


'''simple docstring'''
from __future__ import annotations

from collections.abc import Callable

UpperCamelCase__          = list[list[float | int]]





def    a__			(   lowerCAmelCase__    ,   lowerCAmelCase__		)       ->      List[Any]:
    UpperCAmelCase__     :						int						=							len(snake_case__		)
    UpperCAmelCase__     :						Matrix						=							[[0 for _ in range(size + 1		)] for _ in range(snake_case__		)]
    UpperCAmelCase__     :						int
    UpperCAmelCase__     :						int
    UpperCAmelCase__     :						int
    UpperCAmelCase__     :						int
    UpperCAmelCase__     :						int
    UpperCAmelCase__     :						float

    for row in range(snake_case__		):
        for col in range(snake_case__		):
            UpperCAmelCase__     :						Union[str, Any]						=							matrix[row][col]

        UpperCAmelCase__     :						List[str]						=							vector[row][0]

    UpperCAmelCase__     :						Optional[Any]						=							0
    UpperCAmelCase__     :						Union[str, Any]						=							0
    while row < size and col < size:
        # pivoting
        UpperCAmelCase__     :						Optional[Any]						=							max((abs(augmented[rowa][col]		), rowa) for rowa in range(snake_case__    ,   snake_case__		)		)[
            1
        ]
        if augmented[pivot_row][col] == 0:
            col += 1
            continue
        else:
            UpperCAmelCase__     :						Any						=							augmented[pivot_row], augmented[row]

        for rowa in range(row + 1    ,   snake_case__		):
            UpperCAmelCase__     :						Optional[Any]						=							augmented[rowa][col] / augmented[row][col]
            UpperCAmelCase__     :						int						=							0
            for cola in range(col + 1    ,   size + 1		):
                augmented[rowa][cola] -= augmented[row][cola] * ratio

        row += 1
        col += 1

    # back substitution
    for col in range(1    ,   snake_case__		):
        for row in range(snake_case__		):
            UpperCAmelCase__     :						List[Any]						=							augmented[row][col] / augmented[col][col]
            for cola in range(snake_case__    ,   size + 1		):
                augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
    return [
        [round(augmented[row][size] / augmented[row][row]    ,   10		)] for row in range(snake_case__		)
    ]





def    a__			(   lowerCAmelCase__		)       ->      int:
    UpperCAmelCase__     :						int						=							len(snake_case__		)
    UpperCAmelCase__     :						Matrix						=							[[0 for _ in range(snake_case__		)] for _ in range(snake_case__		)]
    UpperCAmelCase__     :						Matrix						=							[[0] for _ in range(snake_case__		)]
    UpperCAmelCase__     :						Matrix
    UpperCAmelCase__     :						int
    UpperCAmelCase__     :						int
    UpperCAmelCase__     :						int

    for x_val, y_val in enumerate(snake_case__		):
        for col in range(snake_case__		):
            UpperCAmelCase__     :						List[str]						=							(x_val + 1) ** (size - col - 1)
        UpperCAmelCase__     :						List[str]						=							y_val

    UpperCAmelCase__     :						Union[str, Any]						=							solve(snake_case__    ,   snake_case__		)

    def interpolated_func(lowerCAmelCase__		) -> int:
        return sum(
            round(coeffs[x_val][0]		) * (var ** (size - x_val - 1))
            for x_val in range(snake_case__		)		)

    return interpolated_func





def    a__			(   lowerCAmelCase__		)       ->      Tuple:
    return (
        1
        - variable
        + variable**2
        - variable**3
        + variable**4
        - variable**5
        + variable**6
        - variable**7
        + variable**8
        - variable**9
        + variable**10
    )





def    a__			(   lowerCAmelCase__ = question_function    ,   lowerCAmelCase__ = 10		)       ->      str:
    UpperCAmelCase__     :						list[int]						=							[func(snake_case__		) for x_val in range(1    ,   order + 1		)]

    UpperCAmelCase__     :						list[Callable[[int], int]]						=							[
        interpolate(data_points[:max_coeff]		) for max_coeff in range(1    ,   order + 1		)
    ]

