The Antitriangular Factorization of Saddle Point Matrices

Mastronardi and Van Dooren [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 173--196] recently introduced the block antitriangular (``Batman'') decomposition for symmetric indefinite matrices. Here we show the simplification of this factorization for saddle point matrices and demonstrate how it represents the common nullspace method. We show that rank-1 updates to the saddle point matrix can be easily incorporated into the factorization and give bounds on the eigenvalues of matrices important in saddle point theory. We show the relation of this factorization to constraint preconditioning and how it transforms but preserves the structure of block diagonal and block triangular preconditioners.

[1]  Paul Van Dooren,et al.  The Antitriangular Factorization of Symmetric Matrices , 2013, SIAM J. Matrix Anal. Appl..

[2]  Nicholas I. M. Gould,et al.  Implicit-Factorization Preconditioning and Iterative Solvers for Regularized Saddle-Point Systems , 2006, SIAM J. Matrix Anal. Appl..

[3]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[4]  Nicholas I. M. Gould,et al.  On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization , 2001, SIAM J. Sci. Comput..

[5]  Stephen A. Vavasis,et al.  An Iterative Method for Solving Complex-Symmetric Systems Arising in Electrical Power Modeling , 2005, SIAM J. Matrix Anal. Appl..

[6]  Michael A. Saunders,et al.  Large-scale linearly constrained optimization , 1978, Math. Program..

[7]  P. Dooren,et al.  An algorithm for solving the indefinite least squares problem with equality constraints , 2013, BIT Numerical Mathematics.

[8]  Nicholas I. M. Gould,et al.  Constraint Preconditioning for Indefinite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

[9]  Andreas Griewank,et al.  Maintaining factorized KKT systems subject to rank-one updates of Hessians and Jacobians , 2007, Optim. Methods Softw..

[10]  Ilaria Perugia,et al.  Linear Algebra Methods in a Mixed Approximation of Magnetostatic Problems , 1999, SIAM J. Sci. Comput..

[11]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[12]  Jan Vlcek,et al.  Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems , 1998, Numer. Linear Algebra Appl..

[13]  Vivek Sarin,et al.  Parallel iterative methods for dense linear systems in inductance extraction , 2003, Parallel Comput..

[14]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[15]  Daniela di Serafino,et al.  Updating Constraint Preconditioners for KKT Systems in Quadratic Programming Via Low-Rank Corrections , 2013, SIAM J. Optim..

[16]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

[17]  Michael A. Saunders,et al.  A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints , 1982 .

[18]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[19]  M. Benzi,et al.  Some Preconditioning Techniques for Saddle Point Problems , 2008 .

[20]  YU. A. KUZNETSOV,et al.  Efficient iterative solvers for elliptic finite element problems on nonmatching grids , 1995 .

[21]  Ji-guang Sun.  Structured backward errors for KKT systems , 1999 .