title: Evaluating functions of positive-definite matrices using colored-noise thermostats.
abstract: Many applications in computational science require computing the elements of a function of a large matrix. A commonly used approach is based on the the evaluation of the eigenvalue decomposition, a task that, in general, involves a computing time that scales with the cube of the size of the matrix. We present here a method that can be used to evaluate the elements of a function of a positive-definite matrix with a scaling that is linear for sparse matrices and quadratic in the general case. This methodology is based on the properties of the dynamics of a multidimensional harmonic potential coupled with colored-noise, generalized Langevin equation thermostats. This "f-thermostat" approach allows us to calculate directly elements of functions of a positive-definite matrix by carefully tailoring the properties of the stochastic dynamics. We demonstrate the scaling and the accuracy of this approach for both dense and sparse problems and compare the results with other established methodologies.
authors.affiliation: Computational Science, Department of Chemistry and Applied Biosciences, ETH Zurich, USI-Campus, Via Giuseppe Buffi 13, C-6900 Lugano, Switzerland.
authors.forename: Marco
authors.initials: M
authors.lastname: Nava
authors.affiliation: Laboratory of Computational Science and Modeling, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
authors.forename: Michele
authors.initials: M
authors.lastname: Ceriotti
authors.affiliation: Computational Science, Department of Chemistry and Applied Biosciences, ETH Zurich, USI-Campus, Via Giuseppe Buffi 13, C-6900 Lugano, Switzerland.
authors.forename: Chaim
authors.initials: C
authors.lastname: Dryzun
authors.affiliation: Computational Science, Department of Chemistry and Applied Biosciences, ETH Zurich, USI-Campus, Via Giuseppe Buffi 13, C-6900 Lugano, Switzerland.
authors.forename: Michele
authors.initials: M
authors.lastname: Parrinello
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D008433:Mathematics; D015233:Models, Statistical; D009716:Numerical Analysis, Computer-Assisted