



"""simple docstring"""


from __future__ import annotations

from math import pi, pow

def 				_lowerCAmelCase   (     lowercase_		):
						if side_length < 0:
												raise ValueError('vol_cube() only accepts non-negative values'		)
						return pow(lowercase_	,			3		)

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_		):
						if height < 0 or radius < 0:
												raise ValueError('vol_spherical_cap() only accepts non-negative values'		)
						# Volume is 1/3 pi * height squared * (3 * radius - height)
						return 1 / 3 * pi * pow(lowercase_	,			2		) * (3 * radius - height)

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_	,			lowercase_		):
						if radius_a < 0 or radius_a < 0 or centers_distance < 0:
												raise ValueError('vol_spheres_intersect() only accepts non-negative values'		)
						if centers_distance == 0:
												return vol_sphere(min(lowercase_	,			lowercase_		)		)

						UpperCAmelCase    =				(
						    (radius_a - radius_a + centers_distance)
						    * (radius_a + radius_a - centers_distance)
						    / (2 * centers_distance)
						)
						UpperCAmelCase    =				(
						    (radius_a - radius_a + centers_distance)
						    * (radius_a + radius_a - centers_distance)
						    / (2 * centers_distance)
						)

						return vol_spherical_cap(lowercase_	,			lowercase_		) + vol_spherical_cap(lowercase_	,			lowercase_		)

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_	,			lowercase_		):

						if radius_a <= 0 or radius_a <= 0 or centers_distance < 0:
												raise ValueError(
												    'vol_spheres_union() only accepts non-negative values, non-zero radius'		)

						if centers_distance == 0:
												return vol_sphere(max(lowercase_	,			lowercase_		)		)

						return (
						    vol_sphere(lowercase_		)
						    + vol_sphere(lowercase_		)
						    - vol_spheres_intersect(lowercase_	,			lowercase_	,			lowercase_		)
						)

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_	,			lowercase_		):
						if width < 0 or height < 0 or length < 0:
												raise ValueError('vol_cuboid() only accepts non-negative values'		)
						return float(width * height * length		)

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_		):
						if height < 0 or area_of_base < 0:
												raise ValueError('vol_cone() only accepts non-negative values'		)
						return area_of_base * height / 3.0

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_		):
						if height < 0 or radius < 0:
												raise ValueError('vol_right_circ_cone() only accepts non-negative values'		)
						return pi * pow(lowercase_	,			2		) * height / 3.0

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_		):
						if height < 0 or area_of_base < 0:
												raise ValueError('vol_prism() only accepts non-negative values'		)
						return float(area_of_base * height		)

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_		):
						if height < 0 or area_of_base < 0:
												raise ValueError('vol_pyramid() only accepts non-negative values'		)
						return area_of_base * height / 3.0

def 				_lowerCAmelCase   (     lowercase_		):
						if radius < 0:
												raise ValueError('vol_sphere() only accepts non-negative values'		)
						# Volume is 4/3 * pi * radius cubed
						return 4 / 3 * pi * pow(lowercase_	,			3		)

def 				_lowerCAmelCase   (     lowercase_		):
						if radius < 0:
												raise ValueError('vol_hemisphere() only accepts non-negative values'		)
						# Volume is radius cubed * pi * 2/3
						return pow(lowercase_	,			3		) * pi * 2 / 3

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_		):
						if height < 0 or radius < 0:
												raise ValueError('vol_circular_cylinder() only accepts non-negative values'		)
						# Volume is radius squared * height * pi
						return pow(lowercase_	,			2		) * height * pi

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_	,			lowercase_		):
						# Volume - (outer_radius squared - inner_radius squared) * pi * height
						if inner_radius < 0 or outer_radius < 0 or height < 0:
												raise ValueError(
												    'vol_hollow_circular_cylinder() only accepts non-negative values'		)
						if outer_radius <= inner_radius:
												raise ValueError('outer_radius must be greater than inner_radius'		)
						return pi * (pow(lowercase_	,			2		) - pow(lowercase_	,			2		)) * height

