

from __future__ import annotations

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



def    __UpperCAmelCase (	__a      :			List[str]		,__a      :			List[Any]		,__a      :			Optional[Any]		,__a      :			Optional[int]		,)			->     Tuple:

					"""simple docstring"""

					_a	:    Any         =		coefficient_matrix.shape
					_a	:    List[Any]         =		constant_matrix.shape

					if rowsa != colsa:
										_a	:    Tuple         =		F"""Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}"""
										raise ValueError(__a	)

					if colsa != 1:
										_a	:    Any         =		F"""Constant matrix must be nx1 but received {rowsa}x{colsa}"""
										raise ValueError(__a	)

					if rowsa != rowsa:
										_a	:    Tuple         =		(
										    '''Coefficient and constant matrices dimensions must be nxn and nx1 but '''
										    F"""received {rowsa}x{colsa} and {rowsa}x{colsa}"""
										)
										raise ValueError(__a	)

					if len(__a	) != rowsa:
										_a	:    Union[str, Any]         =		(
										    '''Number of initial values must be equal to number of rows in coefficient '''
										    F"""matrix but received {len(__a	)} and {rowsa}"""
										)
										raise ValueError(__a	)

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

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

					_a	:    int         =		table.shape

					strictly_diagonally_dominant(__a	)

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

					return [float(__a	) for i in new_val]



def    __UpperCAmelCase (	__a      :			str	)			->     int:

					"""simple docstring"""

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

					_a	:    int         =		True

					for i in range(0		,__a	):
										_a	:    Optional[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()







def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE       ):
  A_      :				List[Any]		     =					current_set.copy()
  for row_index, row in enumerate(SCREAMING_SNAKE_CASE       ):
    A_      :				List[str]		     =					row[0]
    for column_index, column in enumerate(SCREAMING_SNAKE_CASE       ):
      if magnitude == 0:
        A_      :				Union[str, Any]		     =					column
        continue
      A_      :				Dict		     =					column / magnitude
    # Subtract to cancel term
  A_      :				Union[str, Any]		     =					current_set[0]
  A_      :				Tuple		     =					[first_row]
  A_      :				int		     =					current_set[1::]
  for row in current_set:
    A_      :				Tuple		     =					[]
    # If first term is 0, it is already in form we want, so we preserve it
    if row[0] == 0:
      final_set.append(SCREAMING_SNAKE_CASE       )
      continue
    for column_index in range(len(SCREAMING_SNAKE_CASE       )       ):
      temp_row.append(first_row[column_index] - row[column_index]       )
    final_set.append(SCREAMING_SNAKE_CASE       )
  # Create next recursion iteration set
  if len(final_set[0]       ) != 3:
    A_      :				Optional[Any]		     =					final_set[0]
    A_      :				Any		     =					[]
    A_      :				str		     =					[]
    for row in final_set[1::]:
      current_first_column.append(row[0]       )
      next_iteration.append(row[1::]       )
    A_      :				Optional[Any]		     =					simplify(SCREAMING_SNAKE_CASE       )
    for i in range(len(SCREAMING_SNAKE_CASE       )       ):
      resultant[i].insert(0 ,			current_first_column[i]       )
    resultant.insert(0 ,			SCREAMING_SNAKE_CASE       )
    A_      :				List[Any]		     =					resultant
  return final_set


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE       ):
  if len(SCREAMING_SNAKE_CASE       ) == 0:
    raise IndexError('''solve_simultaneous() requires n lists of length n+1'''       )
  A_      :				str		     =					len(SCREAMING_SNAKE_CASE       ) + 1
  if any(len(SCREAMING_SNAKE_CASE       ) != _length for item in equations       ):
    raise IndexError('''solve_simultaneous() requires n lists of length n+1'''       )
  for row in equations:
    if any(not isinstance(SCREAMING_SNAKE_CASE ,			(int, float)       ) for column in row       ):
      raise ValueError('''solve_simultaneous() requires lists of integers'''       )
  if len(SCREAMING_SNAKE_CASE       ) == 1:
    return [equations[0][-1] / equations[0][0]]
  A_      :				Dict		     =					equations.copy()
  if any(0 in row for row in data_set       ):
    A_      :				Tuple		     =					data_set.copy()
    A_      :				Optional[Any]		     =					[]
    for row_index, row in enumerate(SCREAMING_SNAKE_CASE       ):
      if 0 not in row:
        A_      :				str		     =					data_set.pop(SCREAMING_SNAKE_CASE       )
        break
    if not full_row:
      raise ValueError('''solve_simultaneous() requires at least 1 full equation'''       )
    data_set.insert(0 ,			SCREAMING_SNAKE_CASE       )
  A_      :				int		     =					data_set.copy()
  A_      :				Dict		     =					simplify(SCREAMING_SNAKE_CASE       )
  A_      :				Dict		     =					simplified[::-1]
  A_      :				list		     =					[]
  for row in simplified:
    A_      :				Union[str, Any]		     =					row[-1]
    if not solutions:
      if row[-2] == 0:
        solutions.append(0       )
        continue
      solutions.append(current_solution / row[-2]       )
      continue
    A_      :				Optional[Any]		     =					row.copy()[: len(SCREAMING_SNAKE_CASE       ) - 1 :]
    while temp_row[0] == 0:
      temp_row.pop(0       )
    if len(SCREAMING_SNAKE_CASE       ) == 0:
      solutions.append(0       )
      continue
    A_      :				int		     =					temp_row[1::]
    A_      :				int		     =					temp_row[::-1]
    for column_index, column in enumerate(SCREAMING_SNAKE_CASE       ):
      current_solution -= column * solutions[column_index]
    solutions.append(SCREAMING_SNAKE_CASE       )
  A_      :				Union[str, Any]		     =					[]
  for item in solutions:
    final.append(float(round(SCREAMING_SNAKE_CASE ,			5       )       )       )
  return final[::-1]


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     UpperCamelCase								=	[
         [2, 1, 1, 1, 1, 4],
         [1, 2, 1, 1, 1, 5],
         [1, 1, 2, 1, 1, 6],
         [1, 1, 1, 2, 1, 7],
         [1, 1, 1, 1, 2, 8],
     ]
     print(solve_simultaneous(eq))
     print(solve_simultaneous([[4, 2]]))




