The solution of linear systems of equations has always been and remains a fundamental subproblem of countless computational methods for numerical science and engineering. The method of choice to solve systems is typically guided by their properties, including symmetry, definiteness, sparsity, conditioning, and block structure. Numerous applications, including constrained optimization and fluid dynamics, involve systems with the specific *saddle-point* block structure; i.e., their coefficient has the form

where the block F is square but need not be symmetric, the block G is also square and can be zero, and B_{1} may or may not be equal to B_{2}. The special case of *symmetric saddle-point systems* arises when F is symmetric, B_{1} = B_{2}, and G is either zero or symmetric and negative semidefinite. At the time of this writing, a large number of special cases of saddle-point systems, including the symmetric case, are understood sufficiently well that efficient numerical methods are known to solve them, though it is these authors' opinion that such systems still have much to reveal. The state-of-the-art paper by Benzi, Golub, and Liesen (2005) covers the main properties of general saddle-point systems and families of associated numerical methods. This book does not attempt to give a complete overview of saddle-point systems, of their properties, or of numerical methods, as they are the topic of ongoing research. Rather, we focus our efforts on a special case of symmetric saddle-point systems that arises when numerical methods are some how *regularized*. Thus, we recommend that the interested reader or student who is not yet familiar with saddle-point systems begin with Sections 1–9 of Benzi, Golub, and Liesen (2005) before going further.