• Media type: E-Article
  • Title: Modifiable low‐rank approximation to a matrix
  • Contributor: Barlow, Jesse L.; Erbay, Hasan
  • Published: Wiley, 2009
  • Published in: Numerical Linear Algebra with Applications
  • Extent: 833-860
  • Language: English
  • DOI: 10.1002/nla.651
  • ISSN: 1070-5325; 1099-1506
  • Keywords: Applied Mathematics ; Algebra and Number Theory
  • Abstract: <jats:title>Abstract</jats:title><jats:p>A truncated ULV decomposition (TULVD) of an <jats:italic>m</jats:italic>×<jats:italic>n</jats:italic> matrix <jats:italic>X</jats:italic> of rank <jats:italic>k</jats:italic> is a decomposition of the form <jats:italic>X</jats:italic> = <jats:italic>ULV</jats:italic><jats:sup>T</jats:sup>+<jats:italic>E</jats:italic>, where <jats:italic>U</jats:italic> and <jats:italic>V</jats:italic> are left orthogonal matrices, <jats:italic>L</jats:italic> is a <jats:italic>k</jats:italic>×<jats:italic>k</jats:italic> non‐singular lower triangular matrix, and <jats:italic>E</jats:italic> is an error matrix. Only <jats:italic>U</jats:italic>,<jats:italic>V</jats:italic>, <jats:italic>L</jats:italic>, and ∥<jats:italic>E</jats:italic>∥<jats:sub><jats:italic>F</jats:italic></jats:sub> are stored, but <jats:italic>E</jats:italic> 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 (<jats:italic>SIAM J. Matrix Anal. Appl.</jats:italic> 2005; <jats:bold>27</jats:bold>(1):198–211) that reduces ∥<jats:italic>E</jats:italic>∥<jats:sub><jats:italic>F</jats:italic></jats:sub>, detects rank degeneracy, corrects it, and sharpens the approximation. Copyright © 2009 John Wiley &amp; Sons, Ltd.</jats:p>
  • Description: <jats:title>Abstract</jats:title><jats:p>A truncated ULV decomposition (TULVD) of an <jats:italic>m</jats:italic>×<jats:italic>n</jats:italic> matrix <jats:italic>X</jats:italic> of rank <jats:italic>k</jats:italic> is a decomposition of the form
    <jats:italic>X</jats:italic> = <jats:italic>ULV</jats:italic><jats:sup>T</jats:sup>+<jats:italic>E</jats:italic>, where <jats:italic>U</jats:italic> and <jats:italic>V</jats:italic> are left orthogonal matrices, <jats:italic>L</jats:italic> is a <jats:italic>k</jats:italic>×<jats:italic>k</jats:italic> non‐singular lower triangular matrix, and <jats:italic>E</jats:italic> is an error matrix. Only <jats:italic>U</jats:italic>,<jats:italic>V</jats:italic>, <jats:italic>L</jats:italic>, and ∥<jats:italic>E</jats:italic>∥<jats:sub><jats:italic>F</jats:italic></jats:sub> are stored, but <jats:italic>E</jats:italic> 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 (<jats:italic>SIAM J. Matrix Anal. Appl.</jats:italic> 2005; <jats:bold>27</jats:bold>(1):198–211) that reduces ∥<jats:italic>E</jats:italic>∥<jats:sub><jats:italic>F</jats:italic></jats:sub>, detects rank degeneracy, corrects it, and sharpens the approximation. Copyright © 2009 John Wiley &amp; Sons, Ltd.</jats:p>
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