





from typing import Any

import numpy as np


def      SCREAMING_SNAKE_CASE      (   lowerCAmelCase       ):
       assert np.shape(lowerCAmelCase       )[0] == np.shape(lowerCAmelCase       )[1]

       # If matrix not symmetric, exit right away.
       if np.allclose(lowerCAmelCase     ,							matrix.T       ) is False:
              return False

       # Get eigenvalues and eignevectors for a symmetric matrix.
       _UpperCamelCase							,		_UpperCamelCase														=    np.linalg.eigh(lowerCAmelCase       )

       # Check sign of all eigenvalues.
       # np.all returns a value of type np.bool_
       return bool(np.all(eigen_values > 0       )       )


def      SCREAMING_SNAKE_CASE      (   lowerCAmelCase       ):
       _UpperCamelCase														=    np.random.randn(lowerCAmelCase     ,							lowerCAmelCase       )
       _UpperCamelCase														=    np.dot(lowerCAmelCase     ,							random_matrix.T       )
       assert _is_matrix_spd(lowerCAmelCase       )
       return spd_matrix


def      SCREAMING_SNAKE_CASE      (   lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase = 1_000     ,							lowerCAmelCase = 1E-8     ,							):
       assert np.shape(lowerCAmelCase       )[0] == np.shape(lowerCAmelCase       )[1]
       assert np.shape(lowerCAmelCase       )[0] == np.shape(lowerCAmelCase       )[0]
       assert _is_matrix_spd(lowerCAmelCase       )

       # Initialize solution guess, residual, search direction.
       _UpperCamelCase														=    np.zeros((np.shape(lowerCAmelCase       )[0], 1)       )
       _UpperCamelCase														=    np.copy(lowerCAmelCase       )
       _UpperCamelCase														=    np.copy(lowerCAmelCase       )

       # Set initial errors in solution guess and residual.
       _UpperCamelCase														=    1E9
       _UpperCamelCase														=    1E9
       _UpperCamelCase														=    1E9

       # Set iteration counter to threshold number of iterations.
       _UpperCamelCase														=    0

       while error > tol:
              # Save this value so we only calculate the matrix-vector product once.
              _UpperCamelCase														=    np.dot(lowerCAmelCase     ,							lowerCAmelCase       )

              # The main algorithm.

              # Update search direction magnitude.
              _UpperCamelCase														=    np.dot(ra.T     ,							lowerCAmelCase       ) / np.dot(pa.T     ,							lowerCAmelCase       )
              # Update solution guess.
              _UpperCamelCase														=    xa + alpha * pa
              # Calculate new residual.
              _UpperCamelCase														=    ra - alpha * w
              # Calculate new Krylov subspace scale.
              _UpperCamelCase														=    np.dot(r.T     ,							lowerCAmelCase       ) / np.dot(ra.T     ,							lowerCAmelCase       )
              # Calculate new A conjuage search direction.
              _UpperCamelCase														=    r + beta * pa

              # Calculate errors.
              _UpperCamelCase														=    np.linalg.norm(r - ra       )
              _UpperCamelCase														=    np.linalg.norm(x - xa       )
              _UpperCamelCase														=    np.maximum(lowerCAmelCase     ,							lowerCAmelCase       )

              # Update variables.
              _UpperCamelCase														=    np.copy(lowerCAmelCase       )
              _UpperCamelCase														=    np.copy(lowerCAmelCase       )
              _UpperCamelCase														=    np.copy(lowerCAmelCase       )

              # Update number of iterations.
              iterations += 1
              if iterations > max_iterations:
                     break

       return x


def      SCREAMING_SNAKE_CASE      (   ):
       _UpperCamelCase														=    3
       _UpperCamelCase														=    _create_spd_matrix(lowerCAmelCase       )
       _UpperCamelCase														=    np.random.randn(lowerCAmelCase     ,							1       )
       _UpperCamelCase														=    np.dot(lowerCAmelCase     ,							lowerCAmelCase       )

