
def 		A_       (							snake_case							:      list[list]       )					->			int:



						'''simple docstring'''

						__UpperCamelCase        =	current_set.copy()
						for row_index, row in enumerate(__SCREAMING_SNAKE_CASE       ):
												__UpperCamelCase        =	row[0]
												for column_index, column in enumerate(__SCREAMING_SNAKE_CASE       ):
																		if magnitude == 0:
																								__UpperCamelCase        =	column
																								continue
																		__UpperCamelCase        =	column / magnitude
    # Subtract to cancel term
						__UpperCamelCase        =	current_set[0]
						__UpperCamelCase        =	[first_row]
						__UpperCamelCase        =	current_set[1::]
						for row in current_set:
												__UpperCamelCase        =	[]
												# 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:
												__UpperCamelCase        =	final_set[0]
												__UpperCamelCase        =	[]
												__UpperCamelCase        =	[]
												for row in final_set[1::]:
																		current_first_column.append(row[0]       )
																		next_iteration.append(row[1::]       )
												__UpperCamelCase        =	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       )
												__UpperCamelCase        =	resultant
						return final_set




def 		A_       (							snake_case							:      list[list]       )					->			Any:



						'''simple docstring'''

						if len(__SCREAMING_SNAKE_CASE       ) == 0:
												raise IndexError('''solve_simultaneous() requires n lists of length n+1'''       )
						__UpperCamelCase        =	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]]
						__UpperCamelCase        =	equations.copy()
						if any(0 in row for row in data_set       ):
												__UpperCamelCase        =	data_set.copy()
												__UpperCamelCase        =	[]
												for row_index, row in enumerate(__SCREAMING_SNAKE_CASE       ):
																		if 0 not in row:
																								__UpperCamelCase        =	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       )
						__UpperCamelCase        =	data_set.copy()
						__UpperCamelCase        =	simplify(__SCREAMING_SNAKE_CASE       )
						__UpperCamelCase        =	simplified[::-1]
						__UpperCamelCase        =	[]
						for row in simplified:
												__UpperCamelCase        =	row[-1]
												if not solutions:
																		if row[-2] == 0:
																								solutions.append(0       )
																								continue
																		solutions.append(current_solution / row[-2]       )
																		continue
												__UpperCamelCase        =	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
												__UpperCamelCase        =	temp_row[1::]
												__UpperCamelCase        =	temp_row[::-1]
												for column_index, column in enumerate(__SCREAMING_SNAKE_CASE       ):
																		current_solution -= column * solutions[column_index]
												solutions.append(__SCREAMING_SNAKE_CASE       )
						__UpperCamelCase        =	[]
						for item in solutions:
												final.append(float(round(__SCREAMING_SNAKE_CASE       ,					5       )       )       )
						return final[::-1]


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				lowercase__							:							Optional[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]]))






'''simple docstring'''







import math
import time
from typing import Dict, List, Optional

from torch.utils.data import Dataset

from transformers import SeqaSeqTrainer, is_torch_tpu_available
from transformers.trainer_utils import PredictionOutput, speed_metrics


if is_torch_tpu_available(check_device=False):
							import torch_xla.core.xla_model as xm
							import torch_xla.debug.metrics as met






class  lowerCAmelCase__   (  lowerCamelCase_     ):






					def __init__(   self				,							*__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE=None				,							__SCREAMING_SNAKE_CASE=None				,							**__SCREAMING_SNAKE_CASE		):


											"""simple docstring"""





											super().__init__(*__SCREAMING_SNAKE_CASE				,							**__SCREAMING_SNAKE_CASE		)
											lowercase_	:     Tuple													=       eval_examples
											lowercase_	:     Tuple													=       post_process_function






					def 	_snake_case    (   self				,							__SCREAMING_SNAKE_CASE = None				,							__SCREAMING_SNAKE_CASE=None				,							__SCREAMING_SNAKE_CASE = None				,							__SCREAMING_SNAKE_CASE = "eval"				,							**__SCREAMING_SNAKE_CASE				,							):


											"""simple docstring"""





											lowercase_	:     Optional[int]													=       gen_kwargs.copy()
											lowercase_	:     List[str]													=       (
											    gen_kwargs['''max_length'''] if gen_kwargs.get('''max_length'''		) is not None else self.args.generation_max_length
											)
											lowercase_	:     str													=       (
											    gen_kwargs['''num_beams'''] if gen_kwargs.get('''num_beams'''		) is not None else self.args.generation_num_beams
											)
											lowercase_	:     Dict													=       gen_kwargs

											lowercase_	:     List[Any]													=       self.eval_dataset if eval_dataset is None else eval_dataset
											lowercase_	:     List[str]													=       self.get_eval_dataloader(__SCREAMING_SNAKE_CASE		)
											lowercase_	:     Optional[Any]													=       self.eval_examples if eval_examples is None else eval_examples

