





from __future__ import annotations

from collections.abc import Callable

__A							:			int         =			list[list[float | int]]





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)							-> Matrix:




    """simple docstring"""




    _A     =							len(_SCREAMING_SNAKE_CASE				)
    _A     =							[[0 for _ in range(size + 1				)] for _ in range(_SCREAMING_SNAKE_CASE				)]
    _A     =							42
    _A     =							42
    _A     =							42
    _A     =							42
    _A     =							42
    _A     =							42

    for row in range(_SCREAMING_SNAKE_CASE				):
        for col in range(_SCREAMING_SNAKE_CASE				):
            _A     =							matrix[row][col]

        _A     =							vector[row][0]

    _A     =							0
    _A     =							0
    while row < size and col < size:
        # pivoting
        _A     =							max((abs(augmented[rowa][col]				), rowa) for rowa in range(_SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)				)[
            1
        ]
        if augmented[pivot_row][col] == 0:
            col += 1
            continue
        else:
            _A,			_A     =							augmented[pivot_row], augmented[row]

        for rowa in range(row + 1							,     _SCREAMING_SNAKE_CASE				):
            _A     =							augmented[rowa][col] / augmented[row][col]
            _A     =							0
            for cola in range(col + 1							,     size + 1				):
                augmented[rowa][cola] -= augmented[row][cola] * ratio

        row += 1
        col += 1

    # back substitution
    for col in range(1							,     _SCREAMING_SNAKE_CASE				):
        for row in range(_SCREAMING_SNAKE_CASE				):
            _A     =							augmented[row][col] / augmented[col][col]
            for cola in range(_SCREAMING_SNAKE_CASE							,     size + 1				):
                augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
    return [
        [round(augmented[row][size] / augmented[row][row]							,     10				)] for row in range(_SCREAMING_SNAKE_CASE				)
    ]





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE				)							-> Callable[[int], int]:




    """simple docstring"""




    _A     =							len(_SCREAMING_SNAKE_CASE				)
    _A     =							[[0 for _ in range(_SCREAMING_SNAKE_CASE				)] for _ in range(_SCREAMING_SNAKE_CASE				)]
    _A     =							[[0] for _ in range(_SCREAMING_SNAKE_CASE				)]
    _A     =							42
    _A     =							42
    _A     =							42
    _A     =							42

    for x_val, y_val in enumerate(_SCREAMING_SNAKE_CASE				):
        for col in range(_SCREAMING_SNAKE_CASE				):
            _A     =							(x_val + 1) ** (size - col - 1)
        _A     =							y_val

    _A     =							solve(_SCREAMING_SNAKE_CASE							,     _SCREAMING_SNAKE_CASE				)

    def interpolated_func(_SCREAMING_SNAKE_CASE				) -> int:
        return sum(
            round(coeffs[x_val][0]				) * (var ** (size - x_val - 1))
            for x_val in range(_SCREAMING_SNAKE_CASE				)				)

    return interpolated_func





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE				)							-> int:




    """simple docstring"""




    return (
        1
        - variable
        + variable**2
        - variable**3
        + variable**4
        - variable**5
        + variable**6
        - variable**7
        + variable**8
        - variable**9
        + variable**10
    )





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE = question_function							,     _SCREAMING_SNAKE_CASE = 10				)							-> int:




    """simple docstring"""




    _A     =							[func(_SCREAMING_SNAKE_CASE				) for x_val in range(1							,     order + 1				)]

    _A     =							[
        interpolate(data_points[:max_coeff]				) for max_coeff in range(1							,     order + 1				)
    ]

    _A     =							0
    _A     =							42
    _A     =							42

    for poly in polynomials:
        _A     =							1
        while func(_SCREAMING_SNAKE_CASE				) == poly(_SCREAMING_SNAKE_CASE				):
            x_val += 1

        ret += poly(_SCREAMING_SNAKE_CASE				)

    return ret


if __name__ == "__main__":
 print(f"{solution() = }")













from transformers import BertTokenizer, EncoderDecoderModel, SeqaSeqTrainer, SeqaSeqTrainingArguments
from transformers.testing_utils import TestCasePlus, require_torch, slow
from transformers.utils import is_datasets_available


if is_datasets_available():
							import datasets
class  _UpperCAmelCase   (			A__      ):

				"""simple docstring"""




				@slow
				@require_torch
				def 							lowercase__  (	self		:       Optional[int]    ):