       # Numpy solution.
       _UpperCamelCase														=    np.linalg.solve(lowerCAmelCase     ,							lowerCAmelCase       )

       # Our implementation.
       _UpperCamelCase														=    conjugate_gradient(lowerCAmelCase     ,							lowerCAmelCase       )

       # Ensure both solutions are close to x_true (and therefore one another).
       assert np.linalg.norm(x_numpy - x_true       ) <= 1E-6
       assert np.linalg.norm(x_conjugate_gradient - x_true       ) <= 1E-6


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       test_conjugate_gradient()













import numpy as np
import requests


def      SCREAMING_SNAKE_CASE      (   ):
       _UpperCamelCase														=    requests.get(
           '''https://raw.githubusercontent.com/yashLadha/The_Math_of_Intelligence/'''
           '''master/Week1/ADRvsRating.csv'''       )
       _UpperCamelCase														=    response.text.splitlines()
       _UpperCamelCase														=    []
       for item in lines:
              _UpperCamelCase														=    item.split(''','''       )
              data.append(lowerCAmelCase       )
       data.pop(0       )  # This is for removing the labels from the list
       _UpperCamelCase														=    np.matrix(lowerCAmelCase       )
       return dataset


def      SCREAMING_SNAKE_CASE      (   lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase       ):
       _UpperCamelCase														=    len_data

       _UpperCamelCase														=    np.dot(lowerCAmelCase     ,							data_x.transpose()       )
       prod -= data_y.transpose()
       _UpperCamelCase														=    np.dot(lowerCAmelCase     ,							lowerCAmelCase       )
       _UpperCamelCase														=    theta - (alpha / n) * sum_grad
       return theta


def      SCREAMING_SNAKE_CASE      (   lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase       ):
       _UpperCamelCase														=    np.dot(lowerCAmelCase     ,							data_x.transpose()       )
       prod -= data_y.transpose()
       _UpperCamelCase														=    np.sum(np.square(lowerCAmelCase       )       )
       _UpperCamelCase														=    sum_elem / (2 * len_data)
       return error


def      SCREAMING_SNAKE_CASE      (   lowerCAmelCase     ,							lowerCAmelCase       ):
       _UpperCamelCase														=    100_000
       _UpperCamelCase														=    0.0_0_0_1_5_5_0

       _UpperCamelCase														=    data_x.shape[1]
       _UpperCamelCase														=    data_x.shape[0] - 1

       _UpperCamelCase														=    np.zeros((1, no_features)       )

       for i in range(0     ,							lowerCAmelCase       ):
              _UpperCamelCase														=    run_steep_gradient_descent(lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase       )
              _UpperCamelCase														=    sum_of_square_error(lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase     ,							lowerCAmelCase       )
              print(f'''At Iteration {i + 1} - Error is {error:.5f}'''       )

       return theta


def      SCREAMING_SNAKE_CASE      (   lowerCAmelCase     ,							lowerCAmelCase       ):
       _UpperCamelCase														=    sum(abs(y - predicted_y[i]       ) for i, y in enumerate(lowerCAmelCase       )       )
       return total / len(lowerCAmelCase       )


def      SCREAMING_SNAKE_CASE      (   ):
       _UpperCamelCase														=    collect_dataset()

       _UpperCamelCase														=    data.shape[0]
       _UpperCamelCase														=    np.c_[np.ones(lowerCAmelCase       ), data[:, :-1]].astype(lowerCAmelCase       )
       _UpperCamelCase														=    data[:, -1].astype(lowerCAmelCase       )

       _UpperCamelCase														=    run_linear_regression(lowerCAmelCase     ,							lowerCAmelCase       )
       _UpperCamelCase														=    theta.shape[1]
       print('''Resultant Feature vector : '''       )
       for i in range(0     ,							lowerCAmelCase       ):
              print(f'''{theta[0, i]:.5f}'''       )


if __name__ == "__main__":
       main()






