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