Yazar "Barlow, Jesse L." seçeneğine göre listele

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    An alternative algorithm for the refinement of ULV decompositions
    (Siam Publications, 2005) Barlow, Jesse L.; Erbay, Hasan; Slapnicar, Ivan
    The ULV decomposition ( ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition ( SVD). It is useful in applications of the SVD such as principal components where we are interested in approximating a matrix by one of lower rank. It can be updated and downdated much more quickly than an SVD. In many instances, the ULVD must be refined to improve the approximation it gives for the important right singular subspaces or to improve the matrix approximation. Present algorithms to perform this refinement require O( mn) operations if the rank of the matrix is k, where k is very close to 0 or n, but these algorithms require O( mn(2)) operations otherwise. Presented here is an alternative refinement algorithm that requires O( mn) operations no matter what the rank is. Our tests show that this new refinement algorithm produces similar improvement in matrix approximation and in the subspaces. We also propose slight improvements on the error bounds on subspaces and singular values computed by the ULVD.
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    Improved Gram-Schmidt type downdating methods
    (Springer, 2005) Barlow, Jesse L.; Smoktunowicz, Alicja; Erbay, Hasan
    The problem of deleting a row from a Q-R factorization (called downdating) using Gram-Schmidt orthogonalization is intimately connected to using classical iterative methods to solve a least squares problem with the orthogonal factor as the coefficient matrix. Past approaches to downdating have focused upon accurate computation of the residual of that least squares problem, then finding a unit vector in the direction of the residual that becomes a new column for the orthogonal factor. It is also important to compute the solution vector of the related least squares problem accurately, as that vector must be used in the downdating process to maintain good backward error in the new factorization. Using this observation, new algorithms are proposed. One of the new algorithms proposed is a modification of one due to Yoo and Park [BIT, 36:161-181, 1996]. That algorithm is shown to be a Gram-Schmidt procedure. Also presented are new results that bound the loss of orthogonality after downdating. An error analysis shows that the proposed algorithms' behavior in floating point arithmetic is close to their behavior in exact arithmetic. Experiments show that the changes proposed in this paper can have a dramatic impact upon the accuracy of the downdated Q-R decomposition.
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    Modifiable low-rank approximation to a matrix
    (Wiley, 2009) Barlow, Jesse L.; Erbay, Hasan
    A truncated ULV decomposition (TULVD) of an m x n matrix X of rank k is a decomposition of the form X=ULVT + E, where U and V are left orthogonal matrices, L is a k x k non-singular lower triangular matrix and E is an error matrix. Only U,V, L and parallel to E parallel to(F) are stored, but E is not stored. We propose algorithms for updating and downdating the TULVD. To construct these modification algorithms, we also use a refinement algorithm based upon that in (SIAM J. Matrix Anal. Appl. 2005; 27(1):198-211) that reduces parallel to E parallel to(F), detects rank degeneracy, corrects it, and sharpens the approximation. Copyright (C) 2009 John Wiley & Sons, Ltd.