




from __future__ import annotations

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







def 			_a   (		SCREAMING_SNAKE_CASE_							:       NDArray[floataa]   ,   SCREAMING_SNAKE_CASE_							:       NDArray[floataa]   ,   SCREAMING_SNAKE_CASE_							:       list[int]   ,   SCREAMING_SNAKE_CASE_							:       int   ,   ):
					__lowerCAmelCase     ,							__lowerCAmelCase            = coefficient_matrix.shape
					__lowerCAmelCase     ,							__lowerCAmelCase            = constant_matrix.shape

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

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

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

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

					__lowerCAmelCase     ,							__lowerCAmelCase            = table.shape

					strictly_diagonally_dominant(SCREAMING_SNAKE_CASE_  )

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

					return [float(SCREAMING_SNAKE_CASE_  ) for i in new_val]







def 			_a   (		SCREAMING_SNAKE_CASE_							:       NDArray[floataa]  ):
					__lowerCAmelCase     ,							__lowerCAmelCase            = table.shape

					__lowerCAmelCase            = True

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










from math import sqrt







def 			_a   (		SCREAMING_SNAKE_CASE_							:       int  ):

					if 1 < number < 4:
										# 2 and 3 are primes
										return True
					elif number < 2 or number % 2 == 0 or number % 3 == 0:
										# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
										return False

					# All primes number are in format of 6k +/- 1
					for i in range(5   ,   int(sqrt(SCREAMING_SNAKE_CASE_  ) + 1  )   ,   6  ):
										if number % i == 0 or number % (i + 2) == 0:
															return False
					return True







def 			_a   (		SCREAMING_SNAKE_CASE_							:       int = 1_00_01  ):
					__lowerCAmelCase            = 0
					__lowerCAmelCase            = 1
					while count != nth and number < 3:
										number += 1
										if is_prime(SCREAMING_SNAKE_CASE_  ):
															count += 1
					while count != nth:
										number += 2
										if is_prime(SCREAMING_SNAKE_CASE_  ):
															count += 1
					return number


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




