


"""simple docstring"""



from __future__ import annotations

import math
import random
from collections.abc import Collection
from typing import overload
class 					__magic_name__		:






       def __init__(   self  :    Tuple  ,     snake_case__  :    Collection[float] | None = None				):

              '''simple docstring'''






              if components is None:
                     lowercase		:str       							=				[]
              lowercase		:Dict       							=				list(snake_case__				)






       def __len__(   self  :    Optional[Any]				):

              '''simple docstring'''






              return len(self.__components				)






       def __str__(   self  :    Any				):

              '''simple docstring'''






              return "(" + ",".join(map(snake_case__  ,     self.__components				)				) + ")"






       def __add__(   self  :    List[Any]  ,     snake_case__  :    Vector				):

              '''simple docstring'''






              lowercase		:Union[str, Any]       							=				len(self				)
              if size == len(snake_case__				):
                     lowercase		:List[Any]       							=				[self.__components[i] + other.component(snake_case__				) for i in range(snake_case__				)]
                     return Vector(snake_case__				)
              else:
                     raise Exception('''must have the same size'''				)






       def __sub__(   self  :    Optional[int]  ,     snake_case__  :    Vector				):

              '''simple docstring'''






              lowercase		:List[str]       							=				len(self				)
              if size == len(snake_case__				):
                     lowercase		:str       							=				[self.__components[i] - other.component(snake_case__				) for i in range(snake_case__				)]
                     return Vector(snake_case__				)
              else:  # error case
                     raise Exception('''must have the same size'''				)






       @overload
       def __mul__(   self  :    Any  ,     snake_case__  :    float				):

              '''simple docstring'''






              ...






       @overload
       def __mul__(   self  :    Optional[int]  ,     snake_case__  :    Vector				):

              '''simple docstring'''






              ...






       def __mul__(   self  :    Tuple  ,     snake_case__  :    float | Vector				):

              '''simple docstring'''






              if isinstance(snake_case__  ,     (float, int)				):
                     lowercase		:Dict       							=				[c * other for c in self.__components]
                     return Vector(snake_case__				)
              elif isinstance(snake_case__  ,     snake_case__				) and len(self				) == len(snake_case__				):
                     lowercase		:Union[str, Any]       							=				len(self				)
                     lowercase		:List[str]       							=				[self.__components[i] * other.component(snake_case__				) for i in range(snake_case__				)]
                     return sum(snake_case__				)
              else:  # error case
                     raise Exception('''invalid operand!'''				)






       def  __snake_case						(   self  :    Optional[Any]				):

              '''simple docstring'''






              return Vector(self.__components				)






       def  __snake_case						(   self  :    Optional[int]  ,     snake_case__  :    int				):

              '''simple docstring'''






              if isinstance(snake_case__  ,     snake_case__				) and -len(self.__components				) <= i < len(self.__components				):
                     return self.__components[i]
              else:
                     raise Exception('''index out of range'''				)






       def  __snake_case						(   self  :    Dict  ,     snake_case__  :    int  ,     snake_case__  :    float				):

              '''simple docstring'''






              assert -len(self.__components				) <= pos < len(self.__components				)
              lowercase		:List[Any]       							=				value






       def  __snake_case						(   self  :    Optional[int]				):

              '''simple docstring'''






              if len(self.__components				) == 0:
                     raise Exception('''Vector is empty'''				)
              lowercase		:Union[str, Any]       							=				[c**2 for c in self.__components]
              return math.sqrt(sum(snake_case__				)				)







       def  __snake_case						(   self  :    Optional[int]  ,     snake_case__  :    Vector  ,     snake_case__  :    bool = False				):

              '''simple docstring'''






              lowercase		:Optional[int]       							=				self * other
              lowercase		:List[str]       							=				self.euclidean_length() * other.euclidean_length()
              if deg:
                     return math.degrees(math.acos(num / den				)				)
              else:
                     return math.acos(num / den				)







def       lowerCamelCase					(a_						:int)     ->							Vector:
       assert isinstance(a_				,				a_)
       return Vector([0] * dimension)






def       lowerCamelCase					(a_						:int				,				a_						:int)     ->							Vector:
       assert isinstance(a_				,				a_) and (isinstance(a_				,				a_))
       lowercase		:str       							=				[0] * dimension
       lowercase		:Optional[Any]       							=				1
       return Vector(a_)






def       lowerCamelCase					(a_						:float				,				a_						:Vector				,				a_						:Vector)     ->							Vector:
       assert (
           isinstance(a_				,				a_)
           and isinstance(a_				,				a_)
           and (isinstance(a_				,				(int, float)))
       )
       return x * scalar + y






def       lowerCamelCase					(a_						:int				,				a_						:int				,				a_						:int)     ->							Vector:
       random.seed(a_)
       lowercase		:Tuple       							=				[random.randint(a_				,				a_) for _ in range(a_)]
       return Vector(a_)
class 					__magic_name__		:






       def __init__(   self  :    Optional[Any]  ,     snake_case__  :    list[list[float]]  ,     snake_case__  :    int  ,     snake_case__  :    int				):

