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Browsing by Author "Mach, Thomas"

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Adaptive cross approximation for ill-posed problems

Mach, Thomas; Reichel, Lothar; Van Barel, Marc; Vandebril, R. (Journal of Computational and Applied Mathematics, 2016-09-01)

Integral equations of the first kind with a smooth kernel and perturbed right-hand side, which represents available contaminated data, arise in many applications. Discretization gives rise to linear systems of equations ...
An extended Hessenberg form for Hamiltonian matrices

Ferranti, Micol; Iannazzo, Bruno; Mach, Thomas; Vandebril, Raf (Calcolo, 2016-06-01)

A unitary symplectic similarity transformation for a special class of Hamiltonian matrices to extended Hamiltonian Hessenberg form is presented. Whereas the classical Hessenberg form links to Krylov subspaces, the extended ...
Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices

Mach, Thomas; Pranić, Miroslav S.; Vandebril, Raf (Electronic Transactions on Numerical Analysis, 2014)

It has been shown that approximate extended Krylov subspaces can be computed, under certain assumptions, without any explicit inversion or system solves. Instead, the vectors spanning the extended Krylov space are retrieved ...
Computing the eigenvalues of symmetric H2-matrices by slicing the spectrum

Benner, Peter; Börm, Steffen; Mach, Thomas; Reimer, Knut (Computing and Visualization in Science, 2015-03-04)

The computation of eigenvalues of large-scale matrices arising from finite element discretizations has gained significant interest in the last decade (Knyazev et al. in Numerical solution of PDE eigenvalue problems, vol ...
Convergence rates for inverse-free rational approximation of matrix functions

Jagels, Carl; Mach, Thomas; Reichel, Lothar; Vandebril, Raf (Linear Algebra and Its Applications, 2016-12-01)

This article deduces geometric convergence rates for approximating matrix functions via inverse-free rational Krylov methods. In applications one frequently encounters matrix functions such as the matrix exponential or ...
Fast and backward stable computation of roots of polynomials

Aurentz, Jared L.; Mach, Thomas; Vandebril, Raf; Watkins, David S. (SIAM Journal on Matrix Analysis and Applications, 2015)

A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis's implicitly shifted QR algorithm. A companion matrix is ...
Fast and backward stable computation of the eigenvalues of matrix polynomials

Aurentz, Jared; Mach, Thomas; Robol, Leonardo; Vandebril, Raf; Watkins, David S. (arXiv, 2016-11-30)

In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues. In this article we propose a new backward stable ...
Fast and stable unitary QR algorithm

Aurentz, Jared L.; Mach, Thomas; Vandebril, Raf; Watkins, David S. (Electronic Transactions on Numerical Analysis, 2015)

A fast Fortran implementation of a variant of Gragg's unitary Hessenberg QR algorithm is presented. It is proved, moreover, that all QR- And QZ-like algorithms for the unitary eigenvalue problems are equivalent. The algorithm ...
Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

Mach, Thomas; Van Barel, Marc; Vandebril, Raf (Journal of Computational and Applied Mathematics, 2014-12-15)

In inverse eigenvalue problems one tries to reconstruct a matrix, satisfying some constraints, given some spectral information. Here, two inverse eigenvalue problems are solved. First, given the eigenvalues and the first ...
On deflations in extended QR algorithms

Mach, Thomas; Vandebril, Raf (SIAM Journal on Matrix Analysis and Applications, 2014)

In this paper we discuss the deflation criterion used in the extended QR algorithm based on the chasing of rotations. We provide absolute and relative perturbation bounds for this deflation criterion. Further, we present ...
On the QR decomposition of backslashfancyscript H -matrices

Benner, Peter; Mach, Thomas (Computing (Vienna/New York), 2010-06-09)

The hierarchical ( backslashfancyscriptH -) matrix format allows storing a variety of dense matrices from certain applications in a special data-sparse way with linear-polylogarithmic complexity. Many operations from linear ...
Roots of Polynomials: on twisted QR methods for companion matrices and pencils

Aurentz, Jared L.; Mach, Thomas; Robol, Leonardo; Vandebril, Raf; Watkins, David S. (arXiv, 2016-11-08)

Two generalizations of the companion QR algorithm by J.L. Aurentz, T. Mach, R. Vandebril, and D.S. Watkins, SIAM Journal on Matrix Analysis and Applications, 36(3): 942--973, 2015, to compute the roots of a polynomial are ...