Address

Department of Mathematics, School of Science and Technology
Nazarbayev University
53 Kabanbay Batyr Ave, Block 7, Office 7.519
010000 Astana
Kazakhstan



Phone: +7 (7172) 70 4664
Office: 7.519
Office hours: Tuesday 10:00–11:15, Wednesday 09:00–12:00, and Thursday 10:00–11:15
E-mail: thomas.mach(at)nu.edu.kz

Featured Publication

Conformal J13 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

Publications

3 Submitted Articles

Companion matrix
S3 Roots of Polynomials: on twisted QR methods for companion matrices and pencils;
Aurentz, Jared L.; Mach, Thomas; Robol, Leonardo; Vandebril, Raf; Watkins, David S.;
Submitted for publication; 2016.
Available as preprint arXiv:1611.02435, 28 pages.
Companion matrix
S2 Fast and backward stable computation of the eigenvalues of matrix polynomials;
Aurentz, Jared L.; Mach, Thomas; Robol, Leonardo; Vandebril, Raf; Watkins, David S.;
Submitted for publication; 2016.
Available as preprint arXiv:1611.10142, 21 pages.
Cayley transform
S1 Yet another algorithm for the symmetric eigenvalue problem;
Aurentz, Jared L.; Mach, Thomas; Vandebril, Raf; Watkins, David S.;
Submitted to ETNA Electronic Transaction in Numerical Analysis; 2016.

15 Articles in Journals

Bulge exchange
J15 An extended Hamiltonian QR algorithm;
Ferranti, Micol; Iannazzo, Bruno; Mach, Thomas; Vandebril, Raf;
Calcolo  :  Vol. 54(3), pp.  1097–1120;
Springer; 2017.
DOI: 10.1007/s10092-017-0220-9
Also available as Report TW 667, 21 pages.
Ritz value plot
J14 An extended Hessenberg form for Hamiltonian matrices;
Ferranti, Micol; Iannazzo, Bruno; Mach, Thomas; Vandebril, Raf;
Calcolo  :  Vol. 54, pp. 423–453;
Springer; 2017.
DOI: 10.1007/s10092-016-0192-1
Also available as Report TW 665, 22 pages.
Conformal J13 Convergence rates for inverse-free rational approximation of matrix functions;
Jagels, Carl; Mach, Thomas; Reichel, Lothar; Vandebril, Raf;
Linear Algebra and Its Applications  :  Vol. 510, pp. 291–310;
Springer; 2016.
DOI: 10.1016/j.laa.2016.08.029
Also available from Lothar Reichel's website, 16 pages.
Solution vs exact J12 Adaptive cross approximation for ill-posed problems;
Mach, Thomas; Reichel, Lothar; Van Barel, Marc; Vandebril, Raf;
Journal of Computational and Applied Mathematics  :  Vol. 303, pp. 206–217;
Wiley InterScience; 2016.
DOI: 10.1016/j.cam.2016.02.020
Unitary matrix with bulge J11 Fast and stable unitary QR algorithm;
Aurentz, Jared L.; Mach, Thomas; Vandebril, Raf; Watkins, David S.;
ETNA Electronic Transaction in Numerical Analysis  :  Vol. 44, pp. 327–341;
ETNA, Kent State University; 2015.
The related software package EISCOR is available from Github.
Companion matrix
J10 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  :  Vol. 36, No. 3, pp. 942–973
SIAM; 2015.
DOI: 10.1137/140983434
Also available as Report TW 654, 36 pages. The related software is available from Raf Vandebril's website or as julia package from https://github.com/andreasnoack/AMVW.jl written by Andreas Noack.
J9 Counting the eigenvalues of symmetric ℋ²-matrices by slicing the spectrum;
Benner, Peter; Börm, Steffen; Mach, Thomas; Reimer, Knut;
Computing and Visualization in Science  :  Vol. 16, No. 6, pp. 271–282;
Springer; 2015.
DOI: 10.1007/s00791-015-0238-y
Also available as preprint arXiv:1403.4142 and MPIMD/14-06, 22 pages.
J8 Computing Approximate (Block) Rational Krylov Subspaces without Explicit Inversion with Extensions to Symmetric Matrices;
Mach, Thomas; Pranić, Miroslav S.; Vandebril, Raf;
ETNA Electronic Transaction in Numerical Analysis  :  Vol. 43, pp. 100–124;
ETNA, Kent State University; 2014.
Also available as Report TW 636, 24 pages.
QR Algorithm
J7 On Deflations in Extended QR Algorithms;
Mach, Thomas; Vandebril, Raf;
SIAM Journal on Matrix Analysis and Applications  :  Vol. 35, No. 2, pp. 559–579;
SIAM; 2014.
DOI: 10.1137/130935665
Also available as Report TW 634, 20 pages.
QR factorization
J6 Inverse Eigenvalue Problems Linked to Rational Arnoldi, and Rational (Non)Symmetric Lanczos;
Mach, Thomas; Van Barel, Marc; Vandebril, Raf;
Journal of Computational and Applied Mathematics  :  Vol. 272, pp. 377–398;
Elsevier; 2014.
DOI: 10.1016/j.cam.2014.03.015
Also available as Report TW 629, 22 pages.
J5 Computing Approximate Extended Krylov Subspaces without Explicit Inversion;
Mach, Thomas; Pranić, Miroslav S.; Vandebril, Raf;
ETNA Electronic Transaction in Numerical Analysis  :  Vol. 40, pp. 414–435;
ETNA, Kent State University; 2013.
Also available as Report TW 623, 22 pages.
complexity plot
J4 The Preconditioned Inverse Iteration for Hierarchical Matrices;
Benner, Peter; Mach, Thomas;
Numerical Linear Algebra with Applications  :   Vol. 20, No. 1, pp. 150–166;
Wiley; 2013.
DOI: 10.1002/nla.1830
Also available as preprint MPIMD/11-01, 17 pages.
deflation example
J3 The LR Cholesky Algorithm for Symmetric Hierarchical Matrices;
Benner, Peter; Mach, Thomas;
Linear Algebra and Its Applications  :  Vol. 439, No. 4, pp. 1150–1166;
Elsevier; 2013.
DOI: 10.1016/j.laa.2013.03.001
Also available as preprint MPIMD/12-05, 17 pages.
Error plot
J2 Computing all or some Eigenvalues of symmetric ℋ-Matrices;
Benner, Peter; Mach, Thomas;
SIAM Journal of Scientific Computing  :  Vol. 34, No. 1, pp. A485–A496;
SIAM; 2012.
DOI: 10.1137/100815323
Also available as preprint MPIMD/10-01, 12 pages.
HQR
J1 On the QR Decomposition of ℋ-Matrices;
Benner, Peter; Mach, Thomas;
Computing  :  Vol. 88, No. 3–4, pp. 111–129;
Springer; 2010.
DOI: 10.1007/s00607-010-0087-y
Also available as CSC preprint 09-04, 19 pages.