              '''simple docstring'''






              lowercase		:int       							=				matrix
              lowercase		:Union[str, Any]       							=				w
              lowercase		:List[Any]       							=				h






       def __str__(   self  :    int				):

              '''simple docstring'''






              lowercase		:Optional[int]       							=				''''''
              for i in range(self.__height				):
                     ans += "|"
                     for j in range(self.__width				):
                            if j < self.__width - 1:
                                   ans += str(self.__matrix[i][j]				) + ","
                            else:
                                   ans += str(self.__matrix[i][j]				) + "|\n"
              return ans






       def __add__(   self  :    Tuple  ,     snake_case__  :    Matrix				):

              '''simple docstring'''






              if self.__width == other.width() and self.__height == other.height():
                     lowercase		:List[Any]       							=				[]
                     for i in range(self.__height				):
                            lowercase		:Optional[Any]       							=				[
                                self.__matrix[i][j] + other.component(snake_case__  ,     snake_case__				)
                                for j in range(self.__width				)
                            ]
                            matrix.append(snake_case__				)
                     return Matrix(snake_case__  ,     self.__width  ,     self.__height				)
              else:
                     raise Exception('''matrix must have the same dimension!'''				)






       def __sub__(   self  :    Tuple  ,     snake_case__  :    Matrix				):

              '''simple docstring'''






              if self.__width == other.width() and self.__height == other.height():
                     lowercase		:Tuple       							=				[]
                     for i in range(self.__height				):
                            lowercase		:List[Any]       							=				[
                                self.__matrix[i][j] - other.component(snake_case__  ,     snake_case__				)
                                for j in range(self.__width				)
                            ]
                            matrix.append(snake_case__				)
                     return Matrix(snake_case__  ,     self.__width  ,     self.__height				)
              else:
                     raise Exception('''matrices must have the same dimension!'''				)






       @overload
       def __mul__(   self  :    int  ,     snake_case__  :    float				):

              '''simple docstring'''






              ...






       @overload
       def __mul__(   self  :    Optional[Any]  ,     snake_case__  :    Vector				):

              '''simple docstring'''






              ...






       def __mul__(   self  :    int  ,     snake_case__  :    float | Vector				):

              '''simple docstring'''






              if isinstance(snake_case__  ,     snake_case__				):  # matrix-vector
                     if len(snake_case__				) == self.__width:
                            lowercase		:Any       							=				zero_vector(self.__height				)
                            for i in range(self.__height				):
                                   lowercase		:Dict       							=				[
                                       self.__matrix[i][j] * other.component(snake_case__				)
                                       for j in range(self.__width				)
                                   ]
                                   ans.change_component(snake_case__  ,     sum(snake_case__				)				)
                            return ans
                     else:
                            raise Exception(
                                '''vector must have the same size as the '''
                                '''number of columns of the matrix!'''				)
              elif isinstance(snake_case__  ,     (int, float)				):  # matrix-scalar
                     lowercase		:Tuple       							=				[
                         [self.__matrix[i][j] * other for j in range(self.__width				)]
                         for i in range(self.__height				)
                     ]
                     return Matrix(snake_case__  ,     self.__width  ,     self.__height				)
              return None






       def  __snake_case						(   self  :    Union[str, Any]				):

              '''simple docstring'''






              return self.__height






       def  __snake_case						(   self  :    List[str]				):

              '''simple docstring'''






              return self.__width






       def  __snake_case						(   self  :    str  ,     snake_case__  :    int  ,     snake_case__  :    int				):

              '''simple docstring'''






              if 0 <= x < self.__height and 0 <= y < self.__width:
                     return self.__matrix[x][y]
              else:
                     raise Exception('''change_component: indices out of bounds'''				)






       def  __snake_case						(   self  :    str  ,     snake_case__  :    int  ,     snake_case__  :    int  ,     snake_case__  :    float				):

              '''simple docstring'''






              if 0 <= x < self.__height and 0 <= y < self.__width:
                     lowercase		:Tuple       							=				value
              else:
                     raise Exception('''change_component: indices out of bounds'''				)






       def  __snake_case						(   self  :    Tuple  ,     snake_case__  :    int  ,     snake_case__  :    int				):

              '''simple docstring'''






              if self.__height != self.__width:
                     raise Exception('''Matrix is not square'''				)
              lowercase		:Dict       							=				self.__matrix[:x] + self.__matrix[x + 1 :]
              for i in range(len(snake_case__				)				):
                     lowercase		:Optional[int]       							=				minor[i][:y] + minor[i][y + 1 :]
              return Matrix(snake_case__  ,     self.__width - 1  ,     self.__height - 1				).determinant()






       def  __snake_case						(   self  :    List[Any]  ,     snake_case__  :    int  ,     snake_case__  :    int				):

