:     List[str]						)					->      Any:






    """simple docstring"""


    A__         =    Matrix(self.n						)
    for i in range(self.n						):
       for j in range(self.n						):
          for k in range(self.n						):
             matrix.t[i][j] += self.t[i][k] * b.t[k][j]
    return matrix







def 	_A			(   UpperCAmelCase			,UpperCAmelCase	):






   '''simple docstring'''







   A__         =    Matrix([[1, 0], [0, 1]]	)
   while b > 0:
      if b & 1:
         matrix *= a
      a *= a
      b >>= 1
   return matrix







def 	_A			(   UpperCAmelCase			,UpperCAmelCase			,UpperCAmelCase	):






   '''simple docstring'''







   if n == 1:
      return fa
   elif n == 2:
      return fa
   A__         =    Matrix([[1, 1], [1, 0]]	)
   A__         =    modular_exponentiation(lowerCAmelCase__			,n - 2	)
   return fa * matrix.t[0][0] + fa * matrix.t[0][1]







def 	_A			(   UpperCAmelCase			,UpperCAmelCase			,UpperCAmelCase	):






   '''simple docstring'''







   if n == 1:
      return fa
   elif n == 2:
      return fa

   A__         =    fa
   A__         =    fa
   n -= 2

   while n > 0:
      A__         =    fn_a + fn_a, fn_a
      n -= 1

   return fn_a







def 	_A			(   ):






   '''simple docstring'''







   A__         =    '\nfrom random import randint\nfrom __main__ import fibonacci_with_matrix_exponentiation\n'
   A__         =    'fibonacci_with_matrix_exponentiation(randint(1,70000), 1, 1)'
   A__         =    timeit.timeit(setup=lowerCAmelCase__			,stmt=lowerCAmelCase__			,number=100	)
   print('With matrix exponentiation the average execution time is '			,exec_time / 100	)
   return exec_time







def 	_A			(   ):






   '''simple docstring'''







   A__         =    '\nfrom random import randint\nfrom __main__ import simple_fibonacci\n'
   A__         =    'simple_fibonacci(randint(1,70000), 1, 1)'
   A__         =    timeit.timeit(setup=lowerCAmelCase__			,stmt=lowerCAmelCase__			,number=100	)
   print(
       'Without matrix exponentiation the average execution time is '			,exec_time / 100	)
   return exec_time







def 	_A			(   ):






   '''simple docstring'''







   matrix_exponentiation_time()
   simple_fibonacci_time()


if __name__ == "__main__":
    main()








'''simple docstring'''







from typing import Any

import numpy as np

class   _snake_case:






 def __init__(self					:     int						,		a					:     np.ndarray						,		a					:     int						)					->      None:






    """simple docstring"""


    if np.any(tableau[:, -1]						,		where=tableau[:, -1] < 0						):
       raise ValueError('RHS must be > 0'						)

    A__         =    tableau
    A__     , A__         =    tableau.shape

    # Number of decision variables x1, x2, x3...
    A__         =    n_vars

    # Number of artificial variables to be minimised
    A__         =    len(np.where(tableau[self.n_vars : -1] == -1						)[0]						)

    # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
    A__         =    (self.n_art_vars > 0) + 1

    # Number of slack variables added to make inequalities into equalities
    A__         =    self.n_rows - self.n_stages

    # Objectives for each stage
    A__         =    ['max']

    # In two stage simplex, first minimise then maximise
    if self.n_art_vars:
       self.objectives.append('min'						)

    A__         =    ['']

    # Index of current pivot row and column
    A__         =    None
    A__         =    None

    # Does objective row only contain (non)-negative values?
    A__         =    False






 @staticmethod
 def 	_UpperCamelCase   (*a					:     int						)					->      list[str]:






    """simple docstring"""


    if len(a						) != 3:
       raise ValueError('Must provide n_vars, n_slack, and n_art_vars'						)

    if not all(x >= 0 and isinstance(a						,		a						) for x in args						):
       raise ValueError('All arguments must be non-negative integers'						)

    # decision | slack | artificial
    A__         =    ['x', 's', 'a']
    A__         =    []
    for i in range(3						):
       for j in range(args[i]						):
          titles.append(string_starts[i] + str(j + 1						)						)
    titles.append('RHS'						)
    return titles






 def 	_UpperCamelCase   (self					:     Optional[Any]						,		a					:     np.ndarray						)					->      tuple[Any, Any]:






