Shur-Complement Method

When the matrix G is positive definite, it is possible to use the technique described in the Vol. 1 (Buzzi-Ferraris and Manenti, 2010a) in which the original KKT matrix W is split into four submatrices Wi,i, Wi,2, W2,i, and W2,2-Given the system 

This criterion can be applied whenever the matrix Wn has a particular structure that makes factorization easy. It also requires the matrix W2,2 — Wa.i Wf j Wi to be well conditioned. 

The matrix EG is calculated as follows. First of all, the following system is solved 

The system (11.32) can be solved by either factorizing the symmetric matrix EV or using iterative methods such as the CG method or PCG method. In the latter case, it is not necessary to actually build the matrix EV since it appears in multiplications with vectors only (see Chapter 4). 

Note that it is possible to apply this procedure also when only a portion of the matrix G is positive definite. In fact, the relations (11.26) and (11.27) can also be applied using the positive definite portion of G as matrix Wn. 

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The Shur-complement method for solving the KKT system has the following... 

The matrix G is sparse, while E is dense. In this case, the Shur-complement method is preferable when G is positive definite otherwise, however, the overall system solution is preferable. 

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