teaching:methcalchim:system_of_linear_equations

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teaching:methcalchim:system_of_linear_equations [2018/10/08 13:27] – villersdteaching:methcalchim:system_of_linear_equations [2018/10/18 10:10] (Version actuelle) – villersd
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     * 2.2 Gaussian Elimination with Backsubstitution     * 2.2 Gaussian Elimination with Backsubstitution
     * 2.3 LU Decomposition and Its Application     * 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   * Time complexity analysis
     * Hint : in Python, use the timeit module     * 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 and applications =====
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   * Using [[wp>Tridiagonal_matrix_algorithm|tridiagonal Thomas algorithm]] allows to save computational time thanks to n complexity   * Using [[wp>Tridiagonal_matrix_algorithm|tridiagonal Thomas algorithm]] allows to save computational time thanks to n complexity
   * ? Python library with Thomas algorithm    * ? 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 : ===== ===== References : =====
  • teaching/methcalchim/system_of_linear_equations.1538998066.txt.gz
  • Dernière modification : 2018/10/08 13:27
  • de villersd