    UpperCAmelCase__     :						int						=							0
    UpperCAmelCase__     :						Callable[[int], int]
    UpperCAmelCase__     :						int

    for poly in polynomials:
        UpperCAmelCase__     :						Dict						=							1
        while func(snake_case__		) == poly(snake_case__		):
            x_val += 1

        ret += poly(snake_case__		)

    return ret


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


'''simple docstring'''




import numpy

# List of input, output pairs
lowercase     :							Any       						=     (
    ((5, 2, 3), 15),
    ((6, 5, 9), 25),
    ((11, 12, 13), 41),
    ((1, 1, 1), 8),
    ((11, 12, 13), 41),
)
lowercase     :							str       						=     (((5_15, 22, 13), 5_55), ((61, 35, 49), 1_50))
lowercase     :							Union[str, Any]       						=     [2, 4, 1, 5]
lowercase     :							Any       						=     len(train_data)
lowercase     :							Optional[int]       						=     0.0_09







def 						lowerCAmelCase_							(  snake_case__						,							snake_case__="train"		):




    '''simple docstring'''



    return calculate_hypothesis_value(snake_case__						,							snake_case__		) - output(
        snake_case__						,							snake_case__		)







def 						lowerCAmelCase_							(  snake_case__		):




    '''simple docstring'''



    A			:  Dict   =		0
    for i in range(len(snake_case__		) - 1		):
        hyp_val += data_input_tuple[i] * parameter_vector[i + 1]
    hyp_val += parameter_vector[0]
    return hyp_val







def 						lowerCAmelCase_							(  snake_case__						,							snake_case__		):




    '''simple docstring'''



    if data_set == "train":
        return train_data[example_no][1]
    elif data_set == "test":
        return test_data[example_no][1]
    return None







def 						lowerCAmelCase_							(  snake_case__						,							snake_case__		):




    '''simple docstring'''



    if data_set == "train":
        return _hypothesis_value(train_data[example_no][0]		)
    elif data_set == "test":
        return _hypothesis_value(test_data[example_no][0]		)
    return None







def 						lowerCAmelCase_							(  snake_case__						,							snake_case__=m		):




    '''simple docstring'''



    A			:  List[Any]   =		0
    for i in range(snake_case__		):
        if index == -1:
            summation_value += _error(snake_case__		)
        else:
            summation_value += _error(snake_case__		) * train_data[i][0][index]
    return summation_value







def 						lowerCAmelCase_							(  snake_case__		):




    '''simple docstring'''



    A			:  List[Any]   =		summation_of_cost_derivative(snake_case__						,							snake_case__		) / m
    return cost_derivative_value







def 						lowerCAmelCase_							(  ):




    '''simple docstring'''



    global parameter_vector
    # Tune these values to set a tolerance value for predicted output
    A			:  Dict   =		0.00_00_02
    A			:  Optional[Any]   =		0
    A			:  int   =		0
    while True:
        j += 1
        A			:  List[str]   =		[0, 0, 0, 0]
        for i in range(0						,							len(snake_case__		)		):
            A			:  Union[str, Any]   =		get_cost_derivative(i - 1		)
            A			:  Tuple   =		(
                parameter_vector[i] - LEARNING_RATE * cost_derivative
            )
        if numpy.allclose(
            snake_case__						,							snake_case__						,							atol=snake_case__						,							rtol=snake_case__						,							):
            break
        A			:  List[Any]   =		temp_parameter_vector
    print(('''Number of iterations:''', j)		)







def 						lowerCAmelCase_							(  ):




    '''simple docstring'''



    for i in range(len(snake_case__		)		):
        print(('''Actual output value:''', output(snake_case__						,							'''test'''		))		)
        print(('''Hypothesis output:''', calculate_hypothesis_value(snake_case__						,							'''test'''		))		)


if __name__ == "__main__":
      run_gradient_descent()
      print('\nTesting gradient descent for a linear hypothesis function.\n')
      test_gradient_descent()




