Abstract
A highly regarded method to obtain an orthonormal basis,Z, for the null space of a matrixA T is theQR decomposition ofA, whereQ is the product of Householder matrices. In several optimization contextsA(x) varies continuously withx and it is desirable thatZ(x) vary continuously also. In this note we demonstrate that thestandard implementation of theQR decomposition doesnot yield an orthonormal basisZ(x) whose elements vary continuously withx. We suggest three possible remedies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.H. Bartels and A.R. Conn, “An approach to nonlinearl 1 data fitting”, Numerical Analysis Proceedings, 3rd IIMAS Workshop (Cocoyoc, Mexico, 1982).
T.F. Coleman and A.R. Conn, “Nonlinear programming via an exact penalty function: Asymptotic analysis”,Mathematial Programming 24 (1982a) 123–136.
T.F. Coleman and A.R. Conn, “On the local convergence of a quasi-Newton method for the nonlinear programming problem,” Technical Report 82-509, Department of Computer Science, Cornell University, Ithaca, NY, 1982b).
J.A. George and M.T. Health, “Solution of sparse linear least squares problems using Givens rotations”,Linear Algebra and Its Applications 34 (1980) 69–83.
P. Gill and W. Murray,Numerical methods for constrained optimization (Academic Press, London, 1974).
P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, “On computing the null space in nonlinear optimization algorithms”, Technical Report SOL 83-19, Department of Operations Research, Stanford University (1983).
L. Kaufman, “A variable projection method for solving separable nonlinear least squares problems”,BIT 15 (1975) 49–57.
W. Murray and M. Overton, “A projected Lagrangian algorithm for nonlinear minimax optimization”,SIAM Journal of Scientific and Statistical Computing 1 (1980) 345–370.
W. Murray and M. Wright, “Projected Lagrangian methods based on the trajectories of penalty and barrier functions”, Technical Report 72-8. Stanford Optimization Laboratory (Stanford, CA, 1978).
B.N. Parlett, “Analysis of algorithms for reflections in bisectors”,SIAM Review 13 (1971) 197–208.
B.N. Parlett,The symmetric eigenvalue problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).
K. Tanabe, “Feasibility-improving gradient-acute-projection methods: a unified approach to nonlinear programming”,Lecture Notes in Numerical and Applied Analysis 3 (1981) 57–76.
M. Wright, “Algorithms for nonlinearly constrained optimization” Technical Report 79-24, Stanford Optimization Laboratory (Stanford, CA, 1979).
Author information
Authors and Affiliations
Additional information
Work supported in part by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38.
Rights and permissions
About this article
Cite this article
Coleman, T.F., Sorensen, D.C. A note on the computation of an orthonormal basis for the null space of a matrix. Mathematical Programming 29, 234–242 (1984). https://doi.org/10.1007/BF02592223
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02592223