from __future__ import annotations

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


def    a__       (						__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   ,					):
      SCREAMING_SNAKE_CASE_   ,       SCREAMING_SNAKE_CASE_      =     coefficient_matrix.shape
      SCREAMING_SNAKE_CASE_   ,       SCREAMING_SNAKE_CASE_      =     constant_matrix.shape

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

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

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

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

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

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

      SCREAMING_SNAKE_CASE_   ,       SCREAMING_SNAKE_CASE_      =     table.shape

      strictly_diagonally_dominant(__UpperCamelCase )

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

      return [float(__UpperCamelCase ) for i in new_val]


def    a__       (						__UpperCamelCase ):
      SCREAMING_SNAKE_CASE_   ,       SCREAMING_SNAKE_CASE_      =     table.shape

      SCREAMING_SNAKE_CASE_      =     True

      for i in range(0   ,					__UpperCamelCase ):
            SCREAMING_SNAKE_CASE_      =     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()



def    a__       (						__UpperCamelCase = 1_0_0_0 ):
      SCREAMING_SNAKE_CASE_      =     -1
      SCREAMING_SNAKE_CASE_      =     0
      for a in range(1   ,					n // 3 ):
            # Solving the two equations a**2+b**2=c**2 and a+b+c=N eliminating c
            SCREAMING_SNAKE_CASE_      =     (n * n - 2 * a * n) // (2 * n - 2 * a)
            SCREAMING_SNAKE_CASE_      =     n - a - b
            if c * c == (a * a + b * b):
                  SCREAMING_SNAKE_CASE_      =     a * b * c
                  if candidate >= product:
                        SCREAMING_SNAKE_CASE_      =     candidate
      return product


if __name__ == "__main__":
      print(f"{solution() = }")


