Talk:Gauss–Seidel method

Convergence
Is there an example of a linear system such that the Jacobi method converges but the Gauss-Seidel does not?

Bur40win 03:37, 5 October 2007 (UTC)Michael Harris

The answer to your question wouldn't be completely informative, it would only show that some corner cases exist.

Supposing a system is chosen such that A is positive semi-definite, then both Gauss-Seidel and Jacobi are guaranteed to converge. However, if A is not PSD, then convergence is solely based on the choice of initial guesses.

GS and Jacobi both follow different paths to a solution, and it's certainly possible to find divergent behaviors but it'll be entirely based on the initial guess and how well behaved the system is. —Preceding unsigned comment added by 76.27.5.236 (talk) 06:09, 22 February 2011 (UTC)

One has more than one way to write $$ Ax=b $$ in an iterative form $$ x^{(k+1)}=Bx^{(k)}+c $$. The iteration yields $$ x=(I+B+B^2+B^3+.....)c $$. Convergence is guaranteed by the norm of B less than 1. One does not have to be at the mercy of an initial guess. 70.31.163.151 (talk) 12:04, 3 August 2015 (UTC)

example mistake
$$ C = \begin{bmatrix} 0.0625 & 0.0000 \\           0.0398 & -0.0909      \end{bmatrix} \times \begin{bmatrix} 11 \\          13      \end{bmatrix} =     \begin{bmatrix} 0.6875 \\         -0.7439      \end{bmatrix}. $$

Matlab says it's

$$ C = \begin{bmatrix} 0.0625 & 0.0000 \\           0.0398 & -0.0909      \end{bmatrix} \times \begin{bmatrix} 11 \\          13      \end{bmatrix} =     \begin{bmatrix} 0.6875 \\         -0.7443      \end{bmatrix}. $$

notice the second row number in the resulting matrix ..::Gradient93::.. 13:26, 8 February 2014 (UTC)


 * Fixed.—Anita5192 (talk) 18:28, 27 February 2019 (UTC)

Matrix T wrong?
I just spent a lot of time trying to figure out where my matrix multiplication algorithm went wrong, until I tried an online matrix multiplicator (https://matrixcalc.org/en) and it gave me the same result for matrix T in the example: 0.0,0.1875],[0.0,0.11931818181818182 Not -0.1875, but 0.1875. Same with the lower right number. So I guess the example here is wrong? I still get the wrong result, but that might be in a different part of my program. Fabian42 (talk) 11:03, 27 February 2019 (UTC)


 * Try doing the matrix multiplication by hand and don't neglect any of the minus signs, particularly the sign at the very beginning. I got the result listed in the article.—Anita5192 (talk) 18:20, 27 February 2019 (UTC)

MATLAB Example
It'd be nice if the program included a condition to check if a solution has been found within a given margin of error, and if so stop and avoid unnecessary loops. — Preceding unsigned comment added by 2600:1700:8A10:54A0:B583:5578:BA52:2293 (talk) 02:35, 21 November 2021 (UTC)

Algorithm
This is my first Wikipedia contribution. Sorry if I do anything wrong. I believe the pseudo-code is incorrect. It seems to me that it describes the Jacobi method rather than the Gauss-Seidel Method. RickyMcMuffin (talk) 18:24, 17 April 2024 (UTC)