NDLI: On the QR decomposition of $${\fancyscript {H}}$$ -matrices

Content Provider	Springer Nature Link
Author	Benner, Peter Mach, Thomas
Copyright Year	2010
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.
Ending Page	129
Page Count	19
Starting Page	111
File Format	PDF
ISSN	0010485X
e-ISSN	14365057
Journal	Computing
Issue Number	3-4
Volume Number	88
Language	English
Publisher	Springer Vienna
Publisher Date	2010-06-09
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	HQR decomposition Hierarchical matrices Computational Mathematics and Numerical Analysis Orthogonalization QR decomposition Factorization of matrices Computer Science Sparse matrices Orthogonalisation
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Computational Theory and Mathematics Computational Mathematics Numerical Analysis Computer Science Applications Software

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