An Improved Arc Algorithm for Detecting Definite Hermitian Pairs
SIAM Journal on Matrix Analysis and Applications
Preconditioning Saddle-Point Systems with Applications in Optimization
SIAM Journal on Scientific Computing
Eigenvalue Estimates for Preconditioned Nonsymmetric Saddle Point Matrices
SIAM Journal on Matrix Analysis and Applications
Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations
SIAM Journal on Scientific Computing
Solving Hermitian positive definite systems using indefinite incomplete factorizations
Journal of Computational and Applied Mathematics
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Linear systems in saddle point form are usually highly indefinite,which often slows down iterative solvers such as Krylov subspace methods. It has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix $${{\mathcal A}}$$ whose spectrum is entirely contained in the right half plane. In this paper we study conditions so that $${{\mathcal A}}$$ is diagonalizable with a real and positive spectrum. These conditions are based on necessary and sufficient conditions for positive definiteness of a certain bilinear form,with respect to which $${{\mathcal A}}$$ is symmetric. In case the latter conditions are satisfied, there exists a well defined conjugate gradient (CG) method for solving linear systems with $${{\mathcal A}}$$. We give an efficient implementation of this method, discuss practical issues such as error bounds, and present numerical experiments.