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On optimal backward perturbation bounds for the linear least squares problem
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap.
1997 (engelsk)Inngår i: Bit, Vol. 37, nr 1, 179-188 s.Artikkel i tidsskrift (Fagfellevurdert) Published
Abstract [en]

Consider the linear least squares problem min(x) parallel to b-Ax parallel to(2), where A is an m x n (m > n) matrix, and b is an n-dimensional vector. Let y be an n-dimensional vector, and let eta(LS)(y) be the optimal backward perturbation bound defined by eta(LS)(y) = inf{parallel to F parallel to F : y is a solution to min(x) parallel to b - (A + F)x parallel to 2}. An explicit expression of eta(LS)(y) (y not equal 0) has been given in [8]. However, if we define the optimal backward perturbation bounds eta(MLS)(y) by eta(MLS)(y) = inf{parallel to F parallel to F : y is the minimum 2-norm solution to min(x) parallel to b - (A + F)x parallel to 2}, it may well be asked: How to derive an explicit expression of eta(MLS)(y)? This note gives an answer. The main result is: If b not equal 0 and y not equal 0, then eta(MLS)(y) = eta(LS)(y).

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1997. Vol. 37, nr 1, 179-188 s.
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URN: urn:nbn:se:umu:diva-22004ISBN: 0006-3835 OAI: oai:DiVA.org:umu-22004DiVA: diva2:212267
Tilgjengelig fra: 2009-04-21 Laget: 2009-04-21 Sist oppdatert: 2009-04-21

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