Abstract
SummaryNull‐space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified that we can interpret a null‐space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null‐space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems.
Highlights
A saddle point system is an indefinite linear system of equations of the form [ ] [ ][ ] [ ] x y =A BT B0 x y f g (1)Here, we will assume that B ∈ Rm×n(n > m) has full rank and A ∈ Rn×n is symmetric positive definite on the null space of B
We investigate the suitability of using null-space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems
One approach for solving Equation 1 is to use a null-space method.[1, section 6] These methods, which we will describe in detail below, have been used in the fields of optimization, structural mechanics, fluid mechanics, and electrical engineering
Summary
A saddle point system is an indefinite linear system of equations of the form [ ] [ ][ ] [ ]. One approach for solving Equation 1 is to use a null-space method.[1, section 6] These methods, which we will describe in detail below, have been used in the fields of optimization (where they are known as reduced Hessian methods), structural mechanics (where they are known as a “force” method or “direct elimination”), fluid mechanics (where they are known as the “dual variable” method), and electrical engineering (where they are known as “loop analysis”) This approach remains popular in the large-scale optimization literature.[2,3,4,5,6,7].
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