In this chapter we give a brief overview of direct techniques used for solution of saddle-point problems. They are all based on some factorization of the saddle-point matrix. Therefore, we first recall two main factorizations of general indefinite matrices that preserve the symmetry of the original system matrix. Then we discuss cases when the 2-by-2 block matrix in the form (1.2) admits an LDLT factorization with the diagonal factor and gives its relation to the class of symmetric quasi-definite matrices. In particular, we look at the conditioning of factors in this Cholesky-like factorization in terms of the conditioning of the matrix (1.2) and its (1, 1)-block A. Then, we show that the two main direct approaches for solving the saddle-point problems via the Schur complement method or the null-space method can be seen not just a solution procedures but also as two different factorizations of the saddle-point matrix, namely, the block LDLT factorization and the block QTQT factorization, respectively.
This is a preview of subscription content, log in to check access.
M. Arioli, The use of QR factorization in sparse quadratic programming and backward error issues. SIAM J. Matrix Anal. Appl. 21, 825–839 (2000)MathSciNetCrossRefGoogle Scholar
M. Arioli, L. Baldini, A backward error analysis of a null space algorithm in sparse quadratic programming. SIAM J. Matrix Anal. Appl. 23, 425–442 (2001)MathSciNetCrossRefGoogle Scholar
M. Arioli, G. Manzini, A network programming approach in solving Darcy’s equations by mixed finite-element methods. ETNA 22, 41–70 (2006)MathSciNetzbMATHGoogle Scholar
N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd edn. (SIAM, Philadelphia, 2002)CrossRefGoogle Scholar
D. Orban, M. Arioli, Iterative Solution of Symmetric Quasi-Definite Linear Systems. SIAM Spotlight Series (SIAM, Philadelphia, 2017)CrossRefGoogle Scholar
J. Pestana, A.J. Wathen, The antitriangular factorization of saddle point matrices. SIAM J. Matrix Anal. Appl. 35(2), 339–353 (2014)MathSciNetCrossRefGoogle Scholar
T. Rees, J. Scott, The null-space method and its relationship with matrix factorizations for sparse saddle point systems. Numer. Linear Algebra Appl. 25(1), 1–17 (2018)CrossRefGoogle Scholar
M. Rozložník, F. Okulicka-Dłużewska, A. Smoktunowicz, Cholesky-like factorization of symmetric indefinite matrices and orthogonalization with respect to bilinear forms. SIAM J. Matrix Anal. Appl. 36(2), 727–751 (2015)MathSciNetCrossRefGoogle Scholar