Introductory Example

Abstract

The fact that a matrix can be simplified by compressing its submatrices is not new. The panel clustering method (cf. [149], [225, §7]), the multipole method (cf. [115], [225, §7.1.3.2]), mosaic approximation (cf. [238]), and matrix compression techniques by wavelets (cf. [76]) are based on the same concept. However, in none of these cases is it possible to efficiently perform matrix operations other than matrix-vector multiplication. Therefore, in this chapter we want to illustrate how matrix operations are performed and how costly they are. In particular, it will turn out that all matrix operations can be computed with almost linear work (instead of O(n²) or O(n³) for the full matrix representation). On the other hand, we have to notice that, in general, the results contain approximation errors (these will be analysed later). In the following model example we use a very simple and regularly structured block partition. We remark that in realistic applications the block partition will be adapted individually to the actual problem (cf. Chapter 5).