Rectangular Full Packed Format for Cholesky's Algorithm: Factorization, Solution and Inversion

We describe a new data format for storing triangular, symmetric, and Hermitian matrices called RFPF (Rectangular Full Packed Format). The standard two dimensional arrays of Fortran and C (also known as full format) that are used to represent triangular and symmetric matrices waste nearly half of the storage space but provide high performance via the use of Level 3 BLAS. Standard packed format arrays fully utilize storage (array space) but provide low performance as there is no Level 3 packed BLAS. We combine the good features of packed and full storage using RFPF to obtain high performance via using Level 3 BLAS as RFPF is a standard full format representation. Also, RFPF requires exactly the same minimal storage as packed format. Each LAPACK full and/or packed triangular, symmetric, and Hermitian routine becomes a single new RFPF routine based on eight possible data layouts of RFPF. This new RFPF routine usually consists of two calls to the corresponding LAPACK full format routine and two calls to Level 3 BLAS routines. This means {\it no} new software is required. As examples, we present LAPACK routines for Cholesky factorization, Cholesky solution and Cholesky inverse computation in RFPF to illustrate this new work and to describe its performance on several commonly used computer platforms. Performance of LAPACK full routines using RFPF versus LAPACK full routines using standard format for both serial and SMP parallel processing is about the same while using half the storage. Performance gains are roughly one to a factor of 43 for serial and one to a factor of 97 for SMP parallel times faster using vendor LAPACK full routines with RFPF than with using vendor and/or reference packed routines

Sensitive gravity-gradiometry with atom interferometry: progress towards an improved determination of the gravitational constant

We here present a high sensitivity gravity-gradiometer based on atom interferometry. In our apparatus, two clouds of laser-cooled rubidium atoms are launched in fountain configuration and interrogated by a Raman interferometry sequence to probe the gradient of gravity field. We recently implemented a high-flux atomic source and a newly designed Raman lasers system in the instrument set-up. We discuss the applications towards a precise determination of the Newtonian gravitational constant G. The long-term stability of the instrument and the signal-to-noise ratio demonstrated here open interesting perspectives for pushing the measurement precision below the 100 ppm level

Stability of axial orbits in galactic potentials

We investigate the dynamics in a galactic potential with two reflection symmetries. The phase-space structure of the real system is approximated with a resonant detuned normal form constructed with the method based on the Lie transform. Attention is focused on the stability properties of the axial periodic orbits that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of bifurcations and compare them with numerical results available in the literature.Comment: 20 pages, accepted for publication on Celestial Mechanics and Dynamical Astronom

Parallel computation of echelon forms

International audienceWe propose efficient parallel algorithms and implementations on shared memory architectures of LU factorization over a finite field. Compared to the corresponding numerical routines, we have identified three main difficulties specific to linear algebra over finite fields. First, the arithmetic complexity could be dominated by modular reductions. Therefore, it is mandatory to delay as much as possible these reductions while mixing fine-grain parallelizations of tiled iterative and recursive algorithms. Second, fast linear algebra variants, e.g., using Strassen-Winograd algorithm, never suffer from instability and can thus be widely used in cascade with the classical algorithms. There, trade-offs are to be made between size of blocks well suited to those fast variants or to load and communication balancing. Third, many applications over finite fields require the rank profile of the matrix (quite often rank deficient) rather than the solution to a linear system. It is thus important to design parallel algorithms that preserve and compute this rank profile. Moreover, as the rank profile is only discovered during the algorithm, block size has then to be dynamic. We propose and compare several block decomposition: tile iterative with left-looking, right-looking and Crout variants, slab and tile recursive. Experiments demonstrate that the tile recursive variant performs better and matches the performance of reference numerical software when no rank deficiency occur. Furthermore, even in the most heterogeneous case, namely when all pivot blocks are rank deficient, we show that it is possbile to maintain a high efficiency

