





from __future__ import annotations





def  __magic_name__  (				lowercase				,       lowercase				,       lowercase				)				-> float:

							"""simple docstring"""



							if days_between_payments <= 0:
														raise ValueError("""days_between_payments must be > 0"""				)
							if daily_interest_rate < 0:
														raise ValueError("""daily_interest_rate must be >= 0"""				)
							if principal <= 0:
														raise ValueError("""principal must be > 0"""				)
							return principal * daily_interest_rate * days_between_payments





def  __magic_name__  (				lowercase				,       lowercase				,       lowercase				,       )				-> float:

							"""simple docstring"""



							if number_of_compounding_periods <= 0:
														raise ValueError("""number_of_compounding_periods must be > 0"""				)
							if nominal_annual_interest_rate_percentage < 0:
														raise ValueError("""nominal_annual_interest_rate_percentage must be >= 0"""				)
							if principal <= 0:
														raise ValueError("""principal must be > 0"""				)

							return principal * (
							    (1 + nominal_annual_interest_rate_percentage) ** number_of_compounding_periods
							    - 1
							)





def  __magic_name__  (				lowercase				,       lowercase				,       lowercase				,       )				-> float:

							"""simple docstring"""



							if number_of_years <= 0:
														raise ValueError("""number_of_years must be > 0"""				)
							if nominal_annual_percentage_rate < 0:
														raise ValueError("""nominal_annual_percentage_rate must be >= 0"""				)
							if principal <= 0:
														raise ValueError("""principal must be > 0"""				)

							return compound_interest(
							    lowercase				,       nominal_annual_percentage_rate / 365				,       number_of_years * 365				)


if __name__ == "__main__":
							import doctest

							doctest.testmod()






def  __magic_name__  (				lowercase				)				-> list[list]:

							"""simple docstring"""



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





def  __magic_name__  (				lowercase				)				-> list:

							"""simple docstring"""



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