    """simple docstring"""


    A__         =    self.objectives[-1]

    # Find entries of highest magnitude in objective rows
    A__         =    (objective == 'min') - (objective == 'max')
    A__         =    np.argmax(sign * tableau[0, : self.n_vars]						)

    # Choice is only valid if below 0 for maximise, and above for minimise
    if sign * self.tableau[0, col_idx] <= 0:
       A__         =    True
       return 0, 0

    # Pivot row is chosen as having the lowest quotient when elements of
    # the pivot column divide the right-hand side

    # Slice excluding the objective rows
    A__         =    slice(self.n_stages						,		self.n_rows						)

    # RHS
    A__         =    tableau[s, -1]

    # Elements of pivot column within slice
    A__         =    tableau[s, col_idx]

    # Array filled with nans
    A__         =    np.full(self.n_rows - self.n_stages						,		np.nan						)

    # If element in pivot column is greater than zeron_stages, return
    # quotient or nan otherwise
    A__         =    np.divide(a						,		a						,		out=a						,		where=divisor > 0						)

    # Arg of minimum quotient excluding the nan values. n_stages is added
    # to compensate for earlier exclusion of objective columns
    A__         =    np.nanargmin(a						) + self.n_stages
    return row_idx, col_idx






 def 	_UpperCamelCase   (self					:     Optional[Any]						,		a					:     np.ndarray						,		a					:     int						,		a					:     int						)					->      np.ndarray:






    """simple docstring"""


    A__         =    tableau[row_idx].copy()

    A__         =    piv_row[col_idx]

    # Entry becomes 1
    piv_row *= 1 / piv_val

    # Variable in pivot column becomes basic, ie the only non-zero entry
    for idx, coeff in enumerate(tableau[:, col_idx]						):
       tableau[idx] += -coeff * piv_row
    A__         =    piv_row
    return tableau






 def 	_UpperCamelCase   (self					:     Union[str, Any]						,		a					:     np.ndarray						)					->      np.ndarray:






    """simple docstring"""


    self.objectives.pop()

    if not self.objectives:
       return tableau

    # Slice containing ids for artificial columns
    A__         =    slice(-self.n_art_vars - 1						,		-1						)

    # Delete the artificial variable columns
    A__         =    np.delete(a						,		a						,		axis=1						)

    # Delete the objective row of the first stage
    A__         =    np.delete(a						,		0						,		axis=0						)

    A__         =    1
    self.n_rows -= 1
    A__         =    0
    A__         =    False
    return tableau






 def 	_UpperCamelCase   (self					:     Optional[Any]						)					->      dict[Any, Any]:






    """simple docstring"""


    for _ in range(1_00						):
       # Completion of each stage removes an objective. If both stages
       # are complete, then no objectives are left
       if not self.objectives:
          A__         =    self.generate_col_titles(
              self.n_vars						,		self.n_slack						,		self.n_art_vars						)

          # Find the values of each variable at optimal solution
          return self.interpret_tableau(self.tableau						,		self.col_titles						)

       A__     , A__         =    self.find_pivot(self.tableau						)

       # If there are no more negative values in objective row
       if self.stop_iter:
          # Delete artificial variable columns and rows. Update attributes
          A__         =    self.change_stage(self.tableau						)
       else:
          A__         =    self.pivot(self.tableau						,		a						,		a						)
    return {}






 def 	_UpperCamelCase   (self					:     Optional[Any]						,		a					:     np.ndarray						,		a					:     list[str]						)					->      dict[str, float]:






    """simple docstring"""


    A__         =    {'P': abs(tableau[0, -1]						)}

    for i in range(self.n_vars						):
       # Gives ids of nonzero entries in the ith column
       A__         =    np.nonzero(tableau[:, i]						)
       A__         =    len(nonzero[0]						)

       # First entry in the nonzero ids
       A__         =    nonzero[0][0]
       A__         =    tableau[nonzero_rowidx, i]

       # If there is only one nonzero value in column, which is one
       if n_nonzero == nonzero_val == 1:
          A__         =    tableau[nonzero_rowidx, -1]
          A__         =    rhs_val

        # Check for basic variables
    for title in col_titles:
       # Don't add RHS or slack variables to output dict
       if title[0] not in "R-s-a":
          output_dict.setdefault(a						,		0						)
    return output_dict


if __name__ == "__main__":
    import doctest

    doctest.testmod()


