Hierarchical Matrices

In this chapter let T _I and T _J be binary cluster trees for the index sets I and J, respectively. A generalization to arbitrary cluster trees is possible; see Sect. 4.5 for nested dissection cluster trees, which are ternary. The block cluster tree T _I×J is assumed to be generated using a given admissibility condition as described in the previous chapter. We will define the set of hierarchical matrices originally introduced by Hackbusch [127] and Hackbusch and Khoromskij [133, 132]; see also [128]. The elements of this set can be stored with logarithmic-linear complexity and provide data-sparse representations of fully populated matrices. Additionally, combining the hierarchical partition and the efficient structure of low-rank matrices, an approximate algebra can be defined which is based on divide-and-conquer versions of the usual block operations. The efficient replacements for matrix addition and matrix multiplication can be used to define substitutes for higher level matrix operations such as inversion, LU factorization, and QR factorization.