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If we assume that the coefficient matrix is symmetric, then the Symmetric Successive Overrelaxation method, or SSOR, combines two SOR sweeps together in such a way that the resulting iteration matrix is similar to a symmetric matrix. Specifically, the first SOR sweep is carried out as in (), but in the second sweep the unknowns are updated in the reverse order. That is, SSOR is a forward SOR sweep followed by a backward SOR sweep. The similarity of the SSOR iteration matrix to a symmetric matrix permits the application of SSOR as a preconditioner for other iterative schemes for symmetric matrices. Indeed, this is the primary motivation for SSOR since its convergence rate, with an optimal value of , is usually slower than the convergence rate of SOR with optimal (see Young , page 462). For details on using SSOR as a preconditioner, see Chapter .
In matrix terms, the SSOR iteration can be expressed as follows:
Note that is simply the iteration matrix for SOR from (), and that is the same, but with the roles of and reversed.
The pseudocode for the SSOR algorithm is given in Figure .