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The symmetric successive overrelaxation method features the iteration matrix

$$P=\left(\frac{D}{\omega}+L\right)\frac{\omega}{2-\omega}D^{-1}\left(\frac{D}{\omega}+U\right)$$

Either as a stationary method or as a preconditioner, a system in that matrix is solved at every iteration.

The method depends on a parameter $\omega \in (0,2)$. While it converges for every $\omega$ in that parameter range, there is little information in standard textbooks on how to find the optimal parameter in practice. Often, $\omega = 1$ is a choice without further discussion.

What are practical ways of picking a relaxation parameter? Or is picking the parameter a purely theoretical issue?

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  • $\begingroup$ the book by Axelsson & Barker has a discussion. It comes down to the fact that for SSOR the optimal parameter is much less important than for SOR. $\endgroup$ Sep 21 at 19:27

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In fact, $\omega=1.2$ is often used in practice when using SSOR as a preconditioner in an outer iteration. The way to find which value is optimal is simply one of trying many different values and recording how many iterations you need to solve a linear system to a certain accuracy. It's a one-dimensional optimization problem (the parameter is $\omega$, the objective function is the number of iterations), so easy enough to solve by simply sampling many different values of $\omega$.

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  • $\begingroup$ So in practice it comes down to trial and error? That sounds bleak. $\endgroup$
    – shuhalo
    Sep 24 at 9:25
  • $\begingroup$ It's a 1d optimization problem, that really isn't all that hard. Or you could just accept the value of 1.2 that others have found as gospel. $\endgroup$ Sep 25 at 3:04
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Corless/Fillion book gives a formula for "optimal" $\omega$ of SOR but doesn't say where it comes from. Here's the relevant section:

enter image description here

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Not sure if this helps in some way, but in the book Numerical Recipes, the authors have this to say about the SOR parameter $\omega$ and how it can be iteratively updated:

enter image description here

where

enter image description here

Please note that the context where these conclusions were drawn in the book comes from considerations of a 2D SOR scheme, and they mention nothing about the impact of this update scheme on the SSOR. Also, the intermediate calculation step, denoted with bracketed indices $1/2$ and $n+1/2$ come from the 2D red-black SOR scheme, where first even-even and odd-odd i,j coordinates are solved, and then even-odd and odd-even i,j are solved. i,j here are indices of the grid points along the $x$ and $y$ axes, respectively

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