              '''simple docstring'''






              if self.__height != self.__width:
                     raise Exception('''Matrix is not square'''				)
              if 0 <= x < self.__height and 0 <= y < self.__width:
                     return (-1) ** (x + y) * self.minor(snake_case__  ,     snake_case__				)
              else:
                     raise Exception('''Indices out of bounds'''				)







       def  __snake_case						(   self  :    Tuple				):

              '''simple docstring'''






              if self.__height != self.__width:
                     raise Exception('''Matrix is not square'''				)
              if self.__height < 1:
                     raise Exception('''Matrix has no element'''				)
              elif self.__height == 1:
                     return self.__matrix[0][0]
              elif self.__height == 2:
                     return (
                         self.__matrix[0][0] * self.__matrix[1][1]
                         - self.__matrix[0][1] * self.__matrix[1][0]
                     )
              else:
                     lowercase		:str       							=				[
                         self.__matrix[0][y] * self.cofactor(0  ,     snake_case__				) for y in range(self.__width				)
                     ]
                     return sum(snake_case__				)







def       lowerCamelCase					(a_						:int)     ->							Matrix:
       lowercase		:list[list[float]]       							=				[[0] * n for _ in range(a_)]
       return Matrix(a_				,				a_				,				a_)






def       lowerCamelCase					(a_						:int				,				a_						:int				,				a_						:int				,				a_						:int)     ->							Matrix:
       random.seed(a_)
       lowercase		:list[list[float]]       							=				[
           [random.randint(a_				,				a_) for _ in range(a_)] for _ in range(a_)
       ]
       return Matrix(a_				,				a_				,				a_)





"""simple docstring"""



import unittest

import numpy as np






def       lowerCamelCase					(a_						:np.ndarray				,				a_						:np.ndarray				,				a_						:np.ndarray				,				a_						:np.ndarray | None = None				,				)     ->							np.ndarray:
       lowercase		:str       							=				np.shape(a_)
       lowercase		:Dict       							=				np.shape(a_)
       lowercase		:Optional[int]       							=				np.shape(a_)

       if shape_a[0] != shape_b[0]:
              lowercase		:List[str]       							=				(
                  '''Expected the same number of rows for A and B. '''
                  F"""Instead found A of size {shape_a} and B of size {shape_b}"""
              )
              raise ValueError(a_)

       if shape_b[1] != shape_c[1]:
              lowercase		:List[str]       							=				(
                  '''Expected the same number of columns for B and C. '''
                  F"""Instead found B of size {shape_b} and C of size {shape_c}"""
              )
              raise ValueError(a_)

       lowercase		:Optional[Any]       							=				pseudo_inv
       if a_inv is None:
              try:
                     lowercase		:int       							=				np.linalg.inv(a_)
              except np.linalg.LinAlgError:
                     raise ValueError(
                         '''Input matrix A is not invertible. Cannot compute Schur complement.''')

       return mat_c - mat_b.T @ a_inv @ mat_b
class 					__magic_name__		(			unittest.TestCase     ):






       def  __snake_case						(   self  :    int				):

              '''simple docstring'''






              lowercase		:Dict       							=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
              lowercase		:List[str]       							=				np.array([[0, 3], [3, 0], [2, 3]]				)
              lowercase		:List[str]       							=				np.array([[2, 1], [6, 3]]				)

              lowercase		:Any       							=				schur_complement(snake_case__  ,     snake_case__  ,     snake_case__				)

              lowercase		:int       							=				np.block([[a, b], [b.T, c]]				)

              lowercase		:Tuple       							=				np.linalg.det(snake_case__				)
              lowercase		:Any       							=				np.linalg.det(snake_case__				)
              lowercase		:List[Any]       							=				np.linalg.det(snake_case__				)

              self.assertAlmostEqual(snake_case__  ,     det_a * det_s				)






       def  __snake_case						(   self  :    Optional[int]				):

              '''simple docstring'''






              lowercase		:Union[str, Any]       							=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
              lowercase		:Union[str, Any]       							=				np.array([[0, 3], [3, 0], [2, 3]]				)
              lowercase		:List[str]       							=				np.array([[2, 1], [6, 3]]				)

              with self.assertRaises(snake_case__				):
                     schur_complement(snake_case__  ,     snake_case__  ,     snake_case__				)







       def  __snake_case						(   self  :    List[Any]				):

              '''simple docstring'''






              lowercase		:List[str]       							=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
              lowercase		:int       							=				np.array([[0, 3], [3, 0], [2, 3]]				)
              lowercase		:List[Any]       							=				np.array([[2, 1, 3], [6, 3, 5]]				)

              with self.assertRaises(snake_case__				):
                     schur_complement(snake_case__  ,     snake_case__  ,     snake_case__				)


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 unittest.main()

