Mathematics

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Browsing Mathematics by Author "Benner, Peter"

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    Computing the eigenvalues of symmetric H2-matrices by slicing the spectrum
    (Computing and Visualization in Science, 2015-03-04) Benner, Peter; Börm, Steffen; Mach, Thomas; Reimer, Knut
    The computation of eigenvalues of large-scale matrices arising from finite element discretizations has gained significant interest in the last decade (Knyazev et al. in Numerical solution of PDE eigenvalue problems, vol 56. Mathematisches Forschungsinstitut, Oberwolfach, 2013). Here we present an new algorithm based on slicing the spectrum that takes advantage of the rank structure of resolvent matrices in order to compute (Formula presented.) eigenvalues of the generalized symmetric eigenvalue problem in (Formula presented.) operations, where (Formula presented.) is a small constant.
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    On the QR decomposition of backslashfancyscript H -matrices
    (Computing (Vienna/New York), 2010-06-09) Benner, Peter; Mach, Thomas
    The hierarchical ( backslashfancyscriptH -) 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 backslashfancyscriptH -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 backslashfancyscriptH -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 backslashfancyscriptH -matrix. Like other backslashfancyscriptH -arithmetic operations, the backslashfancyscriptH 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.