Thomas joined the Department of Mathematics in September 2016. Thomas Mach’s research interests lie in numerical linear algebra. He focuses on structured preserving and structure exploiting algorithms. Numerical linear algebra algorithms can become orders of magnitude faster by exploiting the available structure in matrices. Structure preservation on the other hand can also be expensive, but necessary to preserve physical properties. Some years ago, Thomas investigated the solution of boundary element and eigenvalue problems with hierarchical matrices. Recently, he has worked on backward stable algorithm for the unitary and Hamiltonian eigenvalue problem, and for polynomial root finding. Currently, his research further involves inverse eigenvalue problem, adaptive cross approximation, inverse integral equations, and polynomial and structured eigenvalue problems.
Thomas holds a PhD degree from the University of Chemnitz, Germany, where he also obtained an undergraduate degree. During his PhD he worked for the Max Planck Institute for Dynamics of Complex Technical Systems. The last four years Thomas worked as a postdoctoral researcher in Raf Vandebril’s group at KU Leuven, Belgium.
Featured Publication
Recently the paper Convergence rates for inverse-free rational approximation of matrix functions joint work with Carl Jagels, Thomas Mach, Lothar Reichel, and Raf Vandebril got accepted for publication in Linear Algebra and Its Applications (DOI: 10.1016/j.laa.2016.08.029). The paper investigates the convergence behavior of the method developed in Computing Approximate Extended Krylov Subspaces without Explicit Inversion and Computing Approximate (Block) Rational Krylov Subspaces without Explicit Inversion with Extensions to Symmetric Matrices by Thomas Mach, Miroslav Pranić, and Raf Vandebril.
Research Interest
Numerical linear algebra for large or structured matrices, especially eigenvalue algorithms, tensor-structured, hierarchical, data-sparse, and rank structured matrices, unitary eigenvalue problems, polynomial root-finding, inverse eigenvalue problems, adaptive cross approximation, Krylov subspaces, and Givens rotations
Short CV
- 2016—2019: Assistant Professor, School of Science and Technology, Nazarbayev University, Kazakhstan
- 2012—2016: PostDoc, Department Computer Science, KU Leuven, Belgium
- 2010—2012: PhD student, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany
- 2008—2010: PhD student, TU Chemnitz, Germany
Core Chasing Algorithms for the Eigenvalue ProblemAurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.,
2018Society of Industrial and Applied Mathematics,
Fast and backward stable computation of roots of polynomials, Part II: Backward error analysis; companion matrix and companion pencilAurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.,
2018 In : SIAM Journal on Matrix Analysis and Applications. 39, 3, p. 1245-1269
Fast and backward stable computation of eigenvalues and eigenvectors of matrix polynomialsAurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.,
2018 In : Mathematics of Computation. 88, 315, p. 313-347
Computing the eigenvalues of symmetric tridiagonal matrices via a Cayley transformationAurentz, J., Mach, T., Vandebril, R., Watkins, D.,
2017 In : Electronic Transactions on Numerical Analysis. 46, p. 447-459
An extended Hamiltonian QR algorithmFerranti, M., Iannazzo, B., Mach, T., Vandebril, R.,
2017 In : Calcolo. 54, 3, p. 1097
An extended Hessenberg form for Hamiltonian matricesFerranti, M., Iannazzo, B., Mach, T., Vandebril, R.,
2016 In : Calcolo. 54, p. 423-453
Convergence rates for inverse-free rational approximation of matrix functionsJagels, C., Mach, T., Reichel, L., Vandebril, R.,
2016 In : Linear Algebra and Its Applications. 510, p. 291-310
Fast and backward stable computation of the eigenvalues of matrix polynomialsAurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.,
2016 In : arXiv. 1611, 10142,
Roots of Polynomials: on twisted QR methods for companion matrices and pencilsAurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.,
2016 In : arXiv. 1611, 02435,
Adaptive cross approximation for ill-posed problemsMach, T., Reichel, L., Van Barel, M., Vandebril, R.,
2016 In : Journal of Computational and Applied Mathematics. 303, p. 206-217
A note on compacnion pencilsAurentz, J., Mach, T., Vandebril, R., Watkins, D.,
2016 A Panorama of Mathematics: Pure and Applied. American Mathematical Society, 658, p. 91-102
eiscor - EIgenSolvers based on unitary CORe transformationsAurentz, J., Mach, T., Vandebril, R., Watkins, D.,
2015Computing the eigenvalues of symmetric H2-matrices by slicing the spectrumBenner, P., Börm, S., Mach, T., Reimer, K.,
