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  • 1.
    Adlers, M
    et al.
    Linkoping Univ, Dept Math, SE-58183 Linkoping, Sweden.
    Bjorck, A
    Linkoping Univ, Dept Math, SE-58183 Linkoping, Sweden.
    Matrix stretching for sparse least squares problems2000Ingår i: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 7, nr 2, s. 51-65Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For linear least squares problems min(x) parallel to Ax - b parallel to(2) where A is sparse except for a few dense rows, a straightforward application of Cholesky or QR factorization will lead to catastrophic fill in the factor R. We consider handling such problems by a matrix stretching technique, where the dense rows are split into several more sparse rows. We develop both a recursive binary splitting algorithm and a more general splitting method. We show that for both schemes the stretched problem has the same set of solutions as the original least squares problem. Further. the condition number of the stretched problem differs from that of the original by only a modest factor, and hence the approach is numerically stable. Experimental results from applying the recursive binary scheme to a set of modified matrices from the Harwell-Boeing collection are given. We conclude that when A has a small number of dense rows relative to its dimension, there is a significant gain in sparsity of the factor R. A crude estimate of the optimal number of splits is obtained by analysing a simple model problem. Copyright (C) 2000 John Wiley & Sons, Ltd.

  • 2.
    Elden, Lars
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    Savas, Berkant
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    The maximum likelihood estimate in reduced-rank regression2005Ingår i: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 12, nr 8, s. 731-741Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In previous work by Stoica and Viberg the reduced-rank regression problem is solved in a maximum likelihood sense. The present paper proposes an alternative numerical procedure. The solution is written in terms of the principal angles between subspaces spanned by the data matrices. It is demonstrated that the solution is meaningful also in the case when the maximum likelihood criterion is not valid. A numerical example is given. Copyright (c) 2005 John Wiley & Sons, Ltd.

  • 3.
    Eldén, Lars
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Ahmadi-Asl, Salman
    Skolkovo Inst Sci and Technol Skoltech, Russia.
    Solving bilinear tensor least squares problems and application to Hammerstein identification2019Ingår i: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 26, nr 2, artikel-id e2226Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Bilinear tensor least squares problems occur in applications such as Hammerstein system identification and social network analysis. A linearly constrained problem of medium size is considered, and nonlinear least squares solvers of Gauss-Newton-type are applied to numerically solve it. The problem is separable, and the variable projection method can be used. Perturbation theory is presented and used to motivate the choice of constraint. Numerical experiments with Hammerstein models and random tensors are performed, comparing the different methods and showing that a variable projection method performs best.

  • 4.
    Leon, Steven J.
    et al.
    University of Massachusetts, MA 02747 USA .
    Björck, Åke
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska högskolan.
    Gander, Walter
    ETH, Switzerland .
    Gram-Schmidt orthogonalization: 100 years and more2013Ingår i: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506, Vol. 20, nr 3, s. 492-532Artikel, forskningsöversikt (Refereegranskat)
    Abstract [en]

    In 1907, Erhard Schmidt published a paper in which he introduced an orthogonalization algorithm that has since become known as the classical Gram-Schmidt process. Schmidt claimed that his procedure was essentially the same as an earlier one published by J. P.Gram in 1883. The Schmidt version was the first to become popular and widely used. An algorithm related to a modified version of the process appeared in an 1820 treatise by P. S. Laplace. Although related algorithms have been around for almost 200years, it is the Schmidt paper that led to the popularization of orthogonalization techniques. The year 2007 marked the 100th anniversary of that paper. In celebration of that anniversary, we present a comprehensive survey of the research on Gram-Schmidt orthogonalization and its related QR factorization. Its application for solving least squares problems and in Krylov subspace methods are also reviewed. Software and implementation aspects are also discussed.

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