The SSOR preconditioner like the Jacobi preconditioner, can be derived from the coefficient matrix without any work.
If the original, symmetric, matrix is decomposed as
in its diagonal, lower, and upper triangular part, the SSOR matrix is defined as
or, parameterized by
The optimal value of the parameter, like the parameter in the SOR method, will reduce the number of iterations to a lower order. Specifically, for second order elliptic problems a spectral condition number is attainable (see Axelsson and Barker [.4]AxBa:febook). In practice, however, the spectral information needed to calculate the optimal is prohibitively expensive to compute.
The SSOR matrix is given in factored form, so this preconditioner shares many properties of other factorization-based methods (see below). For instance, its suitability for vector processors or parallel architectures depends strongly on the ordering of the variables. On the other hand, since this factorization is given a priori, there is no possibility of breakdown as in the construction phase of incomplete factorization methods.