7 Articles in Conference Proceedings

dynamical system
CA7 Extended Hamiltonian Hessenberg Matrices arise in Projection based Model Order Reduction;
Ferranti, Micol; Mach, Thomas; Vandebril, Raf;
Proceedings in Applied Mathematics and Mechanics  :  Vol. 15, No. 1, pp. 583–584;
Wiley InterScience; 2015.
DOI: 10.1002/pamm.201510281
Companion matrix
CA6 A note on compenion pencils;
Aurentz, Jared L.; Mach, Thomas; Vandebril, Raf; Watkins, David S.;
AMS Contemporary Mathematics  :  Vol. 658, pp. 91–102
AMS; 2016.
QR Algorithm
CA5 A numerical example showing that deflations based on rotation leads to higher relative accuracy in QR algorithm;
Mach, Thomas; Vandebril, Raf;
Proceedings in Applied Mathematics and Mechanics  :  Vol. 14, No. 1, pp. 823–824;
Wiley InterScience; 2014.
DOI: 10.1002/pamm.201410392
Tensor Ranks
CA4 How Competitive is the ADI for Tensor Structured Equations?;
Mach, Thomas; Saak, Jens;
Proceedings in Applied Mathematics and Mechanics  :  Vol. 12, No. 1, pp. 635–636;
Wiley InterScience; 2012.
DOI: 10.1002/pamm.201210306.
tensor train matrix tensor train vector product
CA3 Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format;
Mach, Thomas;
in A. Cangiani, R.L. Davidchack, E.H. Georgoulis, A. Gorban, J. Levesley, M.V. Tretyakov: Numerical Mathematics and Advanced Applications 2011 - Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics, Leicester
Springer; 2011.
ISBN 978-3642331336, pages 781–789.
Also available as preprint MPIMD/11-09, 11 pages.
complexity plot
CA2 Locally Optimal Block Preconditioned Conjugate Gradient Method for Hierarchical Matrices;
Benner, Peter; Mach, Thomas;
Proceedings in Applied Mathematics and Mechanics  :  Vol. 11, No. 1, pp. 741–742;
Wiley InterScience; 2011.
DOI: 10.1002/pamm.201110360.
Deflation Example
CA1 Computing the Eigenvalues of Hierarchical Matrices by LR-Cholesky Transformations;
Benner, Peter; Mach, Thomas;
Mathematisches Forschungsinstitut Oberwolfach, Report No. 37/2009, pp. 325–328;
Mathematisches Forschungsinstitut Oberwolfach; 2009.
DOI: 10.4171/OWR/2009/37.
Also available as preprint, 3 pages.

1 Technical Report

Tensor Ranks
TR1 Towards an ADI iteration for Tensor Structured Equations;
Mach, Thomas; Saak, Jens;
; 2011.
Available as preprint MPIMD/11-12, 30 pages.

2 Thesis

Titlepage
T2 Eigenvalue Algorithms for Symmetric Hierarchical Matrices;
Mach, Thomas;
Dissertation; TU Chemnitz; 2012.
175 pages. Also available on Google Books.
Charge density
T1 Lösung von Randintegralgleichungen zur Bestimmung der Kapazitätsmatrix von Elektrodenanordnungen mittels ℋ-Arithmetik (in German);
Mach, Thomas;
Diplomarbeit; TU Chemnitz; 2008.
73 pages.

1 Other Publication

Photo
O1 Control of a shell and tube heat exchanger;
Andrea Hajdu, Thomas Mach, Paul Medina, Eric Yu;
in Proceedings of the European Student Workshop on Mathematical Modelling in Industry; 2006.
ISBN 84-611-0873-6, pp. 117–134.

See also my Google Scholar profile.