13,189 research outputs found
Efficient numerical diagonalization of hermitian 3x3 matrices
A very common problem in science is the numerical diagonalization of
symmetric or hermitian 3x3 matrices. Since standard "black box" packages may be
too inefficient if the number of matrices is large, we study several
alternatives. We consider optimized implementations of the Jacobi, QL, and
Cuppen algorithms and compare them with an analytical method relying on
Cardano's formula for the eigenvalues and on vector cross products for the
eigenvectors. Jacobi is the most accurate, but also the slowest method, while
QL and Cuppen are good general purpose algorithms. The analytical algorithm
outperforms the others by more than a factor of 2, but becomes inaccurate or
may even fail completely if the matrix entries differ greatly in magnitude.
This can mostly be circumvented by using a hybrid method, which falls back to
QL if conditions are such that the analytical calculation might become too
inaccurate. For all algorithms, we give an overview of the underlying
mathematical ideas, and present detailed benchmark results. C and Fortran
implementations of our code are available for download from
http://www.mpi-hd.mpg.de/~globes/3x3/ .Comment: 13 pages, no figures, new hybrid algorithm added, matches published
version, typo in Eq. (39) corrected; software library available at
http://www.mpi-hd.mpg.de/~globes/3x3
Sensitivity analysis and approximation methods for general eigenvalue problems
Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought
An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations
An a posteriori verification method is proposed for the generalized
real-symmetric eigenvalue problem and is applied to densely clustered
eigenvalue problems in large-scale electronic state calculations. The proposed
method is realized by a two-stage process in which the approximate solution is
computed by existing numerical libraries and is then verified in a moderate
computational time. The procedure returns intervals containing one exact
eigenvalue in each interval. Test calculations were carried out for organic
device materials, and the verification method confirms that all exact
eigenvalues are well separated in the obtained intervals. This verification
method will be integrated into EigenKernel (https://github.com/eigenkernel/),
which is middleware for various parallel solvers for the generalized eigenvalue
problem. Such an a posteriori verification method will be important in future
computational science.Comment: 15 pages, 7 figure
Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems
In these lecture notes, we present a pedagogical review of a number of
related {\it numerically exact} approaches to quantum many-body problems. In
particular, we focus on methods based on the exact diagonalization of the
Hamiltonian matrix and on methods extending exact diagonalization using
renormalization group ideas, i.e., Wilson's Numerical Renormalization Group
(NRG) and White's Density Matrix Renormalization Group (DMRG). These methods
are standard tools for the investigation of a variety of interacting quantum
systems, especially low-dimensional quantum lattice models. We also survey
extensions to the methods to calculate properties such as dynamical quantities
and behavior at finite temperature, and discuss generalizations of the DMRG
method to a wider variety of systems, such as classical models and quantum
chemical problems. Finally, we briefly review some recent developments for
obtaining a more general formulation of the DMRG in the context of matrix
product states as well as recent progress in calculating the time evolution of
quantum systems using the DMRG and the relationship of the foundations of the
method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical
review appearing in the proceedings of the "IX. Training Course in the
Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri
sul Mare (Salerno, Italy, October 2004
Contemporary continuum QCD approaches to excited hadrons
Amongst the bound states produced by the strong interaction, radially excited
meson and nucleon states offer an important phenomenological window into the
long-range behavior of the coupling constant in Quantum Chromodynamics. We here
report on some technical details related to the computation of the bound
state's eigenvalue spectrum in the framework of Bethe-Salpeter and Faddeev
equations.Comment: Proceedings of the 21st International Conference on Few-body Problems
in Physics to be published in EPJ Web of Conference
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