610 research outputs found
Convergence of LR algorithm for a one-point spectrum tridiagonal matrix
We proved convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum - the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case
On the parallel solution of parabolic equations
Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented
Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential
The overlap operator in lattice QCD requires the computation of the sign
function of a matrix, which is non-Hermitian in the presence of a quark
chemical potential. In previous work we introduced an Arnoldi-based Krylov
subspace approximation, which uses long recurrences. Even after the deflation
of critical eigenvalues, the low efficiency of the method restricts its
application to small lattices. Here we propose new short-recurrence methods
which strongly enhance the efficiency of the computational method. Using
rational approximations to the sign function we introduce two variants, based
on the restarted Arnoldi process and on the two-sided Lanczos method,
respectively, which become very efficient when combined with multishift
solvers. Alternatively, in the variant based on the two-sided Lanczos method
the sign function can be evaluated directly. We present numerical results which
compare the efficiencies of a restarted Arnoldi-based method and the direct
two-sided Lanczos approximation for various lattice sizes. We also show that
our new methods gain substantially when combined with deflation.Comment: 14 pages, 4 figures; as published in Comput. Phys. Commun., modified
data in Figs. 2,3 and 4 for improved implementation of FOM algorithm,
extended discussion of the algorithmic cos
Density Matrix Renormalization Group and Reaction-Diffusion Processes
The density matrix renormalization group (DMRG) is applied to some
one-dimensional reaction-diffusion models in the vicinity of and at their
critical point. The stochastic time evolution for these models is given in
terms of a non-symmetric ``quantum Hamiltonian'', which is diagonalized using
the DMRG method for open chains of moderate lengths (up to about 60 sites). The
numerical diagonalization methods for non-symmetric matrices are reviewed.
Different choices for an appropriate density matrix in the non-symmetric DMRG
are discussed. Accurate estimates of the steady-state critical points and
exponents can then be found from finite-size scaling through standard
finite-lattice extrapolation methods. This is exemplified by studying the
leading relaxation time and the density profiles of diffusion-annihilation and
of a branching-fusing model in the directed percolation universality class.Comment: 16 pages, latex, 5 PostScript figures include
Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach
The real symmetric tridiagonal eigenproblem is of outstanding importance in
numerical computations; it arises frequently as part of eigensolvers for
standard and generalized dense Hermitian eigenproblems that are based on a
reduction to tridiagonal form. For its solution, the algorithm of Multiple
Relatively Robust Representations (MRRR) is among the fastest methods. Although
fast, the solvers based on MRRR do not deliver the same accuracy as competing
methods like Divide & Conquer or the QR algorithm. In this paper, we
demonstrate that the use of mixed precisions leads to improved accuracy of
MRRR-based eigensolvers with limited or no performance penalty. As a result, we
obtain eigensolvers that are not only equally or more accurate than the best
available methods, but also -in most circumstances- faster and more scalable
than the competition
Stabilizing Quantum States by Constructive Design of Open Quantum Dynamics
Based on recent work on the asymptotic behavior of controlled quantum
Markovian dynamics, we show that any generic quantum state can be stabilized by
devising constructively a simple Lindblad-GKS generator that can achieve global
asymptotic stability at the desired state. The applications of such result is
demonstrated by designing a direct feedback strategy that achieves global
stabilization of a qubit state encoded in a noise-protected subspace.Comment: Revised version with stronger proofs showing uniqueness can be
achieved in all cases by using the freedom to the choose diagonal elements of
both the Hamiltonian and Lindblad operator, and exploiting the fact that the
non-existence of two orthogonal eigenvectors of the Lindblad operator is
sufficient but not necessary for global asymptotic stability of the target
stat
From qd to LR, or, how were the qd and LR algorithms discovered?
Perhaps, the most astonishing idea in eigenvalue computation is Rutishauser's idea of applying the LR transform to a matrix for generating a sequence of similar matrices that become more and more triangular. The same idea is the foundation of the ubiquitous QR algorithm. It is well known that this idea originated in Rutishauser's qd algorithm, which precedes the LR algorithm and can be understood as applying LR to a tridiagonal matrix. But how did Rutishauser discover qd and when did he find the qd-LR connection? We checked some of the early sources and have come up with an explanatio
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