
def 						UpperCAmelCase      (     a_		,       a_						)		->     tuple[float, float]:







				"""simple docstring"""



				if not len(__UpperCAmelCase						) == len(__UpperCAmelCase						) == 3:
								raise ValueError("Please enter a valid equation."						)
				if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
								raise ValueError("Both a & b of two equations can\'t be zero."						)

				# Extract the coefficients
				__A             =			equationa
				__A             =			equationa

				# Calculate the determinants of the matrices
				__A             =			aa * ba - aa * ba
				__A             =			ca * ba - ca * ba
				__A             =			aa * ca - aa * ca

				# Check if the system of linear equations has a solution (using Cramer's rule)
				if determinant == 0:
								if determinant_x == determinant_y == 0:
												raise ValueError("Infinite solutions. (Consistent system)"						)
								else:
												raise ValueError("No solution. (Inconsistent system)"						)
				else:
								if determinant_x == determinant_y == 0:
												# Trivial solution (Inconsistent system)
												return (0.0, 0.0)
								else:
												__A             =			determinant_x / determinant
												__A             =			determinant_y / determinant
												# Non-Trivial Solution (Consistent system)
												return (x, y)


def 						UpperCAmelCase      (     )		->     list[list[int]]:







				"""simple docstring"""



				return [list(range(1_0_0_0 - i		,       -1_0_0_0 - i		,       -1						)						) for i in range(1_0_0_0						)]


SCREAMING_SNAKE_CASE							:List[str]										=    generate_large_matrix()
SCREAMING_SNAKE_CASE							:str										=    (
    [[4, 3, 2, -1], [3, 2, 1, -1], [1, 1, -1, -2], [-1, -1, -2, -3]],
    [[3, 2], [1, 0]],
    [[7, 7, 6]],
    [[7, 7, 6], [-1, -2, -3]],
    grid,
)


def 						UpperCAmelCase      (     a_						)		->     None:







				"""simple docstring"""



				assert all(row == sorted(a_		,       reverse=a_						) for row in grid						)
				assert all(list(a_						) == sorted(a_		,       reverse=a_						) for col in zip(*a_						)						)


def 						UpperCAmelCase      (     a_						)		->     int:







				"""simple docstring"""



				__A             =			0
				__A             =			len(a_						) - 1

				# Edge cases such as no values or all numbers are negative.
				if not array or array[0] < 0:
								return 0

				while right + 1 > left:
								__A             =			(left + right) // 2
								__A             =			array[mid]

								# Num must be negative and the index must be greater than or equal to 0.
								if num < 0 and array[mid - 1] >= 0:
												return mid

								if num >= 0:
												__A             =			mid + 1
								else:
												__A             =			mid - 1
    # No negative numbers so return the last index of the array + 1 which is the length.
				return len(a_						)


def 						UpperCAmelCase      (     a_						)		->     int:







				"""simple docstring"""



				__A             =			0
				__A             =			len(grid[0]						)

				for i in range(len(a_						)						):
								__A             =			find_negative_index(grid[i][:bound]						)
								total += bound
				return (len(a_						) * len(grid[0]						)) - total


def 						UpperCAmelCase      (     a_						)		->     int:







				"""simple docstring"""



				return len([number for row in grid for number in row if number < 0]						)


def 						UpperCAmelCase      (     a_						)		->     int:







				"""simple docstring"""



				__A             =			0
				for row in grid:
								for i, number in enumerate(a_						):
												if number < 0:
																total += len(a_						) - i
																break
				return total


def 						UpperCAmelCase      (     )		->     None:







				"""simple docstring"""



				from timeit import timeit

				print("Running benchmarks"						)
				__A             =			(
				    "from __main__ import count_negatives_binary_search, "
				    "count_negatives_brute_force, count_negatives_brute_force_with_break, grid"
				)
				for func in (
				    "count_negatives_binary_search",  # took 0.7727 seconds
				    "count_negatives_brute_force_with_break",  # took 4.6505 seconds
				    "count_negatives_brute_force",  # took 12.8160 seconds
				):
								__A             =			timeit(F'''{func}(grid=grid)'''		,       setup=a_		,       number=5_0_0						)
								print(F'''{func}() took {time:0.4f} seconds'''						)


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					benchmark()
