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Computing

, Volume 88, Issue 3–4, pp 111–129 | Cite as

On the QR decomposition of \({\fancyscript {H}}\) -matrices

  • Peter Benner
  • Thomas MachEmail author
Article

Abstract

The hierarchical (\({\fancyscript{H}}\) -) matrix format allows storing a variety of dense matrices from certain applications in a special data-sparse way with linear-polylogarithmic complexity. Many operations from linear algebra like matrix–matrix and matrix–vector products, matrix inversion and LU decomposition can be implemented efficiently using the \({\fancyscript{H}}\) -matrix format. Due to its importance in solving many problems in numerical linear algebra like least-squares problems, it is also desirable to have an efficient QR decomposition of \({\fancyscript{H}}\) -matrices. In the past, two different approaches for this task have been suggested in Bebendorf (Hierarchical matrices: a means to efficiently solve elliptic boundary value problems. Lecture notes in computational science and engineering (LNCSE), vol 63. Springer, Berlin, 2008) and Lintner (Dissertation, Fakultät für Mathematik, TU München. http://tumb1.biblio.tu-muenchen.de/publ/diss/ma/2002/lintner.pdf, 2002). We will review the resulting methods and suggest a new algorithm to compute the QR decomposition of an \({\fancyscript{H}}\) -matrix. Like other \({\fancyscript{H}}\) -arithmetic operations, the \({\fancyscript{H}}\) QR decomposition is of linear-polylogarithmic complexity. We will compare our new algorithm with the older ones by using two series of test examples and discuss benefits and drawbacks of the new approach.

Keywords

Hierarchical matrices QR decomposition Orthogonalisation HQR decomposition 

Mathematics Subject Classification (2000)

65F25 65F50 15A23 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

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