






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

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


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							SCREAMING_SNAKE_CASE__				     =			[
							    [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]]))










import torch
from torch import nn
class    __UpperCamelCase			(       nn.Module				):

 """simple docstring"""



 def __init__(	self		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE=1		,				SCREAMING_SNAKE_CASE=False      ) ->			Tuple:
  super().__init__()

  a__												= n_token
  a__												= d_embed
  a__												= d_proj

  a__												= cutoffs + [n_token]
  a__												= [0] + self.cutoffs
  a__												= div_val

  a__												= self.cutoffs[0]
  a__												= len(self.cutoffs      ) - 1
  a__												= self.shortlist_size + self.n_clusters

  if self.n_clusters > 0:
   a__												= nn.Parameter(torch.zeros(self.n_clusters		,				self.d_embed      )      )
   a__												= nn.Parameter(torch.zeros(self.n_clusters      )      )

  a__												= nn.ModuleList()
  a__												= nn.ParameterList()

  if div_val == 1:
   for i in range(len(self.cutoffs      )      ):
    if d_proj != d_embed:
     self.out_projs.append(nn.Parameter(torch.FloatTensor(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      )      )      )
    else:
     self.out_projs.append(SCREAMING_SNAKE_CASE      )

   self.out_layers.append(nn.Linear(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      )      )
  else:
   for i in range(len(self.cutoffs      )      ):
    a__							,			a__												= self.cutoff_ends[i], self.cutoff_ends[i + 1]
    a__												= d_embed // (div_val**i)

    self.out_projs.append(nn.Parameter(torch.FloatTensor(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      )      )      )

    self.out_layers.append(nn.Linear(SCREAMING_SNAKE_CASE		,				r_idx - l_idx      )      )

  a__												= keep_order

 def 						_UpperCAmelCase       (	self		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      ) ->			str:
  if proj is None:
   a__												= nn.functional.linear(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				bias=SCREAMING_SNAKE_CASE      )
  else:
   # if CUDA_MAJOR <= 9 and CUDA_MINOR <= 1:
   a__												= nn.functional.linear(SCREAMING_SNAKE_CASE		,				proj.t().contiguous()      )
   a__												= nn.functional.linear(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				bias=SCREAMING_SNAKE_CASE      )
   # else:
   #     logit = torch.einsum('bd,de,ev->bv', (hidden, proj, weight.t()))
   #     if bias is not None:
   #         logit = logit + bias

  return logit

 def 						_UpperCAmelCase       (	self		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE=None		,				SCREAMING_SNAKE_CASE=False      ) ->			Optional[int]:

  if labels is not None:
   # Shift so that tokens < n predict n
   a__												= hidden[..., :-1, :].contiguous()
   a__												= labels[..., 1:].contiguous()
   a__												= hidden.view(-1		,				hidden.size(-1      )      )
   a__												= labels.view(-1      )
   if hidden.size(0      ) != labels.size(0      ):
    raise RuntimeError('''Input and labels should have the same size in the batch dimension.'''      )
  else:
   a__												= hidden.view(-1		,				hidden.size(-1      )      )

  if self.n_clusters == 0:
   a__												= self._compute_logit(SCREAMING_SNAKE_CASE		,				self.out_layers[0].weight		,				self.out_layers[0].bias		,				self.out_projs[0]      )
   if labels is not None:
    a__												= labels != -1_0_0
    a__												= torch.zeros_like(SCREAMING_SNAKE_CASE		,				dtype=hidden.dtype		,				device=hidden.device      )
    a__												= (
        -nn.functional.log_softmax(SCREAMING_SNAKE_CASE		,				dim=-1      )[mask].gather(1		,				labels[mask].unsqueeze(1      )      ).squeeze(1      )
    )
   else:
    a__												= nn.functional.log_softmax(SCREAMING_SNAKE_CASE		,				dim=-1      )
  else:
   # construct weights and biases
   a__							,			a__												= [], []
   for i in range(len(self.cutoffs      )      ):
    if self.div_val == 1:
     a__							,			a__												= self.cutoff_ends[i], self.cutoff_ends[i + 1]
     a__												= self.out_layers[0].weight[l_idx:r_idx]
     a__												= self.out_layers[0].bias[l_idx:r_idx]
    else:
     a__												= self.out_layers[i].weight
     a__												= self.out_layers[i].bias

    if i == 0:
     a__												= torch.cat([weight_i, self.cluster_weight]		,				dim=0      )
     a__												= torch.cat([bias_i, self.cluster_bias]		,				dim=0      )

    weights.append(SCREAMING_SNAKE_CASE      )
    biases.append(SCREAMING_SNAKE_CASE      )

