






import numpy as np

def 	__lowerCamelCase      (		snake_case__	,snake_case__	,snake_case__	,snake_case__	,snake_case__		)							->  Optional[Any]:






						"""simple docstring"""





						_SCREAMING_SNAKE_CASE     				=  int(np.ceil((x_end - xa) / h		)		)
						_SCREAMING_SNAKE_CASE     				=  np.zeros((n + 1,)		)
						_SCREAMING_SNAKE_CASE     				=  ya
						_SCREAMING_SNAKE_CASE     				=  xa

						for k in range(SCREAMING_SNAKE_CASE_		):
												_SCREAMING_SNAKE_CASE     				=  f(SCREAMING_SNAKE_CASE_	,y[k]		)
												_SCREAMING_SNAKE_CASE     				=  f(x + 0.5 * h	,y[k] + 0.5 * h * ka		)
												_SCREAMING_SNAKE_CASE     				=  f(x + 0.5 * h	,y[k] + 0.5 * h * ka		)
												_SCREAMING_SNAKE_CASE     				=  f(x + h	,y[k] + h * ka		)
												_SCREAMING_SNAKE_CASE     				=  y[k] + (1 / 6) * h * (ka + 2 * ka + 2 * ka + ka)
												x += h

						return y


if __name__ == "__main__":
		import doctest

		doctest.testmod()









import copy
import re



class 				__UpperCAmelCase		:
			__snake_case    :				Any									=							"hp"
			__snake_case    :				str									=							{}
			__snake_case    :				List[Any]									=							None







			@classmethod
			def 							UpperCamelCase				(     cls:  Optional[Any]				,     UpperCAmelCase_:  int				,     UpperCAmelCase_:  Optional[int]					):







									'''simple docstring'''




									_SCREAMING_SNAKE_CASE     				=  prefix
									_SCREAMING_SNAKE_CASE     				=  defaults
									cls.build_naming_info()







			@staticmethod
			def 							UpperCamelCase				(     UpperCAmelCase_:  Any				,     UpperCAmelCase_:  str					):







									'''simple docstring'''




									if len(UpperCAmelCase_					) == 0:
															return ""
									_SCREAMING_SNAKE_CASE     				=  None
									if any(char.isdigit() for char in word					):
															raise Exception(F'Parameters should not contain numbers: \'{word}\' contains a number'					)
									if word in info["short_word"]:
															return info["short_word"][word]
									for prefix_len in range(1				,     len(UpperCAmelCase_					) + 1					):
															_SCREAMING_SNAKE_CASE     				=  word[:prefix_len]
															if prefix in info["reverse_short_word"]:
																					continue
															else:
																					_SCREAMING_SNAKE_CASE     				=  prefix
																					break

									if short_word is None:
															# Paranoid fallback
															def int_to_alphabetic(UpperCAmelCase_:  List[Any]					):
																					_SCREAMING_SNAKE_CASE     				=  """"""
																					while integer != 0:
																											_SCREAMING_SNAKE_CASE     				=  chr(ord("""A"""					) + integer % 10					) + s
																											integer //= 10
																					return s

															_SCREAMING_SNAKE_CASE     				=  0
															while True:
																					_SCREAMING_SNAKE_CASE     				=  word + """#""" + int_to_alphabetic(UpperCAmelCase_					)
																					if sword in info["reverse_short_word"]:
																											continue
																					else:
																											_SCREAMING_SNAKE_CASE     				=  sword
																											break

									_SCREAMING_SNAKE_CASE     				=  short_word
									_SCREAMING_SNAKE_CASE     				=  word
									return short_word







			@staticmethod
			def 							UpperCamelCase				(     UpperCAmelCase_:  List[str]				,     UpperCAmelCase_:  List[str]					):







									'''simple docstring'''




									_SCREAMING_SNAKE_CASE     				=  param_name.split("""_"""					)

									_SCREAMING_SNAKE_CASE     				=  [TrialShortNamer.shortname_for_word(UpperCAmelCase_				,     UpperCAmelCase_					) for word in words]