								'''simple docstring'''
								lowercase__  					=     EncoderDecoderModel.from_encoder_decoder_pretrained('''prajjwal1/bert-tiny''',    '''prajjwal1/bert-tiny'''    )
								lowercase__  					=     BertTokenizer.from_pretrained('''bert-base-uncased'''    )

								lowercase__  					=     bertabert.config.encoder.vocab_size
								lowercase__  					=     tokenizer.sep_token_id
								lowercase__  					=     tokenizer.cls_token_id
								lowercase__  					=     128

								lowercase__  					=     datasets.load_dataset('''cnn_dailymail''',    '''3.0.0''',    split='''train[:1%]'''    )
								lowercase__  					=     datasets.load_dataset('''cnn_dailymail''',    '''3.0.0''',    split='''validation[:1%]'''    )

								lowercase__  					=     train_dataset.select(range(32    )    )
								lowercase__  					=     val_dataset.select(range(16    )    )

								lowercase__  					=     4

								def _map_to_encoder_decoder_inputs(lowerCamelCase		:       str    ):
												# Tokenizer will automatically set [BOS] <text> [EOS]
												lowercase__  					=     tokenizer(batch['''article'''],    padding='''max_length''',    truncation=lowerCamelCase,    max_length=512    )
												lowercase__  					=     tokenizer(batch['''highlights'''],    padding='''max_length''',    truncation=lowerCamelCase,    max_length=128    )
												lowercase__  					=     inputs.input_ids
												lowercase__  					=     inputs.attention_mask

												lowercase__  					=     outputs.input_ids
												lowercase__  					=     outputs.input_ids.copy()
												lowercase__  					=     [
												    [-100 if token == tokenizer.pad_token_id else token for token in labels] for labels in batch['''labels''']
												]
												lowercase__  					=     outputs.attention_mask

												assert all(len(lowerCamelCase    ) == 512 for x in inputs.input_ids    )
												assert all(len(lowerCamelCase    ) == 128 for x in outputs.input_ids    )

												return batch

								def _compute_metrics(lowerCamelCase		:       Any    ):
												lowercase__  					=     pred.label_ids
												lowercase__  					=     pred.predictions

												# all unnecessary tokens are removed
												lowercase__  					=     tokenizer.batch_decode(lowerCamelCase,    skip_special_tokens=lowerCamelCase    )
												lowercase__  					=     tokenizer.batch_decode(lowerCamelCase,    skip_special_tokens=lowerCamelCase    )

												lowercase__  					=     sum([int(pred_str[i] == label_str[i]    ) for i in range(len(lowerCamelCase    )    )]    ) / len(lowerCamelCase    )

												return {"accuracy": accuracy}

								# map train dataset
								lowercase__  					=     train_dataset.map(
								    _map_to_encoder_decoder_inputs,    batched=lowerCamelCase,    batch_size=lowerCamelCase,    remove_columns=['''article''', '''highlights'''],    )
								train_dataset.set_format(
								    type='''torch''',    columns=['''input_ids''', '''attention_mask''', '''decoder_input_ids''', '''decoder_attention_mask''', '''labels'''],    )

								# same for validation dataset
								lowercase__  					=     val_dataset.map(
								    _map_to_encoder_decoder_inputs,    batched=lowerCamelCase,    batch_size=lowerCamelCase,    remove_columns=['''article''', '''highlights'''],    )
								val_dataset.set_format(
								    type='''torch''',    columns=['''input_ids''', '''attention_mask''', '''decoder_input_ids''', '''decoder_attention_mask''', '''labels'''],    )

								lowercase__  					=     self.get_auto_remove_tmp_dir()

								lowercase__  					=     SeqaSeqTrainingArguments(
								    output_dir=lowerCamelCase,    per_device_train_batch_size=lowerCamelCase,    per_device_eval_batch_size=lowerCamelCase,    predict_with_generate=lowerCamelCase,    evaluation_strategy='''steps''',    do_train=lowerCamelCase,    do_eval=lowerCamelCase,    warmup_steps=0,    eval_steps=2,    logging_steps=2,    )

								# instantiate trainer
								lowercase__  					=     SeqaSeqTrainer(
								    model=lowerCamelCase,    args=lowerCamelCase,    compute_metrics=_compute_metrics,    train_dataset=lowerCamelCase,    eval_dataset=lowerCamelCase,    tokenizer=lowerCamelCase,    )

								# start training
								trainer.train()




