The Equality Constrained Indefinite Least Squares Problem: Theory and Algorithms
 Adam Bojanczyk,
 Nicholas J. Higham,
 Harikrishna Patel
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We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.
 A. Bojanczyk, N. J. Higham, and H. Patel, Solving the indefinite least squares problem by hyperbolic QR factorization, Numerical Analysis Report No. 397, Manchester Centre for Computational Mathematics, Manchester, England, 2002. Revised April 2002. To appear in SIAM J. Matrix Anal. Appl.
 S. Chandrasekaran, M. Gu, and A. H. Sayed, A stable and efficient algorithm for the indefinite linear leastsquares problem, SIAM J. Matrix Anal. Appl., 20:2 (1998), pp. 354362.
 A. J. Cox and N. J. Higham, Accuracy and stability of the null space method for solving the equality constrained least squares problem, BIT, 39:1 (1999), pp. 3450.
 L. Eldén Perturbation theory for the least squares problem with linear equality constraints, SIAM J. Numer. Anal., 17:3 (1980), pp. 338350.
 N. J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2nd ed., 2002.
 N. J. Higham, Jorthogonal matrices: Properties and generation, Numerical Analysis Report No. 408, Manchester Centre for Computational Mathematics, Manchester, England, 2002.
 R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, UK, 1991.
 Title
 The Equality Constrained Indefinite Least Squares Problem: Theory and Algorithms
 Journal

BIT Numerical Mathematics
Volume 43, Issue 3 , pp 505517
 Cover Date
 20030901
 DOI
 10.1023/B:BITN.0000007020.58972.07
 Print ISSN
 00063835
 Online ISSN
 15729125
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 equality constrained indefinite least squares problem
 Jorthogonal matrix
 hyperbolic rotation
 hyperbolic QR factorization
 generalized hyperbolic QR factorization
 rounding error analysis
 forward stability
 perturbation theory
 Cholesky factorization
 Industry Sectors
 Authors

 Adam Bojanczyk ^{(1)}
 Nicholas J. Higham ^{(1)}
 Harikrishna Patel ^{(1)}
 Author Affiliations

 1. School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, 148533801, USA