


"""simple docstring"""



def       lowerCamelCase					(a_						:str)     ->							List[str]:
       if not head:
              return True
       # split the list to two parts
       lowercase		:Optional[int]       							=				head.next, head
       while fast and fast.next:
              lowercase		:Dict       							=				fast.next.next
              lowercase		:Union[str, Any]       							=				slow.next
       lowercase		:Union[str, Any]       							=				slow.next
       lowercase		:Union[str, Any]       							=				None  # Don't forget here! But forget still works!
       # reverse the second part
       lowercase		:Optional[int]       							=				None
       while second:
              lowercase		:Any       							=				second.next
              lowercase		:int       							=				node
              lowercase		:Optional[Any]       							=				second
              lowercase		:Any       							=				nxt
       # compare two parts
       # second part has the same or one less node
       while node:
              if node.val != head.val:
                     return False
              lowercase		:Tuple       							=				node.next
              lowercase		:Optional[int]       							=				head.next
       return True






def       lowerCamelCase					(a_						:Optional[Any])     ->							Tuple:
       if not head or not head.next:
              return True

       # 1. Get the midpoint (slow)
       lowercase		:List[Any]       							=				head
       while fast and fast.next:
              lowercase		:Any       							=				fast.next.next, slow.next

       # 2. Push the second half into the stack
       lowercase		:Optional[Any]       							=				[slow.val]
       while slow.next:
              lowercase		:Optional[int]       							=				slow.next
              stack.append(slow.val)

       # 3. Comparison
       while stack:
              if stack.pop() != cur.val:
                     return False
              lowercase		:Tuple       							=				cur.next

       return True






def       lowerCamelCase					(a_						:Any)     ->							List[str]:
       if not head or not head.next:
              return True
       lowercase		:Optional[int]       							=				{}
       lowercase		:Union[str, Any]       							=				0
       while head:
              if head.val in d:
                     d[head.val].append(snake_case_)
              else:
                     lowercase		:Optional[int]       							=				[pos]
              lowercase		:Dict       							=				head.next
              pos += 1
       lowercase		:Dict       							=				pos - 1
       lowercase		:str       							=				0
       for v in d.values():
              if len(snake_case_) % 2 != 0:
                     middle += 1
              else:
                     lowercase		:List[Any]       							=				0
                     for i in range(0				,				len(snake_case_)):
                            if v[i] + v[len(snake_case_) - 1 - step] != checksum:
                                   return False
                            step += 1
              if middle > 1:
                     return False
       return True





"""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()

