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I want to check the convergence of Jacobi (from now on J) and of Gauss-Seidel (from now on GS) methods applied to a linear system $Ax=b$, where $A$ is square and non-singular.

I was wondering if the convergence of one of the two methods shed light on the convergence of the other. I mean:

When can I say that if J converges then GS converges? When can I say that if GS converges then J converges?

I know that if $A$ is (for example, but this is true with other very similar hypothesis) strictly diagonally dominant then the spectral radius of J and GS matrices is $<1$ and so both J and GS converge. Also I know that for example if $A$ is Hermitian with real positive diagonal elements then $A$ is positive defined iff GS converges.

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    $\begingroup$ The Stein-Rosenberg theorem answers that for certain matrices, see, e.g., here. $\endgroup$ – Algebraic Pavel Jan 8 '18 at 0:38
  • $\begingroup$ @tinlyx It is commendable that you are trying to clean up older posts by fixing the spelling of a few words (even if all you are doing is searching MSE for a few common misspellings and cleaning them up in the quest for a badge). However, please note that when you edit old posts, they become marked as "active," and are brought to the font of the page. Please consider slowing down and spacing your edits out. $\endgroup$ – Xander Henderson Mar 10 '18 at 4:51
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  1. $$ \begin{pmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \\ 1 & 2 & -3 \\ \end{pmatrix} $$

is convergent by J but divergent by GS. and $$ \begin{pmatrix} 1 & 0.5 & 0.5 \\ 0.5 & 1 & 0.5 \\ 0.5 & 0.5 & 1\\ \end{pmatrix} $$ is convergent by GS but divergent by J.

  1. In general,the convergent speed by GS is faster than J if them are also convergent.
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There is an article that addresses exactly this question. The article is found here under the title: The Convergence of Jacobi and Gauss-Seidel Iteration.

Note that Theorem 2 says that for $n=2$ Gauss-Seidel converges if and only if Jacobi converges. For $n>2$ there is no general relation. As shown bt @jean above there are examples where one converges and the other does not, and vice-versa.

If the matrix is diagonal dominant then both converge. I can point to the following important texts that address the convergence of these two methods and of iterative methods for linear systems in general.

R. Varga. Matrix Iterative Analysis. Springer-Verlag Berlin, 2009.

S. Venit. The convergence of Jacobi and Gauss-seidel iteration. Mathematics Magazine, 3:163–167, 1975. (the link above)

D. M. Young. Iterative solution of large linear systems. Academic Press New York, 1971. 347

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