   a__							,			a__							,			a__												= weights[0], biases[0], self.out_projs[0]

   a__												= self._compute_logit(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      )
   a__												= nn.functional.log_softmax(SCREAMING_SNAKE_CASE		,				dim=1      )

   if labels is None:
    a__												= hidden.new_empty((head_logit.size(0      ), self.n_token)      )
   else:
    a__												= torch.zeros_like(SCREAMING_SNAKE_CASE		,				dtype=hidden.dtype		,				device=hidden.device      )

   a__												= 0
   a__												= [0] + self.cutoffs
   for i in range(len(SCREAMING_SNAKE_CASE      ) - 1      ):
    a__							,			a__												= cutoff_values[i], cutoff_values[i + 1]

    if labels is not None:
     a__												= (labels >= l_idx) & (labels < r_idx)
     a__												= mask_i.nonzero().squeeze()

     if indices_i.numel() == 0:
      continue

     a__												= labels.index_select(0		,				SCREAMING_SNAKE_CASE      ) - l_idx
     a__												= head_logprob.index_select(0		,				SCREAMING_SNAKE_CASE      )
     a__												= hidden.index_select(0		,				SCREAMING_SNAKE_CASE      )
    else:
     a__												= hidden

    if i == 0:
     if labels is not None:
      a__												= head_logprob_i.gather(1		,				target_i[:, None]      ).squeeze(1      )
     else:
      a__												= head_logprob[:, : self.cutoffs[0]]
    else:
     a__							,			a__							,			a__												= weights[i], biases[i], self.out_projs[i]

     a__												= self._compute_logit(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      )
     a__												= nn.functional.log_softmax(SCREAMING_SNAKE_CASE		,				dim=1      )
     a__												= self.cutoffs[0] + i - 1  # No probability for the head cluster
     if labels is not None:
      a__												= head_logprob_i[:, cluster_prob_idx] + tail_logprob_i.gather(
          1		,				target_i[:, None]      ).squeeze(1      )
     else:
      a__												= head_logprob[:, cluster_prob_idx, None] + tail_logprob_i
      a__												= logprob_i

    if labels is not None:
     if (hasattr(self		,				'''keep_order'''      ) and self.keep_order) or keep_order:
      out.index_copy_(0		,				SCREAMING_SNAKE_CASE		,				-logprob_i      )
     else:
      out[offset : offset + logprob_i.size(0      )].copy_(-logprob_i      )
     offset += logprob_i.size(0      )

  return out



 def 						_UpperCAmelCase       (	self		,				SCREAMING_SNAKE_CASE      ) ->			List[str]:
  if self.n_clusters == 0:
   a__												= self._compute_logit(SCREAMING_SNAKE_CASE		,				self.out_layers[0].weight		,				self.out_layers[0].bias		,				self.out_projs[0]      )
   return nn.functional.log_softmax(SCREAMING_SNAKE_CASE		,				dim=-1      )
  else:
   # construct weights and biases
   a__							,			a__												= [], []
   for i in range(len(self.cutoffs      )      ):
    if self.div_val == 1:
     a__							,			a__												= self.cutoff_ends[i], self.cutoff_ends[i + 1]
     a__												= self.out_layers[0].weight[l_idx:r_idx]
     a__												= self.out_layers[0].bias[l_idx:r_idx]
    else:
     a__												= self.out_layers[i].weight
     a__												= self.out_layers[i].bias

    if i == 0:
     a__												= torch.cat([weight_i, self.cluster_weight]		,				dim=0      )
     a__												= torch.cat([bias_i, self.cluster_bias]		,				dim=0      )

    weights.append(SCREAMING_SNAKE_CASE      )
    biases.append(SCREAMING_SNAKE_CASE      )

   a__							,			a__							,			a__												= weights[0], biases[0], self.out_projs[0]
   a__												= self._compute_logit(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      )

   a__												= hidden.new_empty((head_logit.size(0      ), self.n_token)      )
   a__												= nn.functional.log_softmax(SCREAMING_SNAKE_CASE		,				dim=1      )

   a__												= [0] + self.cutoffs
   for i in range(len(SCREAMING_SNAKE_CASE      ) - 1      ):
    a__							,			a__												= cutoff_values[i], cutoff_values[i + 1]

    if i == 0:
     a__												= head_logprob[:, : self.cutoffs[0]]
    else:
     a__							,			a__							,			a__												= weights[i], biases[i], self.out_projs[i]

     a__												= self._compute_logit(SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE		,				SCREAMING_SNAKE_CASE      )
     a__												= nn.functional.log_softmax(SCREAMING_SNAKE_CASE		,				dim=1      )

     a__												= head_logprob[:, -i] + tail_logprob_i
     a__												= logprob_i

   return out





