


def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					list[list]				):
     __UpperCamelCase    							=current_set.copy()
     for row_index, row in enumerate(SCREAMING_SNAKE_CASE__				):
          __UpperCamelCase    							=row[0]
          for column_index, column in enumerate(SCREAMING_SNAKE_CASE__				):
               if magnitude == 0:
                    __UpperCamelCase    							=column
                    continue
               __UpperCamelCase    							=column / magnitude
    # Subtract to cancel term
     __UpperCamelCase    							=current_set[0]
     __UpperCamelCase    							=[first_row]
     __UpperCamelCase    							=current_set[1::]
     for row in current_set:
          __UpperCamelCase    							=[]
          # If first term is 0, it is already in form we want, so we preserve it
          if row[0] == 0:
               final_set.append(SCREAMING_SNAKE_CASE__				)
               continue
          for column_index in range(len(SCREAMING_SNAKE_CASE__				)				):
               temp_row.append(first_row[column_index] - row[column_index]				)
          final_set.append(SCREAMING_SNAKE_CASE__				)
     # Create next recursion iteration set
     if len(final_set[0]				) != 3:
          __UpperCamelCase    							=final_set[0]
          __UpperCamelCase    							=[]
          __UpperCamelCase    							=[]
          for row in final_set[1::]:
               current_first_column.append(row[0]				)
               next_iteration.append(row[1::]				)
          __UpperCamelCase    							=simplify(SCREAMING_SNAKE_CASE__				)
          for i in range(len(SCREAMING_SNAKE_CASE__				)				):
               resultant[i].insert(0			,     current_first_column[i]				)
          resultant.insert(0			,     SCREAMING_SNAKE_CASE__				)
          __UpperCamelCase    							=resultant
     return final_set

def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					list[list]				):
     if len(SCREAMING_SNAKE_CASE__				) == 0:
          raise IndexError('solve_simultaneous() requires n lists of length n+1'				)
     __UpperCamelCase    							=len(SCREAMING_SNAKE_CASE__				) + 1
     if any(len(SCREAMING_SNAKE_CASE__				) != _length for item in equations				):
          raise IndexError('solve_simultaneous() requires n lists of length n+1'				)
     for row in equations:
          if any(not isinstance(SCREAMING_SNAKE_CASE__			,     (int, float)				) for column in row				):
               raise ValueError('solve_simultaneous() requires lists of integers'				)
     if len(SCREAMING_SNAKE_CASE__				) == 1:
          return [equations[0][-1] / equations[0][0]]
     __UpperCamelCase    							=equations.copy()
     if any(0 in row for row in data_set				):
          __UpperCamelCase    							=data_set.copy()
          __UpperCamelCase    							=[]
          for row_index, row in enumerate(SCREAMING_SNAKE_CASE__				):
               if 0 not in row:
                    __UpperCamelCase    							=data_set.pop(SCREAMING_SNAKE_CASE__				)
                    break
          if not full_row:
               raise ValueError('solve_simultaneous() requires at least 1 full equation'				)
          data_set.insert(0			,     SCREAMING_SNAKE_CASE__				)
     __UpperCamelCase    							=data_set.copy()
     __UpperCamelCase    							=simplify(SCREAMING_SNAKE_CASE__				)
     __UpperCamelCase    							=simplified[::-1]
     __UpperCamelCase    							=[]
     for row in simplified:
          __UpperCamelCase    							=row[-1]
          if not solutions:
               if row[-2] == 0:
                    solutions.append(0				)
                    continue
               solutions.append(current_solution / row[-2]				)
               continue
          __UpperCamelCase    							=row.copy()[: len(SCREAMING_SNAKE_CASE__				) - 1 :]
          while temp_row[0] == 0:
               temp_row.pop(0				)
          if len(SCREAMING_SNAKE_CASE__				) == 0:
               solutions.append(0				)
               continue
          __UpperCamelCase    							=temp_row[1::]
          __UpperCamelCase    							=temp_row[::-1]
          for column_index, column in enumerate(SCREAMING_SNAKE_CASE__				):
               current_solution -= column * solutions[column_index]
          solutions.append(SCREAMING_SNAKE_CASE__				)
     __UpperCamelCase    							=[]
     for item in solutions:
          final.append(float(round(SCREAMING_SNAKE_CASE__			,     5				)				)				)
     return final[::-1]


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   _A		       =						[
       [2, 1, 1, 1, 1, 4],
       [1, 2, 1, 1, 1, 5],
       [1, 1, 2, 1, 1, 6],
       [1, 1, 1, 2, 1, 7],
       [1, 1, 1, 1, 2, 8],
   ]
   print(solve_simultaneous(eq))
   print(solve_simultaneous([[4, 2]]))









from __future__ import annotations

from math import pi
from typing import Protocol

import matplotlib.pyplot as plt
import numpy as np






class     UpperCAmelCase__       (					A_		):



    """simple docstring"""






    def _a					(			self					,   A_    )  ->       float:
         return 0.0

def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					np.ndarray			,     SCREAMING_SNAKE_CASE__			:					int				):
     __UpperCamelCase    							=min([-20, np.min(fft_results[1 : samplerate // 2 - 1]				)]				)
     __UpperCamelCase    							=max([20, np.max(fft_results[1 : samplerate // 2 - 1]				)]				)
     return lowest, highest

def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					FilterType			,     SCREAMING_SNAKE_CASE__			:					int				):
     __UpperCamelCase    							=5_12
     __UpperCamelCase    							=[1] + [0] * (size - 1)
     __UpperCamelCase    							=[filter_type.process(SCREAMING_SNAKE_CASE__				) for item in inputs]

     __UpperCamelCase    							=[0] * (samplerate - size)  # zero-padding
     outputs += filler
     __UpperCamelCase    							=np.abs(np.fft.fft(SCREAMING_SNAKE_CASE__				)				)
     __UpperCamelCase    							=20 * np.logaa(SCREAMING_SNAKE_CASE__				)

     # Frequencies on log scale from 24 to nyquist frequency
     plt.xlim(24			,     samplerate / 2 - 1				)
     plt.xlabel('Frequency (Hz)'				)
     plt.xscale('log'				)

     # Display within reasonable bounds
     __UpperCamelCase    							=get_bounds(SCREAMING_SNAKE_CASE__			,     SCREAMING_SNAKE_CASE__				)
     plt.ylim(max([-80, bounds[0]]				)			,     min([80, bounds[1]]				)				)
     plt.ylabel('Gain (dB)'				)

     plt.plot(SCREAMING_SNAKE_CASE__				)
     plt.show()





def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					FilterType			,     SCREAMING_SNAKE_CASE__			:					int				):
     __UpperCamelCase    							=5_12
     __UpperCamelCase    							=[1] + [0] * (size - 1)
     __UpperCamelCase    							=[filter_type.process(SCREAMING_SNAKE_CASE__				) for item in inputs]

     __UpperCamelCase    							=[0] * (samplerate - size)  # zero-padding
     outputs += filler
     __UpperCamelCase    							=np.angle(np.fft.fft(SCREAMING_SNAKE_CASE__				)				)

     # Frequencies on log scale from 24 to nyquist frequency
     plt.xlim(24			,     samplerate / 2 - 1				)
     plt.xlabel('Frequency (Hz)'				)
     plt.xscale('log'				)

     plt.ylim(-2 * pi			,     2 * pi				)
     plt.ylabel('Phase shift (Radians)'				)
     plt.plot(np.unwrap(SCREAMING_SNAKE_CASE__			,     -2 * pi				)				)
     plt.show()





