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I am a computer scientist and we typically use Jacobi method to iteratively solve very large linear system of equations (a lot of these turn out to be graph algorithms where vertex values need to be iteratively computed and edges represent dependencies). I am trying to explore whether Gauss-Seidel can be used instead of Jacobi and, to my surprise, on a graph containing several cycles the PageRank algorithm ($pr_i^k = 0.15 + 0.85 \times \sum_{j \in in\_neighbors(i)} pr_j^{k-1}$) gives the exact same result for Gauss-Seidel and Jacobi techniques.

Are there any mathematical properties about the linear equations that allow us to know whether Gauss-Seidel and Jacobi will give the same result?

John Elaine
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    I am not expert for the terms you use, but I can tell you what confuses me: when both (Gauss-Seidel and Jacobi) are methods to solve linear systems of equations, then why are you confused that they give you the same result. In the end the result should be unique, or not? I would be more concerned if the results were different. – M. Winter Apr 24 '18 at 09:25
  • My understanding (mostly based on wikipedia and some online reading) is that Gauss-Seidel allows us to overwrite 'pr' value as it gets computed, and so the RHS of above equation has pr_j^k instead of pr_j^{k-1}. Jacobi doesn't allow this and it strictly requires RHS to be pr_j^{k-1}. So given that the iterative equation has changed across two methods, it was natural for me to assume that the results should be different. – John Elaine Apr 25 '18 at 07:26
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    I checked again on the terms, and it seems both methods solve a linear system of equations. Hence they should give the same result because the result is hopefully unique. There are always different looking algorithms for the same task, so this should be no criterion to judge on their behavior. – M. Winter Apr 25 '18 at 07:38
  • This is hard to understand. Let's say that Jacobi and Gauss-Seidel are proceeding in iterations 1, 2, 3, 4, ... Now, at the end of iteration 1, values of 'pr' may not match for both cases (since \sum in RHS is receiving different values); at the end of iteration 2, values of 'pr' may again not match; and so on .. Then how do they converge to the same result? – John Elaine Apr 25 '18 at 08:03
  • Maybe of interest https://www.maa.org/press/periodicals/loci/joma/iterative-methods-for-solving-iaxi-ibi-analysis-of-jacobi-and-gauss-seidel-methods and https://math.stackexchange.com/questions/270181 – cgiovanardi Aug 18 '18 at 15:45

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