Rank modifications of semidefinite matrices associated with a secant update formula

SIAM Journal on Matrix Analysis and Applications

Volumn 20, Issue 2, 1998, Pages 428-436

  Chu, Moody T ^a   Funderlic, R E ^b   Golub, Gene H ^c  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b North Carolina State University   (United States)

^c Stanford University   (United States)

Author keywords

BFGS update; DFP update; Quasi Newton methods; Rank additivity; Rank subtractivity; Rank one reduction; Wedderburn theorem

Indexed keywords

EID: 0032216893     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479896306021     Document Type: Article

References (18)

1. A. BEN-ISRAEL AND T. N. E. GREVILLE, Generalized Inverses: Theory and Applications, John Wiley, New York, 1974.
2. K. W. BRODLIE, A. R. GOURLAY, AND J. GREENSTADT, Rank-one and rank-two corrections to positive definite matrices expressed in product form, J. Inst. Math. Appl., 11 (1973), pp. 73-82.
3. M. T. CHU, R. E. FUNDERLIC, AND G. H. GOLUB, A rank-one reduction formula and its applications to matrix factorizations, SIAM Rev., 37 (1995), pp. 512-530.
4. R. E. CLINE AND R. E. FUNDERLIC, The rank of a difference of matrices and associated generalized inverses, Linear Algebra Appl., 24 (1979), pp. 185-215.
5. J. E. DENNIS, JR. AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983.
6. A. DEUFLHARD AND R. W. FREUND, Fast secant methods for the iterative solution of nonsymmetric linear equations, Sci. Engrg., 2 (1990), pp. 244-276.
7. E. EGERVÁRY, On rank-diminishing operators and their applications to the solution of linear equations, Z. Angew. Math. Phys., 11 (1960), pp. 376-386.
8. R. FLETCHER, Practical Methods of Optimization, 2nd ed., John Wiley, Chichester, UK, 1987.
9. R. R. GERBER AND F. T. LUK, A generalized Broyden's method for solving simultaneous linear equations, SIAM J. Numer. Anal., 18 (1981), pp. 882-890.
10. L. GUTTMAN, General theory and methods for matrix factoring, Psychometrika, 9 (1944), pp. 1-16.
11. L. GUTTMAN, Multiple group methods for common-factor analysis: Their basis, computation, and interpretation, Psychometrika, 17 (1952), pp. 209-222.
12. L. GUTTMAN, A necessary and sufficient formula for matrix factoring, Psychometrika, 22 (1957), pp. 79-81.
13. A. S. HOUSEHOLDER, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964, and Dover, New York, 1975.
14. L. HUBERT, J. MEULMAN, AND W. HEISER, A Brief Tale of Two Purposes for Matrix Factorization, Preprint, Department of Psychology, University of Illinois, 1998.
15. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, San Diego, CA, 1970.
16. G. W. STEWART AND J. G. SUN, Matrix Perturbation Theory, Academic Press, Boston, 1990.
17. J. H. M. WEDDERBURN, Lectures on Matrices, Colloquium Publications, Vol. XVII. American Mathematical Society, New York, 1934, and Dover, New York, 1964.
18. H. WEYL, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann., 71 (1912), pp. 441-479.