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teaching:methcalchim:system_of_linear_equations [2016/11/25 11:15] – créée villersd | teaching:methcalchim:system_of_linear_equations [2018/10/09 09:20] – villersd | ||
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+ | ===== Theory ===== | ||
* [[wp> | * [[wp> | ||
* [[wp> | * [[wp> | ||
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* [[wp> | * [[wp> | ||
* [[wp> | * [[wp> | ||
+ | * Chapter 2 in the book " | ||
+ | * 2.0 Introduction | ||
+ | * 2.1 Gauss-Jordan Elimination | ||
+ | * 2.2 Gaussian Elimination with Backsubstitution | ||
+ | * 2.3 LU Decomposition and Its Application | ||
+ | * Python [[https:// | ||
+ | * [[https:// | ||
* Time complexity analysis | * Time complexity analysis | ||
* Hint : in Python, use the timeit module | * Hint : in Python, use the timeit module | ||
- | References : | + | ===== 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> | ||
+ | * ? Python library with Thomas algorithm | ||
+ | |||
+ | ===== What you must have learned in this chapter ===== | ||
+ | * Except ill-conditionned, | ||
+ | * 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< | ||
+ | * 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< | ||
+ | |||
+ | ===== 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 | * 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:// | * [[http:// | ||
+ | * in C : [[http:// | ||
* [[http:// | * [[http:// | ||
* [[http:// | * [[http:// |