000			02886cam a22003257 4500
001			w0165
003			NBER
005			20211020115532.0
006			m o d
007			cr cnu||||||||
008			210910s1977 mau fo 000 0 eng d
100	1		_aGolub, Gene.
245	1	0	_aRosetak Document 4: _bRank Degeneracies and Least Square Problems / _cGene Golub, Virginia Klema, G. W. Stewart.
260			_aCambridge, Mass. _bNational Bureau of Economic Research _c1977.
300			_a1 online resource: _billustrations (black and white);
490	1		_aNBER working paper series _vno. w0165
500			_aFebruary 1977.
520	3		_aIn 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 causes no essential difficulties. In other applications, however, the presence of degeneracy is not at all obvious, and the failure to detect it can result in meaningless results or even the catastrophic failure of the numerical algorithms being used to solve the problem. The organization of this paper is the following. In the next section we shall give a precise definition of approximate degeneracy in terms of the singular value decomposition of A. In Section 3 we shall show that under certain conditions there is associated with A a subspace that is insensitive to how it is approximated by various choices of the columns of A, and in Section 4 we shall apply this result to the solution of the least squares problem. Sections 5, 6, and 7 will be concerned with algorithms for selecting a basis for the stable subspace from among the columns of A.
530			_aHardcopy version available to institutional subscribers
538			_aSystem requirements: Adobe [Acrobat] Reader required for PDF files.
538			_aMode of access: World Wide Web.
588	0		_aPrint version record
700	1		_aKlema, Virginia.
700	1		_aStewart, G. W.
710	2		_aNational Bureau of Economic Research.
830		0	_aWorking Paper Series (National Bureau of Economic Research) _vno. w0165.
856	4	0	_uhttps://www.nber.org/papers/w0165
856			_yAcceso en línea al DOI _uhttp://dx.doi.org/10.3386/w0165
942			_2ddc _cW-PAPER
999			_c348441 _d307003