


"""simple docstring"""
from __future__ import annotations

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







def   _a   ( _SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			,	)      ->			list[float]:
							snake_case_    ,  snake_case_          =						coefficient_matrix.shape
							snake_case_    ,  snake_case_          =						constant_matrix.shape

							if rowsa != colsa:
														snake_case_          =						f"""Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}"""
														raise ValueError(_SCREAMING_SNAKE_CASE			)

							if colsa != 1:
														snake_case_          =						f"""Constant matrix must be nx1 but received {rowsa}x{colsa}"""
														raise ValueError(_SCREAMING_SNAKE_CASE			)

							if rowsa != rowsa:
														snake_case_          =						(
														    """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:
														snake_case_          =						(
														    """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"""			)

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

							snake_case_    ,  snake_case_          =						table.shape

							strictly_diagonally_dominant(_SCREAMING_SNAKE_CASE			)

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

							return [float(_SCREAMING_SNAKE_CASE			) for i in new_val]







def   _a   ( _SCREAMING_SNAKE_CASE			)      ->			bool:
							snake_case_    ,  snake_case_          =						table.shape

							snake_case_          =						True

							for i in range(0			,	_SCREAMING_SNAKE_CASE			):
														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()







"""simple docstring"""
from collections import defaultdict
from math import gcd







def   _a			( _SCREAMING_SNAKE_CASE = 1_500_000			)      ->			int:
							snake_case_          =						defaultdict(_SCREAMING_SNAKE_CASE			)
							snake_case_          =						2
							while 2 * euclid_m * (euclid_m + 1) <= limit:
														for euclid_n in range((euclid_m % 2) + 1			,	_SCREAMING_SNAKE_CASE			,	2			):
																					if gcd(_SCREAMING_SNAKE_CASE			,	_SCREAMING_SNAKE_CASE			) > 1:
																												continue
																					snake_case_          =						2 * euclid_m * (euclid_m + euclid_n)
																					for perimeter in range(_SCREAMING_SNAKE_CASE			,	limit + 1			,	_SCREAMING_SNAKE_CASE			):
																												frequencies[perimeter] += 1
														euclid_m += 1
							return sum(1 for frequency in frequencies.values() if frequency == 1			)


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