											# Temporarily disable metric computation, we will do it in the loop here.
											lowercase_	:     Union[str, Any]													=       self.compute_metrics
											lowercase_	:     Optional[int]													=       None
											lowercase_	:     Tuple													=       time.time()
											lowercase_	:     Tuple													=       self.prediction_loop if self.args.use_legacy_prediction_loop else self.evaluation_loop
											try:
																	lowercase_	:     str													=       eval_loop(
																	    __SCREAMING_SNAKE_CASE				,							description='''Evaluation'''				,							prediction_loss_only=True if compute_metrics is None else None				,							ignore_keys=__SCREAMING_SNAKE_CASE				,							metric_key_prefix=__SCREAMING_SNAKE_CASE				,							)
											finally:
																	lowercase_	:     Any													=       compute_metrics
											lowercase_	:     Any													=       self.args.eval_batch_size * self.args.world_size
											if F'''{metric_key_prefix}_jit_compilation_time''' in output.metrics:
																	start_time += output.metrics[F'''{metric_key_prefix}_jit_compilation_time''']
											output.metrics.update(
											    speed_metrics(
											        __SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE				,							num_samples=output.num_samples				,							num_steps=math.ceil(output.num_samples / total_batch_size		)				,							)		)

											if self.post_process_function is not None and self.compute_metrics is not None and self.args.should_save:
																	# Only the main node write the results by default
																	lowercase_	:     Optional[Any]													=       self.post_process_function(__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE		)
																	lowercase_	:     Optional[int]													=       self.compute_metrics(__SCREAMING_SNAKE_CASE		)

																	# Prefix all keys with metric_key_prefix + '_'
																	for key in list(metrics.keys()		):
																							if not key.startswith(F'''{metric_key_prefix}_'''		):
																													lowercase_	:     List[Any]													=       metrics.pop(__SCREAMING_SNAKE_CASE		)

																	metrics.update(output.metrics		)
											else:
																	lowercase_	:     List[Any]													=       output.metrics

											if self.args.should_log:
																	# Only the main node log the results by default
																	self.log(__SCREAMING_SNAKE_CASE		)

											if self.args.tpu_metrics_debug or self.args.debug:
																	# tpu-comment: Logging debug metrics for PyTorch/XLA (compile, execute times, ops, etc.)
																	xm.master_print(met.metrics_report()		)

											lowercase_	:     List[Any]													=       self.callback_handler.on_evaluate(self.args				,							self.state				,							self.control				,							__SCREAMING_SNAKE_CASE		)
											return metrics






					def 	_snake_case    (   self				,							__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE=None				,							__SCREAMING_SNAKE_CASE = "test"				,							**__SCREAMING_SNAKE_CASE		):


											"""simple docstring"""





											lowercase_	:     Union[str, Any]													=       gen_kwargs.copy()

											lowercase_	:     Tuple													=       self.get_test_dataloader(__SCREAMING_SNAKE_CASE		)

											# Temporarily disable metric computation, we will do it in the loop here.
											lowercase_	:     Optional[Any]													=       self.compute_metrics
											lowercase_	:     Optional[int]													=       None
											lowercase_	:     List[Any]													=       time.time()
											lowercase_	:     int													=       self.prediction_loop if self.args.use_legacy_prediction_loop else self.evaluation_loop
											try:
																	lowercase_	:     Tuple													=       eval_loop(
																	    __SCREAMING_SNAKE_CASE				,							description='''Prediction'''				,							prediction_loss_only=True if compute_metrics is None else None				,							ignore_keys=__SCREAMING_SNAKE_CASE				,							metric_key_prefix=__SCREAMING_SNAKE_CASE				,							)
											finally:
																	lowercase_	:     Any													=       compute_metrics

											lowercase_	:     Tuple													=       self.args.eval_batch_size * self.args.world_size
											if F'''{metric_key_prefix}_jit_compilation_time''' in output.metrics:
																	start_time += output.metrics[F'''{metric_key_prefix}_jit_compilation_time''']
											output.metrics.update(
											    speed_metrics(
											        __SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE				,							num_samples=output.num_samples				,							num_steps=math.ceil(output.num_samples / total_batch_size		)				,							)		)
											if self.post_process_function is None or self.compute_metrics is None:
																	return output

											lowercase_	:     Any													=       self.post_process_function(__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE				,							'''predict'''		)
											lowercase_	:     str													=       self.compute_metrics(__SCREAMING_SNAKE_CASE		)

											# Prefix all keys with metric_key_prefix + '_'
											for key in list(metrics.keys()		):
																	if not key.startswith(F'''{metric_key_prefix}_'''		):
																							lowercase_	:     Optional[int]													=       metrics.pop(__SCREAMING_SNAKE_CASE		)
											metrics.update(output.metrics		)
											return PredictionOutput(predictions=predictions.predictions				,							label_ids=predictions.label_ids				,							metrics=__SCREAMING_SNAKE_CASE		)



