735 research outputs found
Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation
[EN] A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient to make the Taylor method superior in performance to Pade approximants over a range of values of the matrix norms. An efficient adjustment to make the method robust against overscaling is also introduced. Numerical experiments show the superior performance of our method to have a similar accuracy in comparison with state-of-the-art implementations, and thus, it is especially recommended to be used in conjunction with Lie-group and exponential integrators where preservation of geometric properties is at issue.This work was funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE). P.B. was additionally supported by a contract within the Program Juan de la Cierva Formacion (Spain).Bader, P.; Blanes Zamora, S.; Casas, F. (2019). Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation. Mathematics. 7(12):1-19. https://doi.org/10.3390/math7121174S119712Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P., & Zanna, A. (2000). Lie-group methods. Acta Numerica, 9, 215-365. doi:10.1017/s0962492900002154Blanes, S., Casas, F., Oteo, J. A., & Ros, J. (2009). The Magnus expansion and some of its applications. Physics Reports, 470(5-6), 151-238. doi:10.1016/j.physrep.2008.11.001Casas, F., & Iserles, A. (2006). Explicit Magnus expansions for nonlinear equations. Journal of Physics A: Mathematical and General, 39(19), 5445-5461. doi:10.1088/0305-4470/39/19/s07Celledoni, E., Marthinsen, A., & Owren, B. (2003). Commutator-free Lie group methods. Future Generation Computer Systems, 19(3), 341-352. doi:10.1016/s0167-739x(02)00161-9Crouch, P. E., & Grossman, R. (1993). Numerical integration of ordinary differential equations on manifolds. Journal of Nonlinear Science, 3(1), 1-33. doi:10.1007/bf02429858Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. doi:10.1017/s0962492910000048Najfeld, I., & Havel, T. F. (1995). Derivatives of the Matrix Exponential and Their Computation. Advances in Applied Mathematics, 16(3), 321-375. doi:10.1006/aama.1995.1017Sidje, R. B. (1998). Expokit. ACM Transactions on Mathematical Software, 24(1), 130-156. doi:10.1145/285861.285868Higham, N. J., & Al-Mohy, A. H. (2010). Computing matrix functions. Acta Numerica, 19, 159-208. doi:10.1017/s0962492910000036Paterson, M. S., & Stockmeyer, L. J. (1973). On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials. SIAM Journal on Computing, 2(1), 60-66. doi:10.1137/0202007Ruiz, P., Sastre, J., Ibáñez, J., & Defez, E. (2016). High performance computing of the matrix exponential. Journal of Computational and Applied Mathematics, 291, 370-379. doi:10.1016/j.cam.2015.04.001Sastre, J., Ibán͂ez, J., Defez, E., & Ruiz, P. (2015). New Scaling-Squaring Taylor Algorithms for Computing the Matrix Exponential. SIAM Journal on Scientific Computing, 37(1), A439-A455. doi:10.1137/090763202Sastre, J. (2018). Efficient evaluation of matrix polynomials. Linear Algebra and its Applications, 539, 229-250. doi:10.1016/j.laa.2017.11.010Westreich, D. (1989). Evaluating the matrix polynomial I+A+. . .+A/sup N-1/. IEEE Transactions on Circuits and Systems, 36(1), 162-164. doi:10.1109/31.16591An Efficient Alternative to the Function Expm of Matlab for the Computation of the Exponential of a Matrix http://www.gicas.uji.es/Research/MatrixExp.htmlKenney, C. S., & Laub, A. J. (1998). A Schur--Fréchet Algorithm for Computing the Logarithm and Exponential of a Matrix. SIAM Journal on Matrix Analysis and Applications, 19(3), 640-663. doi:10.1137/s0895479896300334Dieci, L., & Papini, A. (2000). Padé approximation for the exponential of a block triangular matrix. Linear Algebra and its Applications, 308(1-3), 183-202. doi:10.1016/s0024-3795(00)00042-2Higham, N. J., & Tisseur, F. (2000). A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIAM Journal on Matrix Analysis and Applications, 21(4), 1185-1201. doi:10.1137/s0895479899356080Celledoni, E., & Iserles, A. (2000). Approximating the exponential from a Lie algebra to a Lie group. Mathematics of Computation, 69(232), 1457-1481. doi:10.1090/s0025-5718-00-01223-
Spectral properties from Matsubara Green's function approach - application to molecules
We present results for many-body perturbation theory for the one-body Green's
function at finite temperatures using the Matsubara formalism. Our method
relies on the accurate representation of the single-particle states in standard
Gaussian basis sets, allowing to efficiently compute, among other observables,
quasiparticle energies and Dyson orbitals of atoms and molecules. In
particular, we challenge the second-order treatment of the Coulomb interaction
by benchmarking its accuracy for a well-established test set of small
molecules, which includes also systems where the usual Hartree-Fock treatment
encounters difficulties. We discuss different schemes how to extract
quasiparticle properties and assess their range of applicability. With an
accurate solution and compact representation, our method is an ideal starting
point to study electron dynamics in time-resolved experiments by the
propagation of the Kadanoff-Baym equations.Comment: 12 pages, 8 figure
Efficient computation of the sinc matrix function for the integration of second-order differential equations
This work deals with the numerical solution of systems of oscillatory
second-order differential equations which often arise from the
semi-discretization in space of partial differential equations. Since these
differential equations exhibit pronounced or highly) oscillatory behavior,
standard numerical methods are known to perform poorly. Our approach consists
in directly discretizing the problem by means of Gautschi-type integrators
based on matrix functions. The novelty contained here is
that of using a suitable rational approximation formula for the
matrix function to apply a rational Krylov-like
approximation method with suitable choices of poles. In particular, we discuss
the application of the whole strategy to a finite element discretization of the
wave equation
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