#LnormInf corresponds to the absolute value of the greatest element of the vector.

0 Gauss-Seidel method in Python3, at the beginning of every cycle why I have to set to zero the array with most. Print ('The solution vector in iteration', iter1, 'is:', x) Matlab code for Gauss-Seidel and Successive over relaxation iterative methods. The Poisson Equation in Any Space Dimensions def sor_method(A, b, omega, initial_guess, tolerance, max_iterations): See section 3 on the paper The Optimal Relaxation Parameter for the SOR Method Applied to Obviously, with higher omega values the number of iterations should decrease.Īs for a working algorithm on SOR this is what I have computed, where best convergence is reached when the optimal omega is used. Phi = sor_solver(A, b, omega, initial_guess, residual_convergence)įor an extended answer on omega and its uses please refer to my other answer SOR method as what is quoted below is not accurate. #An example case that mirrors the one in the Wikipedia article What mistakes did I make? I think it is in TriSolve method since if I replaced it with regular LU solver such as (np.linalg.solve) it works.Here I have some python script, which solves the system of linear equations using Gauss-Seidel method: import numpy as np

Below is the same implementation in MATLAB which works: function x = gSeidel(A,B,N) Numerical methods in engineering with Python 3.

Same procedures are also done for XY): This 'finding optimum omega' procedure is taken from Kiusalaas, Jaan. I found optimum omega values with GAuss-Seidel (here I am showing this for XX part of the conductivity. In order for the lower triangular matrix D L to be invertible it is necessary and sufficient for aii 0. For the SOR, you need a parameter called omega. The ranges of convergence for both methods for SLAEs in two and three unknowns, as well as the interrelationships of these ranges are obtained. We will leave, as an exercise for the student, the derivation, but the matrix equation for the Gauss-Seidel iteration method is as follows: xk (D L) 1Uxk 1 + (D L) 1b. For some reason it is not converging even after 50000 iterations to the solution even when the matrix A is strict diagonal dominant. The paper presents a comparative analysis of iterative numerical methods of Jacobi and Gauss-Seidel for solving systems of linear algebraic equations (SLAEs) with complex and real matrices. #O(n) per iteration, so overall O(nN), good for large SPD/SDD matricesĪ = np.array(,])Īns = is my Gauss-Seidel method in Python. From scipy.linalg import solve_triangular as triSolve 7.3 The Jacobi and Gauss-Seidel Iterative Methods The Jacobi Method Two assumptions made on Jacobi Method: The system given by 11++ + 112212 Has a unique solution.11,22, 2.The coefficient matrix has no +zeroson itsmain diagonal, namely, are nonzeros.

Gauss seidel python code