A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices

Abstract.

A class of matrices ($\Cal H$-matrices) is introduced which 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 $\Cal H$-matrix format are again of almost linear complexity. (iv) The same statement holds for the inverse of an $\Cal H$-matrix.

This paper is the first of a series and is devoted to the first introduction of the $\Cal H$-matrix concept. Two concret formats are described. The first one is the simplest possible. Nevertheless, it allows the exact inversion of tridiagonal matrices. The second one is able to approximate discrete integral operators.