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  • 1
    Electronic Resource
    Electronic Resource
    Algebraic connections between the least squares and total least squares problems (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 431-449 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 15A18 ; 65F20 CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper the closeness of the total least squares (TLS) and the classical least squares (LS) problem is studied algebraically. Interesting algebraic connections between their solutions, their residuals, their corrections applied to data fitting and their approximate subspaces are proven. All these relationships point out the parameters which mainly determine the equivalences and differences between the two techniques. These parameters also lead to a better understanding of the differences in sensitivity between both approaches with respect to perturbations of the data. In particular, it is shown how the differences between both approaches increase when the equationsAX≈B become less compatible, when the length ofB orX is growing or whenA tends to be rank-deficient. They are maximal whenB is parallel with the singular vector ofA associated with its smallest singular value. Furthermore, it is shown how TLS leads to a weighted LS problem, and assumptions about the underlying perturbation model of both techniques are deduced. It is shown that many perturbation models correspond with the same TLS solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396047
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Efficient reduction algorithms for bordered band matrices (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 95-113 
    Details
    ISSN: 1070-5325
    Keywords: arrowhead matrix ; band matrix ; inverse eigenvalue problem ; givens rotations ; singular value decomposition ; updating ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Various plane rotation patterns are presented, which provide stable algorithms for reducing a b-band matrix bordered by p rows and/or columns to (b + p)-band form. These schemes generalize previously presented O(N2) reduction algorithms for matrices of order N, b = 1, and p = 1 to the reduction of more general b-band, p-bordered matrices where b ≥ 1 and p ≥ 1. Moreover, by splitting the matrix into two similarly structured submatrices and chasing nonzeros to the corners in two directions, the newly proposed patterns reduce the number of required rotations and hence the computational cost by one half compared to the other existing one-way chasing algorithms. Symmetric, as well as more general matrices, are considered. An example of the first type is the symmetric arrowhead matrix that arises in solving inverse eigenvalue problems. Examples of the second type are found in updating the singular value decomposition (SVD) and the partial SVD.
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680020204
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    Paper (German National Licenses)
    Fulltext
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