






"""simple docstring"""





from __future__ import annotations

import numpy as np
from numpy import floataa
from numpy.typing import NDArray







def   lowerCAmelCase_( SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE ,	)					->	list[float]:
    """simple docstring"""

    UpperCamelCase__       ,						UpperCamelCase__         =    coefficient_matrix.shape
    UpperCamelCase__       ,						UpperCamelCase__         =    constant_matrix.shape

    if rowsa != colsa:
        UpperCamelCase__         =    F'Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}'
        raise ValueError(SCREAMING_SNAKE_CASE					)

    if colsa != 1:
        UpperCamelCase__         =    F'Constant matrix must be nx1 but received {rowsa}x{colsa}'
        raise ValueError(SCREAMING_SNAKE_CASE					)

    if rowsa != rowsa:
        UpperCamelCase__         =    (
            'Coefficient and constant matrices dimensions must be nxn and nx1 but '
            F'received {rowsa}x{colsa} and {rowsa}x{colsa}'
        )
        raise ValueError(SCREAMING_SNAKE_CASE					)

    if len(SCREAMING_SNAKE_CASE					) != rowsa:
        UpperCamelCase__         =    (
            'Number of initial values must be equal to number of rows in coefficient '
            F'matrix but received {len(SCREAMING_SNAKE_CASE					)} and {rowsa}'
        )
        raise ValueError(SCREAMING_SNAKE_CASE					)

    if iterations <= 0:
        raise ValueError('Iterations must be at least 1'					)

    UpperCamelCase__         =    np.concatenate(
        (coefficient_matrix, constant_matrix) ,	axis=1					)

    UpperCamelCase__       ,						UpperCamelCase__         =    table.shape

    strictly_diagonally_dominant(SCREAMING_SNAKE_CASE					)

    # Iterates the whole matrix for given number of times
    for _ in range(SCREAMING_SNAKE_CASE					):
        UpperCamelCase__         =    []
        for row in range(SCREAMING_SNAKE_CASE					):
            UpperCamelCase__         =    0
            for col in range(SCREAMING_SNAKE_CASE					):
                if col == row:
                    UpperCamelCase__         =    table[row][col]
                elif col == cols - 1:
                    UpperCamelCase__         =    table[row][col]
                else:
                    temp += (-1) * table[row][col] * init_val[col]
            UpperCamelCase__         =    (temp + val) / denom
            new_val.append(SCREAMING_SNAKE_CASE					)
        UpperCamelCase__         =    new_val

    return [float(SCREAMING_SNAKE_CASE					) for i in new_val]







def   lowerCAmelCase_( SCREAMING_SNAKE_CASE					)					->	bool:
    """simple docstring"""

    UpperCamelCase__       ,						UpperCamelCase__         =    table.shape

    UpperCamelCase__         =    True

    for i in range(0 ,	SCREAMING_SNAKE_CASE					):
        UpperCamelCase__         =    0
        for j in range(0 ,	cols - 1					):
            if i == j:
                continue
            else:
                total += table[i][j]

        if table[i][i] <= total:
            raise ValueError('Coefficient matrix is not strictly diagonally dominant'					)

    return is_diagonally_dominant


# Test Cases
if __name__ == "__main__":
  import doctest

  doctest.testmod()












"""simple docstring"""





from __future__ import annotations

from typing import Generic, TypeVar

A__					: Any=	TypeVar("""T""")
class    __lowerCamelCase    (  Generic[T]  ):

   def __init__(					self ,				snake_case_      )		->       None:
       UpperCamelCase__         =    data
       UpperCamelCase__         =    self
       UpperCamelCase__         =    0







class    __lowerCamelCase    (  Generic[T]  ):

   def __init__(					self      )		->       None:
       # map from node name to the node object
       UpperCamelCase__         =    {}

   def 						SCREAMING_SNAKE_CASE__ (					self ,				snake_case_      )		->       None:
       # create a new set with x as its member
       UpperCamelCase__         =    DisjointSetTreeNode(snake_case_      )

   def 						SCREAMING_SNAKE_CASE__ (					self ,				snake_case_      )		->       DisjointSetTreeNode[T]:
       # find the set x belongs to (with path-compression)
       UpperCamelCase__         =    self.map[data]
       if elem_ref != elem_ref.parent:
           UpperCamelCase__         =    self.find_set(elem_ref.parent.data      )
       return elem_ref.parent

   def 						SCREAMING_SNAKE_CASE__ (					self ,				snake_case_ ,				snake_case_      )		->       None:
       # helper function for union operation
       if nodea.rank > nodea.rank:
           UpperCamelCase__         =    nodea
       else:
           UpperCamelCase__         =    nodea
           if nodea.rank == nodea.rank:
               nodea.rank += 1







   def 						SCREAMING_SNAKE_CASE__ (					self ,				snake_case_ ,				snake_case_      )		->       None:
       # merge 2 disjoint sets
       self.link(self.find_set(snake_case_      ) ,				self.find_set(snake_case_      )      )







class    __lowerCamelCase    (  Generic[T]  ):

   def __init__(					self      )		->       None:
       # connections: map from the node to the neighbouring nodes (with weights)
       UpperCamelCase__         =    {}

   def 						SCREAMING_SNAKE_CASE__ (					self ,				snake_case_      )		->       None:
       # add a node ONLY if its not present in the graph
       if node not in self.connections:
           UpperCamelCase__         =    {}

   def 						SCREAMING_SNAKE_CASE__ (					self ,				snake_case_ ,				snake_case_ ,				snake_case_      )		->       None:
       # add an edge with the given weight
       self.add_node(snake_case_      )
       self.add_node(snake_case_      )
       UpperCamelCase__         =    weight
       UpperCamelCase__         =    weight







   def 						SCREAMING_SNAKE_CASE__ (					self      )		->       GraphUndirectedWeighted[T]:
       UpperCamelCase__         =    []
       UpperCamelCase__         =    set()
       for start in self.connections:
           for end in self.connections[start]:
               if (start, end) not in seen:
                   seen.add((end, start)      )
                   edges.append((start, end, self.connections[start][end])      )
       edges.sort(key=lambda snake_case_      : x[2]      )

       # creating the disjoint set
       UpperCamelCase__         =    DisjointSetTree[T]()
       for node in self.connections:
           disjoint_set.make_set(snake_case_      )

       # MST generation
       UpperCamelCase__         =    0
       UpperCamelCase__         =    0
       UpperCamelCase__         =    GraphUndirectedWeighted[T]()
       while num_edges < len(self.connections      ) - 1:
           UpperCamelCase__       ,						UpperCamelCase__       ,						UpperCamelCase__         =    edges[index]
           index += 1
           UpperCamelCase__         =    disjoint_set.find_set(snake_case_      )
           UpperCamelCase__         =    disjoint_set.find_set(snake_case_      )
           if parent_u != parent_v:
               num_edges += 1
               graph.add_edge(snake_case_ ,				snake_case_ ,				snake_case_      )
               disjoint_set.union(snake_case_ ,				snake_case_      )
       return graph




