31,679 research outputs found
Window Function for Non-Circular Beam CMB Anisotropy Experiment
We develop computationally rapid methods to compute the window function for a
cosmic microwave background anisotropy experiment with a non-circular beam
which scans over large angles on the sky. To concretely illustrate these
methods we compute the window function for the Python V experiment which scans
over large angles on the sky with an elliptical Gaussian beam.Comment: 27 pages, 5 figure
Analytical expressions and numerical evaluation of the luminosity distance in a flat cosmology
Accurate and efficient methods to evaluate cosmological distances are an
important tool in modern precision cosmology. In a flat CDM cosmology,
the luminosity distance can be expressed in terms of elliptic integrals. We
derive an alternative and simple expression for the luminosity distance in a
flat CDM based on hypergeometric functions. Using a timing experiment
we compare the computation time for the numerical evaluation of the various
exact formulae, as well as for two approximate fitting formulae available in
the literature. We find that our novel expression is the most efficient exact
expression in the redshift range . Ideally, it can be combined with
the expression based on Carlson's elliptic integrals in the range
for high precision cosmology distance calculations over the entire redshift
range. On the other hand, for practical work where relative errors of about
0.1% are acceptable, the analytical approximation proposed by Adachi & Kasai
(2012) is a suitable alternative.Comment: 4 pages, 1 figure, accepted for publication in MNRA
Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach
This paper introduces and analyses the new grid-based tensor approach to
approximate solution of the elliptic eigenvalue problem for the 3D
lattice-structured systems. We consider the linearized Hartree-Fock equation
over a spatial lattice for both periodic and
non-periodic problem setting, discretized in the localized Gaussian-type
orbitals basis. In the periodic case, the Galerkin system matrix obeys a
three-level block-circulant structure that allows the FFT-based
diagonalization, while for the finite extended systems in a box (Dirichlet
boundary conditions) we arrive at the perturbed block-Toeplitz representation
providing fast matrix-vector multiplication and low storage size. The proposed
grid-based tensor techniques manifest the twofold benefits: (a) the entries of
the Fock matrix are computed by 1D operations using low-rank tensors
represented on a 3D grid, (b) in the periodic case the low-rank tensor
structure in the diagonal blocks of the Fock matrix in the Fourier space
reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems
in a box with Dirichlet boundary conditions are treated numerically by our
previous tensor solver for single molecules, which makes possible calculations
on rather large lattices due to reduced numerical
cost for 3D problems. The numerical simulations for both box-type and periodic
lattice chain in a 3D rectangular "tube" with up to
several hundred confirm the theoretical complexity bounds for the
block-structured eigenvalue solvers in the limit of large .Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with
arXiv:1408.383
Design of quasi-symplectic propagators for Langevin dynamics
A vector field splitting approach is discussed for the systematic derivation
of numerical propagators for deterministic dynamics. Based on the formalism, a
class of numerical integrators for Langevin dynamics are presented for single
and multiple timestep algorithms
Computationally efficient recursions for top-order invariant polynomials with applications
The top-order zonal polynomials Ck(A),and top-order invariant polynomials Ck1,...,kr(A1,...,Ar)in which each of the partitions of ki,i = 1,..., r,has only one part, occur frequently in multivariate distribution theory, and econometrics - see, for example Phillips (1980, 1984, 1985, 1986), Hillier (1985, 2001), Hillier and Satchell (1986), and Smith (1989, 1993). However, even with the recursive algorithms of Ruben (1962) and Chikuse (1987), numerical evaluation of these invariant polynomials is extremely time consuming. As a result, the value of invariant polynomials has been largely confined to analytic work on distribution theory. In this paper we present new, very much more efficient, algorithms for computing both the top-order zonal and invariant polynomials. These results should make the theoretical results involving these functions much more valuable for direct practical study. We demonstrate the value of our results by providing fast and accurate algorithms for computing the moments of a ratio of quadratic forms in normal random variables.
Detailed ultraviolet asymptotics for AdS scalar field perturbations
We present a range of methods suitable for accurate evaluation of the leading
asymptotics for integrals of products of Jacobi polynomials in limits when the
degrees of some or all polynomials inside the integral become large. The
structures in question have recently emerged in the context of effective
descriptions of small amplitude perturbations in anti-de Sitter (AdS)
spacetime. The limit of high degree polynomials corresponds in this situation
to effective interactions involving extreme short-wavelength modes, whose
dynamics is crucial for the turbulent instabilities that determine the ultimate
fate of small AdS perturbations. We explicitly apply the relevant asymptotic
techniques to the case of a self-interacting probe scalar field in AdS and
extract a detailed form of the leading large degree behavior, including closed
form analytic expressions for the numerical coefficients appearing in the
asymptotics.Comment: v2: 19 pages, expanded version accepted to JHE
Efficient numerical diagonalization of hermitian 3x3 matrices
A very common problem in science is the numerical diagonalization of
symmetric or hermitian 3x3 matrices. Since standard "black box" packages may be
too inefficient if the number of matrices is large, we study several
alternatives. We consider optimized implementations of the Jacobi, QL, and
Cuppen algorithms and compare them with an analytical method relying on
Cardano's formula for the eigenvalues and on vector cross products for the
eigenvectors. Jacobi is the most accurate, but also the slowest method, while
QL and Cuppen are good general purpose algorithms. The analytical algorithm
outperforms the others by more than a factor of 2, but becomes inaccurate or
may even fail completely if the matrix entries differ greatly in magnitude.
This can mostly be circumvented by using a hybrid method, which falls back to
QL if conditions are such that the analytical calculation might become too
inaccurate. For all algorithms, we give an overview of the underlying
mathematical ideas, and present detailed benchmark results. C and Fortran
implementations of our code are available for download from
http://www.mpi-hd.mpg.de/~globes/3x3/ .Comment: 13 pages, no figures, new hybrid algorithm added, matches published
version, typo in Eq. (39) corrected; software library available at
http://www.mpi-hd.mpg.de/~globes/3x3
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