Viriginia Klema

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NBER Working Papers and Publications

February 1977Rosetak Document 4: Rank Degeneracies and Least Square Problems
with Gene Golub, G. W. Stewart: w0165
In this paper we shall be concerned with the following problem. Let A be an m x n matrix with m being greater than or equal to n, and suppose that A is near (in a sense to be made precise later) a matrix B whose rank is less than n. Can one find a set of linearly independent columns of A that span a good approximation to the column space of B? The solution of this problem is important in a number of applications. In this paper we shall be chiefly interested in the case where the columns of A represent factors or carriers in a linear model which is to be fit to a vector of observations b. In some such applications, where the elements of A can be specified exactly (e.g. the analysis of variance), the presence of rank degeneracy in A can be dealt with by explicit mathematical formulas and cau...
July 1974The Singular Value Analysis in Matrix Computation
with Richard A. Becker, Neil Kaden: w0046
This paper discusses the robustness and the computational stability of the singular value decomposition algorithm used at the NBER Computer Research Center. The effect of perturbations on input data is explored. Suggestions are made for using the algorithm to get information about the rank of a real square or rectangular matrix. The algorithm can also be used to compute the best approximate solution of linear system of equations in the least squares sense, to solve linear systems of equations with equality constraints, and to determine dependencies or near dependencies among the rows or columns of a matrix. A copy of the subroutine that is used and some examples on which it has been tested are included in the appendixes.
July 1973A Note On Matrix Factorization
in Annals of Economic and Social Measurement, Volume 2, number 3, Sanford V. Berg, editor
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