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Towards an optimal condition number of certain augmented Lagrangian‐type saddle‐point matrices
Author(s) -
Estrin R.,
Greif C.
Publication year - 2016
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2050
Subject(s) - mathematics , invertible matrix , saddle point , eigenvalues and eigenvectors , augmented lagrangian method , rank (graph theory) , block (permutation group theory) , matrix (chemical analysis) , saddle , type (biology) , singular value decomposition , combinatorics , block matrix , lagrangian , singular value , pure mathematics , mathematical optimization , algorithm , geometry , ecology , physics , materials science , quantum mechanics , composite material , biology
Summary We present an analysis for minimizing the condition number of nonsingular parameter‐dependent 2 × 2 block‐structured saddle‐point matrices with a maximally rank‐deficient (1,1) block. The matrices arise from an augmented Lagrangian approach. Using quasidirect sums, we show that a decomposition akin to simultaneous diagonalization leads to an optimization based on the extremal nonzero eigenvalues and singular values of the associated block matrices. Bounds on the condition number of the parameter‐dependent matrix are obtained, and we demonstrate their tightness on some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.