110 research outputs found
Harmonic and Refined Harmonic Shift-Invert Residual Arnoldi and Jacobi--Davidson Methods for Interior Eigenvalue Problems
This paper concerns the harmonic shift-invert residual Arnoldi (HSIRA) and
Jacobi--Davidson (HJD) methods as well as their refined variants RHSIRA and
RHJD for the interior eigenvalue problem. Each method needs to solve an inner
linear system to expand the subspace successively. When the linear systems are
solved only approximately, we are led to the inexact methods. We prove that the
inexact HSIRA, RHSIRA, HJD and RHJD methods mimic their exact counterparts well
when the inner linear systems are solved with only low or modest accuracy. We
show that (i) the exact HSIRA and HJD expand subspaces better than the exact
SIRA and JD and (ii) the exact RHSIRA and RHJD expand subspaces better than the
exact HSIRA and HJD. Based on the theory, we design stopping criteria for inner
solves. To be practical, we present restarted HSIRA, HJD, RHSIRA and RHJD
algorithms. Numerical results demonstrate that these algorithms are much more
efficient than the restarted standard SIRA and JD algorithms and furthermore
the refined harmonic algorithms outperform the harmonic ones very
substantially.Comment: 15 pages, 4 figure
Some results on condition numbers of the scaled total least squares problem
AbstractUnder the Golub–Van Loan condition for the existence and uniqueness of the scaled total least squares (STLS) solution, a first order perturbation estimate for the STLS solution and upper bounds for condition numbers of a STLS problem have been derived by Zhou et al. recently. In this paper, a different perturbation analysis approach for the STLS solution is presented. The analyticity of the solution to the perturbed STLS problem is explored and a new expression for the first order perturbation estimate is derived. Based on this perturbation estimate, for some STLS problems with linear structure we further study the structured condition numbers and derive estimates for them. Numerical experiments show that the structured condition numbers can be markedly less than their unstructured counterparts
A rounding error analysis of the joint bidiagonalization process with applications to the GSVD computation
The joint bidiagonalization(JBD) process is a useful algorithm for
approximating some extreme generalized singular values and vectors of a large
sparse or structured matrix pair {A,L\}. We present a rounding error analysis
of the JBD process, which establishes connections between the JBD process and
the two joint Lanczos bidiagonalizations. We investigate the loss of
orthogonality of the computed Lanczos vectors. Based on the results of rounding
error analysis, we investigate the convergence and accuracy of the approximate
generalized singular values and vectors of {A,L\}. The results show that
semiorthogonality of the Lanczos vectors is enough to guarantee the accuracy
and convergence of the approximate generalized singular values, which is a
guidance for designing an efficient semiorthogonalization strategy for the JBD
process. We also investigate the residual norm appeared in the computation of
the generalized singular value decomposition (GSVD), and show that its upper
bound can be used as a stopping criterion.Comment: 28 pages, 9 figure
Coherent manipulation of spin wave vector for polarization of photons in an atomic ensemble
We experimentally demonstrate the manipulation of two-orthogonal components
of a spin wave in an atomic ensemble. Based on Raman two-photon transition and
Larmor spin precession induced by magnetic field pulses, the coherent rotations
between the two components of the spin wave is controllably achieved.
Successively, the two manipulated spin-wave components are mapped into two
orthogonal polarized optical emissions, respectively. By measuring Ramsey
fringes of the retrieved optical signals, the \pi/2-pulse fidelity of ~96% is
obtained. The presented manipulation scheme can be used to build an arbitrary
rotation for qubit operations in quantum information processing based on atomic
ensembles
Quantum Interference of Stored Coherent Spin-wave Excitations in a Two-channel Memory
Quantum memories are essential elements in long-distance quantum networks and
quantum computation. Significant advances have been achieved in demonstrating
relative long-lived single-channel memory at single-photon level in cold atomic
media. However, the qubit memory corresponding to store two-channel spin-wave
excitations (SWEs) still faces challenges, including the limitations resulting
from Larmor procession, fluctuating ambient magnetic field, and
manipulation/measurement of the relative phase between the two channels. Here,
we demonstrate a two-channel memory scheme in an ideal tripod atomic system, in
which the total readout signal exhibits either constructive or destructive
interference when the two-channel SWEs are retrieved by two reading beams with
a controllable relative phase. Experimental result indicates quantum coherence
between the stored SWEs. Based on such phase-sensitive storage/retrieval
scheme, measurements of the relative phase between the two SWEs and Rabi
oscillation, as well as elimination of the collapse and revival of the readout
signal, are experimentally demonstrated
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