====== System of linear equations ======
<note tip>Numerical methods used to solve such problem allow to introduce and experiment on [[wp>Time_complexity]], considering cubic time behavior of standard algorithms and //i.e.// quadratic time solutions using LU decomposition.</note>

===== Theory =====
  * [[wp>System_of_linear_equations]]
  * [[wp>Gaussian_elimination]], Gauss and Gauss-Jordan eliminations (diagonalization, triangularization)
  * [[wp>Pivot_element]], pivoting
  * [[wp>LU_decomposition]]
    * [[wp>Triangular_matrix#Forward_and_back_substitution]]
  * Chapter 2 in the book "Numerical Recipes" :
    * 2.0 Introduction
    * 2.1 Gauss-Jordan Elimination
    * 2.2 Gaussian Elimination with Backsubstitution
    * 2.3 LU Decomposition and Its Application
  * Python [[https://docs.scipy.org/doc/numpy/|NumPy]] library : [[https://docs.scipy.org/doc/numpy/reference/index.html|NumPy Reference]]
    * [[https://docs.scipy.org/doc/numpy/reference/routines.linalg.html|Linear algebra (numpy.linalg)]] : [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.solve.html#numpy.linalg.solve|numpy.linalg.solve]]
  * Time complexity analysis
    * Hint : in Python, use the timeit module

===== Jupyter notebooks =====
  * Example file (to be continued) : [[https://notebooks.azure.com/linusable/libraries/samples-public/html/notebooks/calculation_methods_applied_to_chemistry/Gauss-Jordan-01.ipynb]]

===== Exercices and applications =====
  * Exercices :
    * write a python function for diagonalisation with partial pivoting
    * random numbers → linear systems
    * comparison with numpy standard library
    * measurements of execution time to check cubic complexity

==== 1D problems with neigbours ====
  * Thermal diffusion and chemical diffusion (transient or stationary) on a regular 1D space with equidistant steps. ODE equations can be writen such a given evolution equation for node # i only imlies nodes i+1 and i-1
  * Using [[wp>Tridiagonal_matrix_algorithm|tridiagonal Thomas algorithm]] allows to save computational time thanks to n complexity
  * ? Python library with Thomas algorithm 

===== What you must have learned in this chapter =====
  * Except ill-conditionned, linear systems can be solved "exactly" using linear algebra algorithms in a finite and known number of arithmetic operations.
  * The accuracy is determined by the number of numerical figures which are encoded in floating point description
  * For a general system of n equations, diagonalisation requires of the order of n<sup>3</sup> operations. Also for solving a system using these method.
  * If the coefficient matrix is the same for different systems (only the independent coefficients are different), it is possible to solve systems with the order of n<sup>2</sup> operations, if the matrix of coeeficients is decomposed in the product of two triangular matrix (Lower-Upper decomposition). This n<sup>3</sup> step is realised only once.

===== References : =====
  * Numerical recipes, The Art of Scientific Computing 3rd Edition, William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, 2007, isbn: 9780521880688
    * [[http://numerical.recipes/]]
      * in C : [[http://apps.nrbook.com/c/index.html]]
    * [[http://www2.units.it/ipl/students_area/imm2/files/Numerical_Recipes.pdf]]
    * [[http://apps.nrbook.com/empanel/index.html#]]