510 research outputs found
Multiple precision evaluation of the Airy Ai function with reduced cancellation
The series expansion at the origin of the Airy function Ai(x) is alternating
and hence problematic to evaluate for x > 0 due to cancellation. Based on a
method recently proposed by Gawronski, M\"uller, and Reinhard, we exhibit two
functions F and G, both with nonnegative Taylor expansions at the origin, such
that Ai(x) = G(x)/F(x). The sums are now well-conditioned, but the Taylor
coefficients of G turn out to obey an ill-conditioned three-term recurrence. We
use the classical Miller algorithm to overcome this issue. We bound all errors
and our implementation allows an arbitrary and certified accuracy, that can be
used, e.g., for providing correct rounding in arbitrary precision
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Exact Instanton Expansion of ABJM Partition Function
We review recent progress in determining the partition function of the ABJM
theory in the large N expansion, including all of the perturbative and
non-perturbative corrections. Especially, we will focus on how these exact
expansions are obtained from various beautiful relations to Fermi gas system,
topological string theory, integrable model and supergroup.Comment: 47 pages, no figures, a review article submitted to PTE
Exponential representation and consistency checking for M-layer
M-layer,the tropospheric propagation effect prediction program is revised for greater accuracy, speed and stability. This is achieved through converting the extended complex number representation into the representation by complex exponent, improving the accuracy in Airy function computation, introducing a new mode locating algorithm and implementing a consistency checking procedure for determining the proper method to evaluate the height gain function. The revision has been documented and the new program source code has been delivered. It is recommended that the mode search protocol, not just the mode locating algorithm introduced in this revision, be completely revised. Unlike the current approach of blanketing the whole possible region until exhaustion, modes should be searched according to their range attenuation rates one by one along a well defined path. this should result in a faster and even more stable program. The program size can also be reducedNaval Command, Control and Ocean Surveillance Center, RDT&E Division, Code 543http://archive.org/details/exponentialrepre00leehNaval Command, Control and Ocean Surveillance
Center, KDT&E DivisionNAApproved for public release; distribution is unlimited
Numerical applications of the generalized method of steepest descents
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1992.Includes bibliographical references (leaves 268-272).by Jean-Marie Clarisse.Ph.D
Ballistic matter waves with angular momentum: Exact solutions and applications
An alternative description of quantum scattering processes rests on
inhomogeneous terms amended to the Schroedinger equation. We detail the
structure of sources that give rise to multipole scattering waves of definite
angular momentum, and introduce pointlike multipole sources as their limiting
case. Partial wave theory is recovered for freely propagating particles. We
obtain novel results for ballistic scattering in an external uniform force
field, where we provide analytical solutions for both the scattering waves and
the integrated particle flux. Our theory directly applies to p-wave
photodetachment in an electric field. Furthermore, illustrating the effects of
extended sources, we predict some properties of vortex-bearing atom laser beams
outcoupled from a rotating Bose-Einstein condensate under the influence of
gravity.Comment: 42 pages, 8 figures, extended version including photodetachment and
semiclassical theor
Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
We present a systematic short time expansion for the generating function of
the one point height probability distribution for the KPZ equation with droplet
initial condition, which goes much beyond previous studies. The expansion is
checked against a numerical evaluation of the known exact Fredholm determinant
expression. We also obtain the next order term for the Brownian initial
condition. Although initially devised for short time, a resummation of the
series allows to obtain also the \textit{long time large deviation function},
found to agree with previous works using completely different techniques.
Unexpected similarities with stationary large deviations of TASEP with periodic
and open boundaries are discussed. Two additional applications are given. (i)
Our method is generalized to study the linear statistics of the {Airy point
process}, i.e. of the GUE edge eigenvalues. We obtain the generating function
of the cumulants of the empirical measure to a high order. The second cumulant
is found to match the result in the bulk obtained from the Gaussian free field
by Borodin and Ferrari, but we obtain systematic corrections to the Gaussian
free field (higher cumulants, expansion towards the edge). This also extends a
result of Basor and Widom to a much higher order. We obtain {large deviation
functions} for the {Airy point process} for a variety of linear statistics test
functions. (ii) We obtain results for the \textit{counting statistics of
trapped fermions} at the edge of the Fermi gas in both the high and the low
temperature limits.Comment: 84 pages, 3 figure
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