By Robert L. Taylor

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Duff (1997), Sparse numerical linear algebra: Direct methods and preconditioning, in The State of the Art in Numerical Analysis (I. S. Duff and G. A. Watson, eds), Oxford University Press, London, UK, pp. 27-62. L. Elden (1977), 'Algorithms for the regularization of ill-conditioned least squares problems', BIT 17, 134-145. L. Elden (1980), 'Perturbation theory for the least squares problem with linear equality constraints', SIAM J. Numer. Anal. 17, 338-350. 27 on Tue Nov 09 11:59:45 GMT 2010. 001 Cambridge Books Online © Cambridge University Press, 2010 THE CALCULATION OF LINEAR LEAST SQUARES PROBLEMS 49 L.

Then \\Ren\\ = \\PTRQQTw\\ = \\PTRw\\ = an, which shows that the entire last column in R is small. Given w the matrices P and Q can be constructed as a sequence of Givens rotations (see Stewart (1992), where algorithms are also given for updating a URV decomposition when a new row is appended). As for the RRQR decompositions, the URV decomposition yields approximations to the singular values. i, i= l : r , and where /= 1- - WRvWl Hence the smaller the norm of the off-diagonal block R\2, the better the bounds will be.

R. D. Fierro, P. Hansen, and P. S. K. Hansen (1999), 'UTV tools: Matlab templates for rank revealing UTV decompositions', Numer. Algorithms 20, 165-194. W. Gander (1981), 'Least squares with a quadratic constraint', Numer. Math. 36, 291-307. J. A. George and M. T. Heath (1980), 'Solution of sparse linear least squares problems using Givens rotations', Linear Algebra Appl. 34, 69-83. J. R. Gilbert, C. Moler and R. Schreiber (1992), 'Sparse matrices in MATLAB: Design and implementation', SIAM J. Matrix.