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_	,			lowercase_		):
						# Volume is 1/3 * pi * height *
						#           (radius_1 squared + radius_2 squared + radius_1 * radius_2)
						if radius_a < 0 or radius_a < 0 or height < 0:
												raise ValueError('vol_conical_frustum() only accepts non-negative values'		)
						return (
						    1
						    / 3
						    * pi
						    * height
						    * (pow(lowercase_	,			2		) + pow(lowercase_	,			2		) + radius_a * radius_a)
						)

def 				_lowerCAmelCase   (     lowercase_	,			lowercase_		):
						if torus_radius < 0 or tube_radius < 0:
												raise ValueError('vol_torus() only accepts non-negative values'		)
						return 2 * pow(lowercase_	,			2		) * torus_radius * pow(lowercase_	,			2		)

def 				_lowerCAmelCase   (     ):
						print('Volumes:'		)
						print(F"""Cube: {vol_cube(2		) = }"""		)  # = 8
						print(F"""Cuboid: {vol_cuboid(2	,			2	,			2		) = }"""		)  # = 8
						print(F"""Cone: {vol_cone(2	,			2		) = }"""		)  # ~= 1.33
						print(F"""Right Circular Cone: {vol_right_circ_cone(2	,			2		) = }"""		)  # ~= 8.38
						print(F"""Prism: {vol_prism(2	,			2		) = }"""		)  # = 4
						print(F"""Pyramid: {vol_pyramid(2	,			2		) = }"""		)  # ~= 1.33
						print(F"""Sphere: {vol_sphere(2		) = }"""		)  # ~= 33.5
						print(F"""Hemisphere: {vol_hemisphere(2		) = }"""		)  # ~= 16.75
						print(F"""Circular Cylinder: {vol_circular_cylinder(2	,			2		) = }"""		)  # ~= 25.1
						print(F"""Torus: {vol_torus(2	,			2		) = }"""		)  # ~= 157.9
						print(F"""Conical Frustum: {vol_conical_frustum(2	,			2	,			4		) = }"""		)  # ~= 58.6
						print(F"""Spherical cap: {vol_spherical_cap(1	,			2		) = }"""		)  # ~= 5.24
						print(F"""Spheres intersetion: {vol_spheres_intersect(2	,			2	,			1		) = }"""		)  # ~= 21.21
						print(F"""Spheres union: {vol_spheres_union(2	,			2	,			1		) = }"""		)  # ~= 45.81
						print(
						    F"""Hollow Circular Cylinder: {vol_hollow_circular_cylinder(1	,			2	,			3		) = }"""		)  # ~= 28.3


if __name__ == "__main__":
				main()










"""simple docstring"""


from typing import Any

import numpy as np







class    A_    :



						"""simple docstring"""






						def __init__(      self				:List[str]     ,						lowercase_				:np.ndarray     ,						lowercase_				:int      )       ->    None:
												# Check if RHS is negative
												if np.any(tableau[:, -1]     ,						where=tableau[:, -1] < 0      ):
																		raise ValueError('RHS must be > 0'      )

												UpperCAmelCase    =				tableau
												UpperCAmelCase   ,					UpperCAmelCase    =				tableau.shape

												# Number of decision variables x1, x2, x3...
												UpperCAmelCase    =				n_vars

												# Number of artificial variables to be minimised
												UpperCAmelCase    =				len(np.where(tableau[self.n_vars : -1] == -1      )[0]      )

												# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
												UpperCAmelCase    =				(self.n_art_vars > 0) + 1

												# Number of slack variables added to make inequalities into equalities
												UpperCAmelCase    =				self.n_rows - self.n_stages

												# Objectives for each stage
												UpperCAmelCase    =				['max']

												# In two stage simplex, first minimise then maximise
												if self.n_art_vars:
																		self.objectives.append('min'      )

												UpperCAmelCase    =				['']

												# Index of current pivot row and column
												UpperCAmelCase    =				None
												UpperCAmelCase    =				None

												# Does objective row only contain (non)-negative values?
												UpperCAmelCase    =				False






						@staticmethod
						def 			UpperCAmelCase__  (      *lowercase_				:int      )       ->    list[str]:
												if len(lowercase_      ) != 3:
																		raise ValueError('Must provide n_vars, n_slack, and n_art_vars'      )

												if not all(x >= 0 and isinstance(lowercase_     ,						lowercase_      ) for x in args      ):
																		raise ValueError('All arguments must be non-negative integers'      )

												# decision | slack | artificial
												UpperCAmelCase    =				['x', 's', 'a']
												UpperCAmelCase    =				[]
												for i in range(3      ):
																		for j in range(args[i]      ):
																								titles.append(string_starts[i] + str(j + 1      )      )
												titles.append('RHS'      )
												return titles






						def 			UpperCAmelCase__  (      self				:List[str]     ,						lowercase_				:np.ndarray      )       ->    tuple[Any, Any]:
												UpperCAmelCase    =				self.objectives[-1]

												# Find entries of highest magnitude in objective rows
												UpperCAmelCase    =				(objective == 'min') - (objective == 'max')
												UpperCAmelCase    =				np.argmax(sign * tableau[0, : self.n_vars]      )

												# Choice is only valid if below 0 for maximise, and above for minimise
												if sign * self.tableau[0, col_idx] <= 0:
																		UpperCAmelCase    =				True
																		return 0, 0

