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By Semenov K.N.

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Bebendorf, Hierarchical Matrices. Lecture Notes in Computational Science and Engineering 63, c Springer-Verlag Berlin Heidelberg 2008 49 50 2 Hierarchical Matrices purely algebraic way using the hierarchical inverse or the hierarchical LU decomposition. Since we do not want to exploit properties of the underlying operator at this point, the complexity is estimated in terms of the maximum rank among the blocks in the partition. The size of this rank will be analyzed depending on the prescribed accuracy in the second part of this book when more properties of the underlying operator can be accessed.

The elements b ∈ P will be called index blocks or just blocks. It is common to denote the ith component of a vector x ∈ CI by xi . We will use the following generalization. If t ⊂ I, then xt ∈ Ct denotes the restriction of x to the indices in t. Note that Ct is used in contrast to C|t| in order to emphasize the index structure of x ∈ Ct . Analogously, Ats or Ab denotes the restriction of a given matrix A ∈ Cm×n to the indices in b := t × s, where t ⊂ I and s ⊂ J. 22 1 Low-Rank Matrices and Matrix Partitioning The aim of this section is to introduce algorithms for generating partitions P of matrices A ∈ CI×J such that the restriction Ab of A to each block b ∈ P can either be approximated by a matrix of low rank or is small.

Matrix operations such as addition and multiplication and relations between local and global norm estimates were investigated in detail in [116]. We review the results and improve some of the estimates in Sect. 4, Sect. 5, and Sect. 7, since these results will be important for higher matrix operations. Furthermore, we adapt the proofs to our way of clustering and present an improved addition algorithm which can be shown to preserve positivity. An accelerated matrix multiplication can be defined (see Sect.

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