Direct Solution of Saddle-Point Problems

  • Miroslav Rozložník
Part of the Nečas Center Series book series (NECES)


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.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Miroslav Rozložník
    • 1
  1. 1.Institute of MathematicsCzech Academy of SciencesPragueCzech Republic

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