												# Pivot row is chosen as having the lowest quotient when elements of
												# the pivot column divide the right-hand side

												# Slice excluding the objective rows
												UpperCAmelCase    =				slice(self.n_stages     ,						self.n_rows      )

												# RHS
												UpperCAmelCase    =				tableau[s, -1]

												# Elements of pivot column within slice
												UpperCAmelCase    =				tableau[s, col_idx]

												# Array filled with nans
												UpperCAmelCase    =				np.full(self.n_rows - self.n_stages     ,						np.nan      )

												# If element in pivot column is greater than zeron_stages, return
												# quotient or nan otherwise
												UpperCAmelCase    =				np.divide(lowercase_     ,						lowercase_     ,						out=lowercase_     ,						where=divisor > 0      )

												# Arg of minimum quotient excluding the nan values. n_stages is added
												# to compensate for earlier exclusion of objective columns
												UpperCAmelCase    =				np.nanargmin(lowercase_      ) + self.n_stages
												return row_idx, col_idx






						def 			UpperCAmelCase__  (      self				:Union[str, Any]     ,						lowercase_				:np.ndarray     ,						lowercase_				:int     ,						lowercase_				:int      )       ->    np.ndarray:
												UpperCAmelCase    =				tableau[row_idx].copy()

												UpperCAmelCase    =				piv_row[col_idx]

												# Entry becomes 1
												piv_row *= 1 / piv_val

												# Variable in pivot column becomes basic, ie the only non-zero entry
												for idx, coeff in enumerate(tableau[:, col_idx]      ):
																		tableau[idx] += -coeff * piv_row
												UpperCAmelCase    =				piv_row
												return tableau






						def 			UpperCAmelCase__  (      self				:Optional[Any]     ,						lowercase_				:np.ndarray      )       ->    np.ndarray:
												self.objectives.pop()

												if not self.objectives:
																		return tableau

												# Slice containing ids for artificial columns
												UpperCAmelCase    =				slice(-self.n_art_vars - 1     ,						-1      )

												# Delete the artificial variable columns
												UpperCAmelCase    =				np.delete(lowercase_     ,						lowercase_     ,						axis=1      )

												# Delete the objective row of the first stage
												UpperCAmelCase    =				np.delete(lowercase_     ,						0     ,						axis=0      )

												UpperCAmelCase    =				1
												self.n_rows -= 1
												UpperCAmelCase    =				0
												UpperCAmelCase    =				False
												return tableau






						def 			UpperCAmelCase__  (      self				:List[Any]      )       ->    dict[Any, Any]:
												# Stop simplex algorithm from cycling.
												for _ in range(1_00      ):
																		# Completion of each stage removes an objective. If both stages
																		# are complete, then no objectives are left
																		if not self.objectives:
																								UpperCAmelCase    =				self.generate_col_titles(
																								    self.n_vars     ,						self.n_slack     ,						self.n_art_vars      )

																								# Find the values of each variable at optimal solution
																								return self.interpret_tableau(self.tableau     ,						self.col_titles      )

																		UpperCAmelCase   ,					UpperCAmelCase    =				self.find_pivot(self.tableau      )

																		# If there are no more negative values in objective row
																		if self.stop_iter:
																								# Delete artificial variable columns and rows. Update attributes
																								UpperCAmelCase    =				self.change_stage(self.tableau      )
																		else:
																								UpperCAmelCase    =				self.pivot(self.tableau     ,						lowercase_     ,						lowercase_      )
												return {}






						def 			UpperCAmelCase__  (      self				:Union[str, Any]     ,						lowercase_				:np.ndarray     ,						lowercase_				:list[str]      )       ->    dict[str, float]:
												UpperCAmelCase    =				{'P': abs(tableau[0, -1]      )}

												for i in range(self.n_vars      ):
																		# Gives ids of nonzero entries in the ith column
																		UpperCAmelCase    =				np.nonzero(tableau[:, i]      )
																		UpperCAmelCase    =				len(nonzero[0]      )

																		# First entry in the nonzero ids
																		UpperCAmelCase    =				nonzero[0][0]
																		UpperCAmelCase    =				tableau[nonzero_rowidx, i]

																		# If there is only one nonzero value in column, which is one
																		if n_nonzero == nonzero_val == 1:
																								UpperCAmelCase    =				tableau[nonzero_rowidx, -1]
																								UpperCAmelCase    =				rhs_val

        # Check for basic variables
												for title in col_titles:
																		# Don't add RHS or slack variables to output dict
																		if title[0] not in "R-s-a":
																								output_dict.setdefault(lowercase_     ,						0      )
												return output_dict


if __name__ == "__main__":
				import doctest

				doctest.testmod()





