
'''simple docstring'''






from __future__ import annotations

import numpy as np
from numpy import floataa
from numpy.typing import NDArray







def 		A_       (	_lowerCAmelCase   :    NDArray[floataa]				,	_lowerCAmelCase   :    NDArray[floataa]				,	_lowerCAmelCase   :    list[int]				,	_lowerCAmelCase   :    int				,	):



      """simple docstring"""




      _lowerCamelCase				:   Tuple								=   coefficient_matrix.shape
      _lowerCamelCase				:   List[str]								=   constant_matrix.shape

      if rowsa != colsa:
            _lowerCamelCase				:   Optional[Any]								=   F'Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}'
            raise ValueError(__UpperCamelCase							)

      if colsa != 1:
            _lowerCamelCase				:   Any								=   F'Constant matrix must be nx1 but received {rowsa}x{colsa}'
            raise ValueError(__UpperCamelCase							)

      if rowsa != rowsa:
            _lowerCamelCase				:   Union[str, Any]								=   (
                '''Coefficient and constant matrices dimensions must be nxn and nx1 but '''
                F'received {rowsa}x{colsa} and {rowsa}x{colsa}'
            )
            raise ValueError(__UpperCamelCase							)

      if len(__UpperCamelCase							) != rowsa:
            _lowerCamelCase				:   Dict								=   (
                '''Number of initial values must be equal to number of rows in coefficient '''
                F'matrix but received {len(__UpperCamelCase							)} and {rowsa}'
            )
            raise ValueError(__UpperCamelCase							)

      if iterations <= 0:
            raise ValueError("Iterations must be at least 1"							)

      _lowerCamelCase				:   NDArray[floataa]								=   np.concatenate(
          (coefficient_matrix, constant_matrix)				,	axis=1							)

      _lowerCamelCase				:   Union[str, Any]								=   table.shape

      strictly_diagonally_dominant(__UpperCamelCase							)

      # Iterates the whole matrix for given number of times
      for _ in range(__UpperCamelCase							):
            _lowerCamelCase				:   str								=   []
            for row in range(__UpperCamelCase							):
                  _lowerCamelCase				:   Any								=   0
                  for col in range(__UpperCamelCase							):
                        if col == row:
                              _lowerCamelCase				:   List[str]								=   table[row][col]
                        elif col == cols - 1:
                              _lowerCamelCase				:   Union[str, Any]								=   table[row][col]
                        else:
                              temp += (-1) * table[row][col] * init_val[col]
                  _lowerCamelCase				:   List[Any]								=   (temp + val) / denom
                  new_val.append(__UpperCamelCase							)
            _lowerCamelCase				:   Any								=   new_val

      return [float(__UpperCamelCase							) for i in new_val]







def 		A_       (	_lowerCAmelCase   :    NDArray[floataa]							):



      """simple docstring"""




      _lowerCamelCase				:   Union[str, Any]								=   table.shape

      _lowerCamelCase				:   Union[str, Any]								=   True

      for i in range(0				,	__UpperCamelCase							):
            _lowerCamelCase				:   Union[str, Any]								=   0
            for j in range(0				,	cols - 1							):
                  if i == j:
                        continue
                  else:
                        total += table[i][j]

            if table[i][i] <= total:
                  raise ValueError("Coefficient matrix is not strictly diagonally dominant"							)

      return is_diagonally_dominant


# Test Cases
if __name__ == "__main__":
 import doctest

 doctest.testmod()


'''simple docstring'''






import math







def 		A_       (	_lowerCAmelCase   :    int							):



      """simple docstring"""




      _lowerCamelCase				:   Optional[int]								=   math.loga(math.sqrt(4 * positive_integer + 1							) / 2 + 1 / 2							)

      return exponent == int(_lowerCAmelCase							)







def 		A_       (	_lowerCAmelCase   :    float = 1 / 12345							):



      """simple docstring"""




      _lowerCamelCase				:   Optional[int]								=   0
      _lowerCamelCase				:   int								=   0

      _lowerCamelCase				:   List[str]								=   3
      while True:
            _lowerCamelCase				:   List[Any]								=   (integer**2 - 1) / 4
            # if candidate is an integer, then there is a partition for k
            if partition_candidate == int(_lowerCAmelCase							):
                  _lowerCamelCase				:   Any								=   int(_lowerCAmelCase							)
                  total_partitions += 1
                  if check_partition_perfect(_lowerCAmelCase							):
                        perfect_partitions += 1
            if perfect_partitions > 0:
                  if perfect_partitions / total_partitions < max_proportion:
                        return int(_lowerCAmelCase							)
            integer += 1


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