Abstract:A new formulation of the generalized linear least squares problem is given. This is based on some ideas in estimation and allows complete generality in that there are no restrictions on the matrices involved. The formulation leads directly to a numerical algorithm involving orthogonal decompositions for solving the problem. A perturbation analysis of the problem is obtained by using the new formulation and some of the decompositions used in the solution. A rounding error analysis is given to show that the algorithm is numerically stable.
- Ȧke Björck, A uniform numerical method for linear estimation from general Gauss-Markoff models, Compstat 1974 (Proc. Sympos. Computational Statist., Univ. Vienna, Vienna, 1974) Physica Verlag, Vienna, 1974, pp. 131–140. MR 0373173
- Ȧke Björck, Solving linear least squares problems by Gram-Schmidt orthogonalization, Nordisk Tidskr. Informationsbehandling (BIT) 7 (1967), 1–21. MR 214275, DOI 10.1007/bf01934122
- Ȧke Björck, Iterative refinement of linear least squares solutions. I, Nordisk Tidskr. Informationsbehandling (BIT) 7 (1967), 257–278. MR 233494, DOI 10.1007/bf01939321
- Peter Businger and Gene H. Golub, Handbook series linear algebra. Linear least squares solutions by Householder transformations, Numer. Math. 7 (1965), 269–276. MR 176590, DOI 10.1007/BF01436084
- A. K. Cline, An elimination method for the solution of linear least squares problems, SIAM J. Numer. Anal. 10 (1973), 283–289. MR 359294, DOI 10.1137/0710027
- G. Golub, Numerical methods for solving linear least squares problems, Numer. Math. 7 (1965), 206–216. MR 181094, DOI 10.1007/BF01436075
- G. H. Golub and C. Reinsch, Handbook Series Linear Algebra: Singular value decomposition and least squares solutions, Numer. Math. 14 (1970), no. 5, 403–420. MR 1553974, DOI 10.1007/BF02163027
- Gene H. Golub and George P. H. Styan, Numerical computations for univariate linear models, J. Statist. Comput. Simulation 2 (1973), 253–274. MR 375649, DOI 10.1080/00949657308810051
- G. H. Golub and J. H. Wilkinson, Note on the iterative refinement of least squares solution, Numer. Math. 9 (1966), 139–148. MR 212984, DOI 10.1007/BF02166032
- Charles L. Lawson and Richard J. Hanson, Solving least squares problems, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0366019 G. PETERS & J. H. WILKINSON, "The least squares problem and pseudo-inverses," Comput. J., v. 13, 1970, pp. 309-316.
- G. W. Stewart, On the continuity of the generalized inverse, SIAM J. Appl. Math. 17 (1969), 33–45. MR 245583, DOI 10.1137/0117004
- G. W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev. 19 (1977), no. 4, 634–662. MR 461871, DOI 10.1137/1019104 G. GOLUB, V. KLEMA, & G. W. STEWART, Rank Degeneracy and Least Squares Problems, Stanford University Computer Science Report STAN-CS-76-559, August, 1976.
- C. Radhakrishna Rao, Linear statistical inference and its applications, 2nd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1973. MR 0346957
- G. A. F. Seber, Linear regression analysis, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1977. MR 0436482
- C. C. Paige and M. A. Saunders, Least squares estimation of discrete linear dynamic systems using orthogonal transformations, SIAM J. Numer. Anal. 14 (1977), no. 2, 180–193. MR 437197, DOI 10.1137/0714012 S. KOUROUKLIS, Computing Weighted Linear Least Squares Solutions, McGill University School of Computer Science, M.Sc. Project, May 1977.
- G. W. Stewart, Introduction to matrix computations, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0458818
- A. van der Sluis, Stability of the solutions of linear least squares problems, Numer. Math. 23 (1974/75), 241–254. MR 373259, DOI 10.1007/BF01400307
- J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
- Sven Hammarling, A note on modifications to the Givens plane rotation, J. Inst. Math. Appl. 13 (1974), 215–218. MR 343568
- C. C. Paige, Fast numerically stable computations for generalized linear least squares problems, SIAM J. Numer. Anal. 16 (1979), no. 1, 165–171. MR 518691, DOI 10.1137/0716012
- C. C. Paige, Numerically stable computations for general univariate linear models, Comm. Statist. B—Simulation Comput. 7 (1978), no. 5, 437–453. MR 516832, DOI 10.1080/03610917808812090
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 171-183
- MSC: Primary 65D10; Secondary 65F35
- DOI: https://doi.org/10.1090/S0025-5718-1979-0514817-3
- MathSciNet review: 514817