


def        SCREAMING_SNAKE_CASE_						(			__A :			list[list]				)	->		list[list]:

       """simple docstring"""

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







def        SCREAMING_SNAKE_CASE_						(			__A :			list[list]				)	->		list:

       """simple docstring"""

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


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   UpperCAmelCase_      : Any  					=       [
       [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]]))








from __future__ import annotations

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




def 			__magic_name__				( A		:					NDArray[floataa],  A		:					NDArray[floataa],  A		:					list[int],  A		:					int,  ):






			'''simple docstring'''







			a    ,		a    =			coefficient_matrix.shape
			a    ,		a    =			constant_matrix.shape

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

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

			if rowsa != rowsa:
						a    =			(
						    "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    =			(
						    "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    =			np.concatenate(
			    (coefficient_matrix, constant_matrix),  axis=1   )

			a    ,		a    =			table.shape

			strictly_diagonally_dominant(A   )

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

			return [float(A   ) for i in new_val]




def 			__magic_name__				( A		:					NDArray[floataa]   ):






			'''simple docstring'''







			a    ,		a    =			table.shape

			a    =			True

			for i in range(0,  A   ):
						a    =			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()






