Journal ArticleDOI
A projection method for the computation of inner eigenvalues using high degree rational operators
Wolfgang Hackbusch,Wendy Kress +1 more
TLDR
In this article, the authors present a technique to efficiently calculate a projection without knowledge of the spectrum, which requires only few matrix-matrix products and inversions, which for some classes of matrices, like the $${\mathcal{H}}-matrices, can be computed in almost linear complexity.Abstract:
To efficiently calculate only part of the spectrum of a matrix, one can use a projection onto a suitable subspace. In this work, we present a technique to efficiently calculate such a projection without knowledge of the spectrum. The technique requires only few matrix–matrix products and inversions, which for some classes of matrices, like the $${\mathcal{H}}$$-matrices, can be computed in almost linear complexity.read more
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Journal ArticleDOI
Numerical methods for large eigenvalue problems
TL;DR: Over the past decade considerable progress has been made towards the numerical solution of large-scale eigenvalue problems, particularly for nonsymmetric matrices, and the methods and software that have led to these advances are surveyed.
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A sparse matrix arithmetic based on H -matrices. Part I: introduction to H -matrices
TL;DR: This paper is the first of a series and is devoted to the first introduction of the $\Cal H$-matrix concept, which allows the exact inversion of tridiagonal matrices.
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Introduction to Hierarchical Matrices with Applications
TL;DR: A short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems.
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Eigenvalue computation in the 20th century
TL;DR: The intention of this contribution is to sketch the main developments of this century, especially as they relate to one another, and to give an impression of the state of the art at the turn of the authors' century.
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Existence of ℋ-matrix approximants to the inverse FE-matrix of elliptic operators with L ∞ -coefficients
TL;DR: It will be shown that the corresponding Green functions can be approximated by degenerate functions giving rise to the existence of blockwise low-rank approximants of FEM inverses.