2015 In : Computing and Visualization in Science.
Fast and stable unitary QR algorithmAurentz, J., Mach, T., Vandebril, R., Watkins, D.,
2015 In : Electronic Transactions on Numerical Analysis. 44, p. 327-341
Fast and backward stable computation of roots of polynomialsAurentz, J., Mach, T., Vandebril, R., Watkins, D.,
2015 In : SIAM Journal on Matrix Analysis and Applications. 36, 3, p. 942-973
Extended Hamiltonian Hessenberg Matrices arise in Projection based Model Order ReductionFerranti, M., Mach, T., Vandebril, R.,
2015 Proceedings in Applied Mathematics and Mechanics. Wiley-VCH, 15, p. 583-584
Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matricesMach, T., Van Barel, M., Vandebril, R.,
2014 In : Journal of Computational and Applied Mathematics. 272, p. 377-398
On deflations in extended QR algorithmsMacH, T., Vandebril, R.,
2014 In : SIAM Journal on Matrix Analysis and Applications. 35, 2, p. 559-579
Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matricesMach, T., Pranić, M., Vandebril, R.,
2014 In : Electronic Transactions on Numerical Analysis. 43, p. 100-124
A numerical example showing that deflations based on rotation leads to higher relative accuracy in QR algorithmMach, T., Vandebril, R.,
2014 Proceedings in Applied Mathematics and Mechanics. Wiley-VCH, 14, p. 823-824
The preconditioned inverse iteration for hierarchical matricesBenner, P., Mach, T.,
2013 In : Numerical Linear Algebra with Applications. 20, 1, p. 150-166
The LR Cholesky algorithm for symmetric hierarchical matricesBenner, P., Mach, T.,
2013 In : Linear Algebra and Its Applications. 439, 4, p. 1150-1166
Computing Inner Eigenvalues of Matrices in Tensor Train Matrix FormatMach, T.,
2013 ENUMATH 2011 Proceedings Volume. p. 781-789
Computing approximate extended Krylov subspaces without explicit inversionMach, T., Praníc, M., Vandebril, R.,
2013 In : Electronic Transactions on Numerical Analysis. 40, p. 414-435
Eigenvalue Algorithms for Symmetric Hierarchical MatricesMach, T.,
2012Qucosa,
How Competitive is the ADI for Tensor Structured Equations?Mach, T., Saak, J.,
2012 Proceedings in Applied Mathematics and Mechanics. Wiley-VCH, 12, p. 635-636
Computing all or some eigenvalues of symmetric Hl-matricesBenner, P., Mach, T.,
2012 In : SIAM Journal on Scientific Computing. 34, 1,
Extend ADI to tensor structured equationsMach, T., Saak, J.,
2011Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical MatricesBenner, P., Mach, T.,
2011 Proceedings in Applied Mathematics and Mechanics. Wiley-VCH, 11, p. 741-742
On the QR decomposition of backslashfancyscript H -matricesBenner, P., Mach, T.,
2010 In : Computing (Vienna/New York). 88, 3, p. 111-129
Computing the Eigenvalues of Hierarchical Matrices by LR-Cholesky TransformationsBenner, P., Mach, T.,
2009 Mathematisches Forschungsinstitut Oberwolfach, Report No. 37/2009. p. 325-328
Lösung von Randintegralgleichungen zur Bestimmung der Kapazitätsmatrix von Elektrodenanordnungen mittels $H$-ArithmetikMach, T.,
2008Qucosa,
Control of a shell and tube heat exchangerHajdu, A., Mach, T., Medina, P., Yu, E.,
2006 Proceedings of the European Student Workshop on Mathematical Modelling in Industry 2005. p. 117-134
