


'''simple docstring'''





def 			_lowerCAmelCase						(							lowerCAmelCase_							:List[str]							)->str:
   '''simple docstring'''






   if len(__lowerCamelCase							) == 0 or not all(len(__lowerCamelCase							) == 2 for pair in coordinates							):
      raise ValueError("The program cannot work out a fitting polynomial."							)

   if len({tuple(__lowerCamelCase							) for pair in coordinates}							) != len(__lowerCamelCase							):
      raise ValueError("The program cannot work out a fitting polynomial."							)

   snake_case_					    =					{x for x, _ in coordinates}
   if len(__lowerCamelCase							) == 1:
      return F'''x={coordinates[0][0]}'''

   if len(__lowerCamelCase							) != len(__lowerCamelCase							):
      raise ValueError("The program cannot work out a fitting polynomial."							)

   snake_case_					    =					len(__lowerCamelCase							)

   snake_case_					    =					0
   snake_case_					    =					[]
   # put the x and x to the power values in a matrix
   while count_of_line < x:
      snake_case_					    =					0
      snake_case_					    =					coordinates[count_of_line][0]
      snake_case_					    =					[]
      while count_in_line < x:
         count_line.append(a ** (x - (count_in_line + 1))							)
         count_in_line += 1
      matrix.append(__lowerCamelCase							)
      count_of_line += 1

   snake_case_					    =					0
   # put the y values into a vector
   snake_case_					    =					[]
   while count_of_line < x:
      vector.append(coordinates[count_of_line][1]							)
      count_of_line += 1

   snake_case_					    =					0

   while count < x:
      snake_case_					    =					0
      while zahlen < x:
         if count == zahlen:
            zahlen += 1
         if zahlen == x:
            break
         snake_case_					    =					matrix[zahlen][count] / matrix[count][count]
         for counting_columns, item in enumerate(matrix[count]							):
            # manipulating all the values in the matrix
            matrix[zahlen][counting_columns] -= item * bruch
         # manipulating the values in the vector
         vector[zahlen] -= vector[count] * bruch
         zahlen += 1
      count += 1

   snake_case_					    =					0
   # make solutions
   snake_case_					    =					[]
   while count < x:
      solution.append(str(vector[count] / matrix[count][count]							)							)
      count += 1

   snake_case_					    =					0
   snake_case_					    =					"""f(x)="""

   while count < x:
      snake_case_					    =					solution[count].split("E"							)
      if len(__lowerCamelCase							) > 1:
         snake_case_					    =					F'''{remove_e[0]}*10^{remove_e[1]}'''
      solved += F'''x^{x - (count + 1)}*{solution[count]}'''
      if count + 1 != x:
         solved += "+"
      count += 1

   return solved


if __name__ == "__main__":
   print(points_to_polynomial([]))
   print(points_to_polynomial([[]]))
   print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
   print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
   print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
   print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
   print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
   print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
   print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
   print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))








from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


SCREAMING_SNAKE_CASE	:List[str]  		=     {
    '''configuration_prophetnet''': ['''PROPHETNET_PRETRAINED_CONFIG_ARCHIVE_MAP''', '''ProphetNetConfig'''],
    '''tokenization_prophetnet''': ['''ProphetNetTokenizer'''],
}

try:
   if not is_torch_available():
      raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
   pass
else:
   SCREAMING_SNAKE_CASE	:List[str]  		=     [
       '''PROPHETNET_PRETRAINED_MODEL_ARCHIVE_LIST''',
       '''ProphetNetDecoder''',
       '''ProphetNetEncoder''',
       '''ProphetNetForCausalLM''',
       '''ProphetNetForConditionalGeneration''',
       '''ProphetNetModel''',
       '''ProphetNetPreTrainedModel''',
   ]


if TYPE_CHECKING:
   from .configuration_prophetnet import PROPHETNET_PRETRAINED_CONFIG_ARCHIVE_MAP, ProphetNetConfig
   from .tokenization_prophetnet import ProphetNetTokenizer

   try:
      if not is_torch_available():
         raise OptionalDependencyNotAvailable()
   except OptionalDependencyNotAvailable:
      pass
   else:
      from .modeling_prophetnet import (
          PROPHETNET_PRETRAINED_MODEL_ARCHIVE_LIST,
          ProphetNetDecoder,
          ProphetNetEncoder,
          ProphetNetForCausalLM,
          ProphetNetForConditionalGeneration,
          ProphetNetModel,
          ProphetNetPreTrainedModel,
      )

else:
   import sys

   SCREAMING_SNAKE_CASE	:int  		=     _LazyModule(__name__, globals()['''__file__'''], _import_structure, module_spec=__spec__)




