linear_algebra.gaussian_elimination
===================================

.. py:module:: linear_algebra.gaussian_elimination

.. autoapi-nested-parse::

   | Gaussian elimination method for solving a system of linear equations.
   | Gaussian elimination - https://en.wikipedia.org/wiki/Gaussian_elimination



Functions
---------

.. autoapisummary::

   linear_algebra.gaussian_elimination.gaussian_elimination
   linear_algebra.gaussian_elimination.retroactive_resolution


Module Contents
---------------

.. py:function:: gaussian_elimination(coefficients: numpy.typing.NDArray[numpy.float64], vector: numpy.typing.NDArray[numpy.float64]) -> numpy.typing.NDArray[numpy.float64]

   This function performs Gaussian elimination method

   Examples:
       1.
           * 1x1 - 4x2 - 2x3 = -2
           * 5x1 + 2x2 - 2x3 = -3
           * 1x1 - 1x2 + 0x3 = 4
       2.
           * 1x1 + 2x2 = 5
           * 5x1 + 2x2 = 5

   >>> gaussian_elimination([[1, -4, -2], [5, 2, -2], [1, -1, 0]], [[-2], [-3], [4]])
   array([[ 2.3 ],
          [-1.7 ],
          [ 5.55]])
   >>> gaussian_elimination([[1, 2], [5, 2]], [[5], [5]])
   array([[0. ],
          [2.5]])


.. py:function:: retroactive_resolution(coefficients: numpy.typing.NDArray[numpy.float64], vector: numpy.typing.NDArray[numpy.float64]) -> numpy.typing.NDArray[numpy.float64]

   This function performs a retroactive linear system resolution
   for triangular matrix

   Examples:
       1.
           * 2x1 + 2x2 - 1x3 = 5
           * 0x1 - 2x2 - 1x3 = -7
           * 0x1 + 0x2 + 5x3 = 15
       2.
           * 2x1 + 2x2 = -1
           * 0x1 - 2x2 = -1

   >>> gaussian_elimination([[2, 2, -1], [0, -2, -1], [0, 0, 5]], [[5], [-7], [15]])
   array([[2.],
          [2.],
          [3.]])
   >>> gaussian_elimination([[2, 2], [0, -2]], [[-1], [-1]])
   array([[-1. ],
          [ 0.5]])