430 research outputs found
Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures
The QR factorization and the SVD are two fundamental matrix decompositions
with applications throughout scientific computing and data analysis. For
matrices with many more rows than columns, so-called "tall-and-skinny
matrices," there is a numerically stable, efficient, communication-avoiding
algorithm for computing the QR factorization. It has been used in traditional
high performance computing and grid computing environments. For MapReduce
environments, existing methods to compute the QR decomposition use a
numerically unstable approach that relies on indirectly computing the Q factor.
In the best case, these methods require only two passes over the data. In this
paper, we describe how to compute a stable tall-and-skinny QR factorization on
a MapReduce architecture in only slightly more than 2 passes over the data. We
can compute the SVD with only a small change and no difference in performance.
We present a performance comparison between our new direct TSQR method, a
standard unstable implementation for MapReduce (Cholesky QR), and the classic
stable algorithm implemented for MapReduce (Householder QR). We find that our
new stable method has a large performance advantage over the Householder QR
method. This holds both in a theoretical performance model as well as in an
actual implementation
Fast linear algebra is stable
In an earlier paper, we showed that a large class of fast recursive matrix
multiplication algorithms is stable in a normwise sense, and that in fact if
multiplication of -by- matrices can be done by any algorithm in
operations for any , then it can be done
stably in operations for any . Here we extend
this result to show that essentially all standard linear algebra operations,
including LU decomposition, QR decomposition, linear equation solving, matrix
inversion, solving least squares problems, (generalized) eigenvalue problems
and the singular value decomposition can also be done stably (in a normwise
sense) in operations.Comment: 26 pages; final version; to appear in Numerische Mathemati
Single polaron properties of the breathing-mode Hamiltonian
We investigate numerically various properties of the one-dimensional (1D)
breathing-mode polaron. We use an extension of a variational scheme to compute
the energies and wave-functions of the two lowest-energy eigenstates for any
momentum, as well as a scheme to compute directly the polaron Greens function.
We contrast these results with results for the 1D Holstein polaron. In
particular, we find that the crossover from a large to a small polaron is
significantly sharper. Unlike for the Holstein model, at moderate and large
couplings the breathing-mode polaron dispersion has non-monotonic dependence on
the polaron momentum k. Neither of these aspects is revealed by a previous
study based on the self-consistent Born approximation
Quantum Circulant Preconditioner for Linear System of Equations
We consider the quantum linear solver for with the circulant
preconditioner . The main technique is the singular value estimation (SVE)
introduced in [I. Kerenidis and A. Prakash, Quantum recommendation system, in
ITCS 2017]. However, some modifications of SVE should be made to solve the
preconditioned linear system . Moreover, different from
the preconditioned linear system considered in [B. D. Clader, B. C. Jacobs, C.
R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett.,
2013], the circulant preconditioner is easy to construct and can be directly
applied to general dense non-Hermitian cases. The time complexity depends on
the condition numbers of and , as well as the Frobenius norm
From incommensurate correlations to mesoscopic spin resonance in YbRh2Si2
Spin fluctuations are reported near the magnetic field driven quantum
critical point in YbRh2Si2. On cooling, ferromagnetic fluctuations evolve into
incommensurate correlations located at q0=+/- (delta,delta) with delta=0.14 +/-
0.04 r.l.u. At low temperatures, an in plane magnetic field induces a sharp
intra doublet resonant excitation at an energy E0=g muB mu0 H with g=3.8 +/-
0.2. The intensity is localized at the zone center indicating precession of
spin density extending xi=6 +/- 2 A beyond the 4f site.Comment: (main text - 4 pages, 4 figures; supplementary information - 3 pages,
3 figures; to be published in Physical Review Letters
Eigenvalue Estimation of Differential Operators
We demonstrate how linear differential operators could be emulated by a
quantum processor, should one ever be built, using the Abrams-Lloyd algorithm.
Given a linear differential operator of order 2S, acting on functions
psi(x_1,x_2,...,x_D) with D arguments, the computational cost required to
estimate a low order eigenvalue to accuracy Theta(1/N^2) is
Theta((2(S+1)(1+1/nu)+D)log N) qubits and O(N^{2(S+1)(1+1/nu)} (D log N)^c)
gate operations, where N is the number of points to which each argument is
discretized, nu and c are implementation dependent constants of O(1). Optimal
classical methods require Theta(N^D) bits and Omega(N^D) gate operations to
perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby
leads to exponential reduction in memory and polynomial reduction in gate
operations, provided the domain has sufficiently large dimension D >
2(S+1)(1+1/nu). In the case of Schrodinger's equation, ground state energy
estimation of two or more particles can in principle be performed with fewer
quantum mechanical gates than classical gates.Comment: significant content revisions: more algorithm details and brief
analysis of convergenc
Mixed precision bisection
We discuss the implementation of the bisection algorithm for the computation of the eigenvalues of
symmetric tridiagonal matrices in a context of mixed precision arithmetic. This approach is motivated by the
emergence of processors which carry out floating-point operations much faster in single precision than they do in double precision. Perturbation theory results are used to decide when to switch from single to double precision. Numerical examples are presente
Magnetic Excitations in the Ground State of
We report an extensive study on the zero field ground state of a powder
sample of the pyrochlore . A sharp heat capacity anomaly
that labels a low temperature phase transition in this material is observed at
280 mK. Neutron diffraction shows that a \emph{quasi-collinear} ferromagnetic
order develops below with a magnetic moment of
. High resolution inelastic neutron scattering
measurements show, below the phase transition temperature, sharp gapped
low-lying magnetic excitations coexisting with a remnant quasielastic
contribution likely associated with persistent spin fluctuations. Moreover, a
broad inelastic continuum of excitations at meV is observed from the
lowest measured temperature up to at least 2.5 K. At 10 K, the continuum has
vanished and a broad quasielastic conventional paramagnetic scattering takes
place at the observed energy range. Finally, we show that the exchange
parameters obtained within the framework of linear spin-wave theory do not
accurately describe the observed zero field inelastic neutron scattering data.Comment: 11 pages, 9 figures, Phys. Rev. B. (accepted
- …