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According to Wikipedia, the Gauss-Seidel Iteration for solving a linear system of equations Ax=b converges if

  • A is symmetric positive-definite, or

  • A is strictly or irreducibly diagonally dominant.

Does this also hold for the Projected Gauss-Seidel (which simply makes the iterate non-negative by taking max(gauss-Seidel iterate,0)?

Similarly, does the convergence criterion for Jacobi iteration also hold for the Projected Jacobi?

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I believe the general convergence condition for (non-projected) Gauss-Seidel is that $(I-P^{-1}A)$ has spectral radius less than one, where $P = L + D$ is the lower triangular part of $A$. If there were an eigenvalue with at least unit norm, then the component of error in the direction of the corresponding eigenvector would not shrink. Unfortunately, I don't know what projecting does to the spectrum. –  S. Carnahan May 29 '10 at 21:36
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