A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices
Year: 2019
Communications in Computational Physics, Vol. 25 (2019), Iss. 2 : pp. 564–585
Abstract
We address the general mathematical problem of computing the inverse p-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary p-th roots and their inverses of symmetric positive definite matrices is presented. We show that the order of convergence is at least quadratic and that adjusting a parameter q leads to an even faster convergence. In this way, a better performance than with previously known iteration schemes is achieved. The efficiency of the iterative functions is demonstrated for various matrices with different densities, condition numbers and spectral radii.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2018-0053
Communications in Computational Physics, Vol. 25 (2019), Iss. 2 : pp. 564–585
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Matrix $p$-th root iteration function order of convergence symmetric positive definite matrices Newton-Schulz Altman hyperpower method.
-
Breaking the exascale barrier for the electronic structure problem in ab-initio molecular dynamics
Schade, Robert | Kenter, Tobias | Elgabarty, Hossam | Lass, Michael | Kühne, Thomas D. | Plessl, ChristianThe International Journal of High Performance Computing Applications, Vol. 37 (2023), Iss. 5 P.530
https://doi.org/10.1177/10943420231177631 [Citations: 8] -
Towards electronic structure-based ab-initio molecular dynamics simulations with hundreds of millions of atoms
Schade, Robert | Kenter, Tobias | Elgabarty, Hossam | Lass, Michael | Schütt, Ole | Lazzaro, Alfio | Pabst, Hans | Mohr, Stephan | Hutter, Jürg | Kühne, Thomas D. | Plessl, ChristianParallel Computing, Vol. 111 (2022), Iss. P.102920
https://doi.org/10.1016/j.parco.2022.102920 [Citations: 26] -
CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations
Kühne, Thomas D. | Iannuzzi, Marcella | Del Ben, Mauro | Rybkin, Vladimir V. | Seewald, Patrick | Stein, Frederick | Laino, Teodoro | Khaliullin, Rustam Z. | Schütt, Ole | Schiffmann, Florian | Golze, Dorothea | Wilhelm, Jan | Chulkov, Sergey | Bani-Hashemian, Mohammad Hossein | Weber, Valéry | Borštnik, Urban | Taillefumier, Mathieu | Jakobovits, Alice Shoshana | Lazzaro, Alfio | Pabst, Hans | Müller, Tiziano | Schade, Robert | Guidon, Manuel | Andermatt, Samuel | Holmberg, Nico | Schenter, Gregory K. | Hehn, Anna | Bussy, Augustin | Belleflamme, Fabian | Tabacchi, Gloria | Glöß, Andreas | Lass, Michael | Bethune, Iain | Mundy, Christopher J. | Plessl, Christian | Watkins, Matt | VandeVondele, Joost | Krack, Matthias | Hutter, JürgThe Journal of Chemical Physics, Vol. 152 (2020), Iss. 19
https://doi.org/10.1063/5.0007045 [Citations: 1931] -
A Submatrix-Based Method for Approximate Matrix Function Evaluation in the Quantum Chemistry Code CP2K
Lass, Michael | Schade, Robert | Kuhne, Thomas D. | Plessl, ChristianSC20: International Conference for High Performance Computing, Networking, Storage and Analysis, (2020), P.1
https://doi.org/10.1109/SC41405.2020.00084 [Citations: 4] -
A Massively Parallel Algorithm for the Approximate Calculation of Inverse p-th Roots of Large Sparse Matrices
Lass, Michael | Mohr, Stephan | Wiebeler, Hendrik | Kühne, Thomas D. | Plessl, ChristianProceedings of the Platform for Advanced Scientific Computing Conference, (2018), P.1
https://doi.org/10.1145/3218176.3218231 [Citations: 6]