A weighted pseudoinverse, generalized singular values, and constrained least squares problems
Part II Numerical Mathematics Received: 19 February 1981 Revised: 01 April 1982 Abstract
The weighted pseudoinverse providing the minimum semi-norm solution of the weighted linear least squares problem is studied. It is shown that it has properties analogous to those of the Moore-Penrose pseudoinverse. The relation between the weighted pseudoinverse and generalized singular values is explained. The weighted pseudoinverse theory is used to analyse least squares problems with linear and quadratic constraints. A numerical algorithm for the computation of the weighted pseudoinverse is briefly described.
Key words Least squares pseudoinverse weight matrix constraint generalized singular values
This work was supported in part by the Swedish Institute for Applied Mathematics.
Regression and the Moore-Penrose Pseudoinverse
, Academic Press, New York, 1972.
A. Ben-Israel, T. N. E. Greville,
Generalized Inverses: Theory and Applications
, John Wiley & Sons, New York, 1974.
T. L. Boullion, P. L. Odell,
Generalized Inverse Matrices
, Wiley-Interscience, New York, 1971.
A note on weighted pseudoinverses with application to the regularization of Fredholm integral equations of the first kind
, Department of Mathematics, Linköping University, Report LiTH-MATR-1975-11, 1975.
Algorithms for the regularization of ill-conditioned least squares problems
Perturbation theory for the least squares problem with linear equality constraints
, SIAM J. Num. Anal.
(1980), pp. 338–350.
G. H. Golub,
Some modified eigenvalue problems
, SIAM Review 15 (1973), pp. 318–334.
C. L. Lawson, R. J. Hanson,
Solving Least Squares Problems
, Prentice-Hall, Englewood Cliffs, 1974.
D. G. Luenberger,
Introduction to Linear and Nonlinear Programming
, Addison-Wesley, Reading Mass., 1973.
Least squares methods for ill-posed problems with a prescribed bound
, SIAM J. Math. Anal.
(1970), pp. 52–74.
S. K. Mitra, C. R. Rao,
Projections under seminorms and generalized Moore Penrose inverses
, Lin. Alg. Appl.
(1974), pp. 155–167.
S. K. Mitra,
Optimal inverse of a matrix
, Sankhya, Ser. A,
(1975) pp. 550–563.
V. A. Morozov,
Regularization Methods for the Solution of Improperly Posed Problems
, Moscow University Press, Moscow, 1974 (Russian).
A. D. Poley,
Connections between some results on the generalized least squares problem
, Techn. Univ. Eindhoven, Dep. Math., TH-Report 81-WSK-01, 1981.
C. M. Price,
The matrix pseudoinverse and minimal variance estimates
, SIAM Review
(1964), pp. 115–120.
C. R. Rao, S. K. Mitra,
Generalized Inverse of Matrices and its Applications
, John Wiley & Sons, New York, 1971.
V. F. Turchin,
Selection of an ensemble of smooth functions for the solution of the inverse problem
, Zh. Vychisl. Mat. Mat. Fiz.
(1968), 230–238=USSR Comp. Math. Math. Phsy
(1968), pp. 328–339.
C. F. Van Loan,
Generalizing the singular value decomposition
, SIAM J. Numer. Anal.
(1976), pp. 76–83.
J. F. Ward,
On a limit formula for weighted pseudoinverses
, SIAM J. Appl. Math.
(1977), pp. 34–38.
Notes on the constrained least squares problem. A new approach based on generalized inverses
. Report UMINF - 75.79, Department of Numerical Analysis, Umeå University, 1979.