38,319 research outputs found
Comparing efficient computation methods for massless QCD tree amplitudes: Closed Analytic Formulae versus Berends-Giele Recursion
Recent advances in our understanding of tree-level QCD amplitudes in the
massless limit exploiting an effective (maximal) supersymmetry have led to the
complete analytic construction of tree-amplitudes with up to four external
quark-anti-quark pairs. In this work we compare the numerical efficiency of
evaluating these closed analytic formulae to a numerically efficient
implementation of the Berends-Giele recursion. We compare calculation times for
tree-amplitudes with parton numbers ranging from 4 to 25 with no, one, two and
three external quark lines. We find that the exact results are generally faster
in the case of MHV and NMHV amplitudes. Starting with the NNMHV amplitudes the
Berends-Giele recursion becomes more efficient. In addition to the runtime we
also compared the numerical accuracy. The analytic formulae are on average more
accurate than the off-shell recursion relations though both are well suited for
complicated phenomenological applications. In both cases we observe a reduction
in the average accuracy when phase space configurations close to singular
regions are evaluated. We believe that the above findings provide valuable
information to select the right method for phenomenological applications.Comment: 22 pages, 9 figures, Mathematica package GGT.m and example notebook
is included in submissio
Generalized Fermi-Dirac Functions and Derivatives: Properties and Evaluation
The generalized Fermi-Dirac functions and their derivatives are important in
evaluating the thermodynamic quantities of partially degenerate electrons in
hot dense stellar plasmas. New recursion relations of the generalized
Fermi-Dirac functions have been found. An effective numerical method to
evaluate the derivatives of the generalized Fermi-Dirac functions up to third
order with respect to both degeneracy and temperature is then proposed,
following Aparicio. A Fortran program based on this method, together with a
sample test case, is provided. Accuracy and domain of reliability of some
other, popularly used analytic approximations of the generalized Fermi-Dirac
functions for extreme conditions are investigated and compared with our
results.Comment: accepted for publication in Comp. Phys. Com
A static cost analysis for a higher-order language
We develop a static complexity analysis for a higher-order functional
language with structural list recursion. The complexity of an expression is a
pair consisting of a cost and a potential. The former is defined to be the size
of the expression's evaluation derivation in a standard big-step operational
semantics. The latter is a measure of the "future" cost of using the value of
that expression. A translation function tr maps target expressions to
complexities. Our main result is the following Soundness Theorem: If t is a
term in the target language, then the cost component of tr(t) is an upper bound
on the cost of evaluating t. The proof of the Soundness Theorem is formalized
in Coq, providing certified upper bounds on the cost of any expression in the
target language.Comment: Final versio
Markov chain approximations to scale functions of L\'evy processes
We introduce a general algorithm for the computation of the scale functions
of a spectrally negative L\'evy process , based on a natural weak
approximation of via upwards skip-free continuous-time Markov chains with
stationary independent increments. The algorithm consists of evaluating a
finite linear recursion with its (nonnegative) coefficients given explicitly in
terms of the L\'evy triplet of . Thus it is easy to implement and
numerically stable. Our main result establishes sharp rates of convergence of
this algorithm providing an explicit link between the semimartingale
characteristics of and its scale functions, not unlike the one-dimensional
It\^o diffusion setting, where scale functions are expressed in terms of
certain integrals of the coefficients of the governing SDE.Comment: 46 pages, 4 figure
Expansion of Einstein-Yang-Mills Amplitude
In this paper, we provide a thorough study on the expansion of single trace
Einstein-Yang-Mills amplitudes into linear combination of color-ordered
Yang-Mills amplitudes, from various different perspectives. Using the gauge
invariance principle, we propose a recursive construction, where EYM amplitude
with any number of gravitons could be expanded into EYM amplitudes with less
number of gravitons. Through this construction, we can write down the complete
expansion of EYM amplitude in the basis of color-ordered Yang-Mills amplitudes.
As a byproduct, we are able to write down the polynomial form of BCJ numerator,
i.e., numerators satisfying the color-kinematic duality, for Yang-Mills
amplitude. After the discussion of gauge invariance, we move to the BCFW
on-shell recursion relation and discuss how the expansion can be understood
from the on-shell picture. Finally, we show how to interpret the expansion from
the aspect of KLT relation and the way of evaluating the expansion coefficients
efficiently.Comment: 50 pages, 1 figure, Revised versio
Block bond-order potential as a convergent moments-based method
The theory of a novel bond-order potential, which is based on the block
Lanczos algorithm, is presented within an orthogonal tight-binding
representation. The block scheme handles automatically the very different
character of sigma and pi bonds by introducing block elements, which produces
rapid convergence of the energies and forces within insulators, semiconductors,
metals, and molecules. The method gives the first convergent results for
vacancies in semiconductors using a moments-based method with a low number of
moments. Our use of the Lanczos basis simplifies the calculations of the band
energy and forces, which allows the application of the method to the molecular
dynamics simulations of large systems. As an illustration of this convergent
O(N) method we apply the block bond-order potential to the large scale
simulation of the deformation of a carbon nanotube.Comment: revtex, 43 pages, 11 figures, submitted to Phys. Rev.
Efficient Recursion Method for Inverting Overlap Matrix
A new O(N) algorithm based on a recursion method, in which the computational
effort is proportional to the number of atoms N, is presented for calculating
the inverse of an overlap matrix which is needed in electronic structure
calculations with the the non-orthogonal localized basis set. This efficient
inverting method can be incorporated in several O(N) methods for
diagonalization of a generalized secular equation. By studying convergence
properties of the 1-norm of an error matrix for diamond and fcc Al, this method
is compared to three other O(N) methods (the divide method, Taylor expansion
method, and Hotelling's method) with regard to computational accuracy and
efficiency within the density functional theory. The test calculations show
that the new method is about one-hundred times faster than the divide method in
computational time to achieve the same convergence for both diamond and fcc Al,
while the Taylor expansion method and Hotelling's method suffer from numerical
instabilities in most cases.Comment: 17 pages and 4 figure
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