Abstract
A class of matrices (H-matrices) has recently been introduced by one of the authors. These matrices have the following properties: (i) They are sparse in the sense that only few data are needed for their representation, (ii) The matrix-vector multiplication is of almost linear complexity, (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the H-matrix format are again of almost linear complexity, (iv) The same statement holds for the inverse of an H-matrix.
The term “almost linear complexity” used above means that estimates are given by O(nlogα n). The logarithmic factor can be avoided by a further improvement, which is described in the present paper. We prove that the storage requirements and the cost of the matrix-vector multiplication is strictly linear in the dimension n, while still (full) system matrices of the boundary element method can be approximated up to the discretization error.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Hackbusch, W.: Iterative Solution of Large Sparse Systems. Springer Verlag, New York, 1994
Hackbusch, W.: Integral Equations, Theory and Numerical Treatment. ISNM 128.Birkhäuser, Basel, 1995
Hackbusch, W.: A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices. Computing 62 (1999), 89–108
Hackbusch, W., Khoromskij, B.N.: A Sparse H-Matrix Arithmetic. Part II: Application to Multi-Dimensional Problems. Preprint Nr 22/1999, Max-Planck-Institut für Mathematik, Leipzig, 1999, to appear in Computing
Hackbusch, W., Khoromskij, B.N., Sauter, S.A.: On H 2-Matrices. Preprint, Max-Planck-Institut für Mathematik, Leipzig, 1999
Hackbusch, W., Nowak, Z.P.: On the Fast Matrix Multiplication in the Boundary Element Method by Panel Clustering. Numer. Math. 54 (1989), 463–491
Sauter, S.A.: Uber die effiziente Verwendung des Galerkin-Verfahrens zur Losung Fredholmscher Integralgleichungen. Dissertation, Universitat Kiel, 1992
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Professor Karl-Heinz Hoffmann on the ocasion of his 60th birthday
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hackbusch, W., Khoromskij, B., Sauter, S.A. (2000). On H 2-Matrices. In: Bungartz, HJ., Hoppe, R.H.W., Zenger, C. (eds) Lectures on Applied Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59709-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-59709-1_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64094-0
Online ISBN: 978-3-642-59709-1
eBook Packages: Springer Book Archive