System of linear equations

System_of_linear_equations

Gaussian_elimination, Gauss and Gauss-Jordan eliminations (diagonalization, triangularization)

Pivot_element, pivoting

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

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 tridiagonal Thomas algorithm allows to save computational time thanks to n complexity

? Python library with Thomas algorithm

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 n³ 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 n² operations, if the matrix of coeeficients is decomposed in the product of two triangular matrix (Lower-Upper decomposition). This n³ step is realised only once.