									# We try to create a separatorless short name, but if there is a collision we have to fallback
									# to a separated short name
									_SCREAMING_SNAKE_CASE     				=  ["""""", """_"""]

									for separator in separators:
															_SCREAMING_SNAKE_CASE     				=  separator.join(UpperCAmelCase_					)
															if shortname not in info["reverse_short_param"]:
																					_SCREAMING_SNAKE_CASE     				=  shortname
																					_SCREAMING_SNAKE_CASE     				=  param_name
																					return shortname

									return param_name







			@staticmethod
			def 							UpperCamelCase				(     UpperCAmelCase_:  Optional[int]				,     UpperCAmelCase_:  Optional[int]					):







									'''simple docstring'''




									_SCREAMING_SNAKE_CASE     				=  TrialShortNamer.shortname_for_key(UpperCAmelCase_				,     UpperCAmelCase_					)
									_SCREAMING_SNAKE_CASE     				=  short_name
									_SCREAMING_SNAKE_CASE     				=  param_name







			@classmethod
			def 							UpperCamelCase				(     cls:  str					):







									'''simple docstring'''




									if cls.NAMING_INFO is not None:
															return

									_SCREAMING_SNAKE_CASE     				=  {
									    """short_word""": {},
									    """reverse_short_word""": {},
									    """short_param""": {},
									    """reverse_short_param""": {},
									}

									_SCREAMING_SNAKE_CASE     				=  list(cls.DEFAULTS.keys()					)

									for k in field_keys:
															cls.add_new_param_name(UpperCAmelCase_				,     UpperCAmelCase_					)

									_SCREAMING_SNAKE_CASE     				=  info







			@classmethod
			def 							UpperCamelCase				(     cls:  Any				,     UpperCAmelCase_:  Optional[Any]					):







									'''simple docstring'''




									cls.build_naming_info()
									assert cls.PREFIX is not None
									_SCREAMING_SNAKE_CASE     				=  [copy.copy(cls.PREFIX					)]

									for k, v in params.items():
															if k not in cls.DEFAULTS:
																					raise Exception(F'You should provide a default value for the param name {k} with value {v}'					)
															if v == cls.DEFAULTS[k]:
																					# The default value is not added to the name
																					continue

															_SCREAMING_SNAKE_CASE     				=  cls.NAMING_INFO["""short_param"""][k]

															if isinstance(UpperCAmelCase_				,     UpperCAmelCase_					):
																					_SCREAMING_SNAKE_CASE     				=  1 if v else 0

															_SCREAMING_SNAKE_CASE     				=  """""" if isinstance(UpperCAmelCase_				,     (int, float)					) else """-"""
															_SCREAMING_SNAKE_CASE     				=  F'{key}{sep}{v}'
															name.append(UpperCAmelCase_					)

									return "_".join(UpperCAmelCase_					)







			@classmethod
			def 							UpperCamelCase				(     cls:  int				,     UpperCAmelCase_:  int					):







									'''simple docstring'''




									_SCREAMING_SNAKE_CASE     				=  repr[len(cls.PREFIX					) + 1 :]
									if repr == "":
															_SCREAMING_SNAKE_CASE     				=  []
									else:
															_SCREAMING_SNAKE_CASE     				=  repr.split("""_"""					)

									_SCREAMING_SNAKE_CASE     				=  {}

									for value in values:
															if "-" in value:
																					_SCREAMING_SNAKE_CASE				,					_SCREAMING_SNAKE_CASE     				=  value.split("""-"""					)
															else:
																					_SCREAMING_SNAKE_CASE     				=  re.sub("""[0-9.]"""				,     """"""				,     UpperCAmelCase_					)
																					_SCREAMING_SNAKE_CASE     				=  float(re.sub("""[^0-9.]"""				,     """"""				,     UpperCAmelCase_					)					)

															_SCREAMING_SNAKE_CASE     				=  cls.NAMING_INFO["""reverse_short_param"""][p_k]

															_SCREAMING_SNAKE_CASE     				=  p_v

									for k in cls.DEFAULTS:
															if k not in parameters:
																					_SCREAMING_SNAKE_CASE     				=  cls.DEFAULTS[k]

									return parameters

