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Ci-dessous, les différences entre deux révisions de la page.

--- teaching:methcalchim:system_of_linear_equations [2016/11/25 11:15] – créée villersd
+++ teaching:methcalchim:system_of_linear_equations [2018/10/18 10:10] (Version actuelle) – villersd
@@ Ligne 2: / Ligne 2: @@
 <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)
@@ Ligne 7: / Ligne 8: @@
   * [[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
-References :
+===== 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#]]