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Additional info for A structure utilizing inexact primal-dual interior-point method for analysis of linear differential inclusions

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To obtain fast convergence in an iterative solver, the preconditioner is to approximate A with high accuracy. If an approximation with high accuracy is chosen, applying the preconditioner could be computationally demanding. However, if the preconditioner is a poor approximation of A, applying the preconditioner can often be computationally cheap. Naturally a poor approximation leads to more iterations in the solver, but if the iterations have a low computational cost, the overall cost might be reduced.

There is a wide variety of preconditioners in the literature. In Benzi et al. (2005) the general class of saddle point problems are studied and some preconditioners are discussed. In this section only the preconditioners applicable to an indefinite system of equations are discussed. There are many strategies to approximate the linear system of equations to obtain a preconditioner. A popular choice is to approximate the symmetric and positive definite (1, 1)-block of the coefficient matrix with some less complicated structure.

It is important to notice that H∆S and H∆Z are invertible and positive definite. It is easily seen that a non-singular scaling matrix R make the function H invertible. 17), is a system of equations with nx + n(n + 1) variables. The invertibility of the symmetrization operator implies the elimination of either the slack variable ∆S or the dual variable ∆Z. Eliminating the slack variable ∆S and reordering the vectorized variables results in −1 H∆S H∆Z AT A 0 svec(∆Z) vec(∆x) = −1 D1 − H∆S D3 . 19) is referred to as the augmented equations.

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