TY - JOUR

T1 - Improved Gram-Schmidt type downdating methods

AU - Barlow, Jesse L.

AU - Smoktunowicz, Alicja

AU - Erbay, Hasan

N1 - Funding Information:
★★ The research of Jesse L. Barlow and Hasan Erbay was supported by the National Science Foundation under grant no. CCR-9732081. The research of Alicja Smoktunowicz was supported by a grant from the Faculty of Mathematics and Information Science of Warsaw University of Technology. Travel for this research was sponsored by an Eastern European supplement to Jesse Barlow’s NSF contract no.CCR-9732081.

PY - 2005/6

Y1 - 2005/6

N2 - 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.

AB - 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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U2 - 10.1007/s10543-005-0015-2

DO - 10.1007/s10543-005-0015-2

M3 - Article

AN - SCOPUS:27144496091

VL - 45

SP - 259

EP - 285

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 2

ER -