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MSOR can be written as a splitting of matrix ( A ) into diagonal ( D ), strictly lower ( L ), and strictly upper ( U ).
For a 2-group MSOR (red-black), the iteration matrix is ( \mathcalL MSOR ). To convert, we seek a scalar ( \omega ) such that the spectral radius ( \rho(\mathcalL SOR(\omega)) ) approximates ( \rho(\mathcalL_MSOR(\omega_1,\omega_2)) ). convert msor to sor
[ \omega_SOR^(effective) = \frac\omega_1 + \omega_22 \quad \text(experimental, low accuracy) ] MSOR can be written as a splitting of
For many practical problems (e.g., Poisson's equation on a grid), the optimal SOR parameter is related to the spectral radius of the Jacobi method (( \mu )): [ \omega_opt^SOR = \frac21 + \sqrt1-\mu^2 ] strictly lower ( L )
(derived from the eigenvalues of the Jacobi matrix).
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