


from __future__ import annotations

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



def 				UpperCamelCase_(      lowerCamelCase_							, lowerCamelCase_							, lowerCamelCase_							, lowerCamelCase_							, )    ->						list[float]:
    _lowercase		,				_lowercase       :							Union[str, Any]      		=		coefficient_matrix.shape
    _lowercase		,				_lowercase       :							Any      		=		constant_matrix.shape

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

    if colsa != 1:
        _lowercase       :							List[str]      		=		F'''Constant matrix must be nx1 but received {rowsa}x{colsa}'''
        raise ValueError(lowerCamelCase_ )

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

    if len(lowerCamelCase_ ) != rowsa:
        _lowercase       :							Optional[Any]      		=		(
            'Number of initial values must be equal to number of rows in coefficient '
            F'''matrix but received {len(lowerCamelCase_ )} and {rowsa}'''
        )
        raise ValueError(lowerCamelCase_ )

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

    _lowercase       :							NDArray[floataa]      		=		np.concatenate(
        (coefficient_matrix, constant_matrix)							, axis=1 )

    _lowercase		,				_lowercase       :							Optional[int]      		=		table.shape

    strictly_diagonally_dominant(lowerCamelCase_ )

    # Iterates the whole matrix for given number of times
    for _ in range(lowerCamelCase_ ):
        _lowercase       :							List[str]      		=		[]
        for row in range(lowerCamelCase_ ):
            _lowercase       :							List[str]      		=		0
            for col in range(lowerCamelCase_ ):
                if col == row:
                    _lowercase       :							Union[str, Any]      		=		table[row][col]
                elif col == cols - 1:
                    _lowercase       :							Optional[int]      		=		table[row][col]
                else:
                    temp += (-1) * table[row][col] * init_val[col]
            _lowercase       :							Union[str, Any]      		=		(temp + val) / denom
            new_val.append(lowerCamelCase_ )
        _lowercase       :							Any      		=		new_val

    return [float(lowerCamelCase_ ) for i in new_val]



def 				UpperCamelCase_(      lowerCamelCase_ )    ->						bool:
    _lowercase		,				_lowercase       :							Any      		=		table.shape

    _lowercase       :							List[Any]      		=		True

    for i in range(0							, lowerCamelCase_ ):
        _lowercase       :							Optional[Any]      		=		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 __future__ import annotations

from math import ceil, floor, sqrt



def 				UpperCamelCase_(      lowerCamelCase_ = 200_0000 )    ->						int:
    _lowercase       :							list[int]      		=		[0]
    _lowercase       :							int

    for idx in range(1							, ceil(sqrt(target * 2 ) * 1.1 ) ):
        triangle_numbers.append(triangle_numbers[-1] + idx )

    # we want this to be as close as possible to target
    _lowercase       :							int      		=		0
    # the area corresponding to the grid that gives the product closest to target
    _lowercase       :							int      		=		0
    # an estimate of b, using the quadratic formula
    _lowercase       :							float
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the triangle number corresponding to b_floor
    _lowercase       :							int
    # the triangle number corresponding to b_ceil
    _lowercase       :							int

    for idx_a, triangle_a in enumerate(triangle_numbers[1:]							, 1 ):
        _lowercase       :							Optional[int]      		=		(-1 + sqrt(1 + 8 * target / triangle_a )) / 2
        _lowercase       :							List[str]      		=		floor(lowerCamelCase_ )
        _lowercase       :							Dict      		=		ceil(lowerCamelCase_ )
        _lowercase       :							List[str]      		=		triangle_numbers[b_floor]
        _lowercase       :							List[str]      		=		triangle_numbers[b_ceil]

        if abs(target - triangle_b_first_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Union[str, Any]      		=		triangle_b_first_guess * triangle_a
            _lowercase       :							Union[str, Any]      		=		idx_a * b_floor

        if abs(target - triangle_b_second_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Any      		=		triangle_b_second_guess * triangle_a
            _lowercase       :							Optional[Any]      		=		idx_a * b_ceil

    return area


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