Precision Measurement of the Newtonian Gravitational Constant Using Cold Atoms

About 300 experiments have tried to determine the value of the Newtonian gravitational constant, G, so far, but large discrepancies in the results have made it impossible to know its value precisely. The weakness of the gravitational interaction and the impossibility of shielding the effects of gravity make it very difficult to measure G while keeping systematic effects under control. Most previous experiments performed were based on the torsion pendulum or torsion balance scheme as in the experiment by Cavendish in 1798, and in all cases macroscopic masses were used. Here we report the precise determination of G using laser-cooled atoms and quantum interferometry. We obtain the value G=6.67191(99) x 10^(-11) m^3 kg^(-1) s^(-2) with a relative uncertainty of 150 parts per million (the combined standard uncertainty is given in parentheses). Our value differs by 1.5 combined standard deviations from the current recommended value of the Committee on Data for Science and Technology. A conceptually different experiment such as ours helps to identify the systematic errors that have proved elusive in previous experiments, thus improving the confidence in the value of G. There is no definitive relationship between G and the other fundamental constants, and there is no theoretical prediction for its value, against which to test experimental results. Improving the precision with which we know G has not only a pure metrological interest, but is also important because of the key role that G has in theories of gravitation, cosmology, particle physics and astrophysics and in geophysical models.Comment: 3 figures, 1 tabl

Bohr-Sommerfeld Quantization of Periodic Orbits

We show, that the canonical invariant part of $\hbar$ corrections to the Gutzwiller trace formula and the Gutzwiller-Voros spectral determinant can be computed by the Bohr-Sommerfeld quantization rules, which usually apply for integrable systems. We argue that the information content of the classical action and stability can be used more effectively than in the usual treatment. We demonstrate the improvement of precision on the example of the three disk scattering system.Comment: revte

A Nanofiber-Based Optical Conveyor Belt for Cold Atoms

We demonstrate optical transport of cold cesium atoms over millimeter-scale distances along an optical nanofiber. The atoms are trapped in a one-dimensional optical lattice formed by a two-color evanescent field surrounding the nanofiber, far red- and blue-detuned with respect to the atomic transition. The blue-detuned field is a propagating nanofiber-guided mode while the red-detuned field is a standing-wave mode which leads to the periodic axial confinement of the atoms. Here, this standing wave is used for transporting the atoms along the nanofiber by mutually detuning the two counter-propagating fields which form the standing wave. The performance and limitations of the nanofiber-based transport are evaluated and possible applications are discussed

Mach's Principle and the Origin of Inertia

The current status of Mach's principle is discussed within the context of general relativity. The inertial properties of a particle are determined by its mass and spin, since these characterize the irreducible unitary representations of the inhomogeneous Lorentz group. The origin of the inertia of mass and intrinsic spin are discussed and the inertia of intrinsic spin is studied via the coupling of intrinsic spin with rotation. The implications of spin-rotation coupling and the possibility of history dependence and nonlocality in relativistic physics are briefly mentioned.Comment: 14 pages. Dedicated to Carl Brans in honor of his 80th birthday. To appear in the Brans Festschrift; v2: typo corrected, published in: At the Frontier of Spacetime, edited by T. Asselmeyer-Maluga (Springer, 2016), Chapter 10, pp. 177-18

Coherent matter wave inertial sensors for precision measurements in space

We analyze the advantages of using ultra-cold coherent sources of atoms for matter-wave interferometry in space. We present a proof-of-principle experiment that is based on an analysis of the results previously published in [Richard et al., Phys. Rev. Lett., 91, 010405 (2003)] from which we extract the ratio h/m for 87Rb. This measurement shows that a limitation in accuracy arises due to atomic interactions within the Bose-Einstein condensate

Selective nanomanipulation using optical forces

We present a detailed theoretical study of the recent proposal for selective nanomanipulation of nanometric particles above a substrate using near-field optical forces [Chaumet {\it et al.} Phys. Rev. Lett. {\bf 88}, 123601 (2002)]. Evanescent light scattering at the apex of an apertureless near-field probe is used to create an optical trap. The position of the trap is controlled on a nanometric scale via the probe and small objects can be selectively trapped and manipulated. We discuss the influence of the geometry of the particles and the probe on the efficiency of the trap. We also consider the influence of multiple scattering among the particles on the substrate and its effect on the robustness of the trap.Comment: 12 pages, 17 figure