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A sparse matrix arithmetic based on $${\mathfrak H}$$-matrices. I: Introduction to $${\mathfrak H}$$-matrices. (English) Zbl 0927.65063
The concept of hierarchical matrices $$\mathfrak H$$ is introduced. This paper is the first of a series that will be devoted to the $$\mathfrak H$$-matrices. These matrices are not sparse in the sense that there are only few non-zero entries, but they are data-sparse – the matrices are described by only few data. The other properties of such matrices are that the matrix-vector multiplication is of almost linear complexity, and sums and products of the matrices are not in the same set, but their truncation to the $$\mathfrak H$$-matrix format are of almost linear complexity. The same statement holds for the inverse of an $$\mathfrak H$$-matrix. Two new concepts are introduced. These allow the exact inversion of tridiagonal matrices and the approximation of some discrete integral operators.

##### MSC:
 65F30 Other matrix algorithms (MSC2010) 15B57 Hermitian, skew-Hermitian, and related matrices 65F05 Direct numerical methods for linear systems and matrix inversion 65F50 Computational methods for sparse matrices
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