466 research outputs found
Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals
Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of , where is a negative definite matrix and is the exponential function or one of the related `` functions'' such as . Building on previous work by Trefethen and Gutknecht, Gonchar and Rakhmanov, and Lu, we propose two methods for the fast evaluation of that are especially useful when shifted systems can be solved efficiently, e.g. by a sparse direct solver. The first method method is based on best rational approximations to on the negative real axis computed via the Carathéodory-Fejér procedure, and we conjecture that the accuracy scales as , where is the number of complex matrix solves. In particular, three matrix solves suffice to evaluate to approximately six digits of accuracy. The second method is an application of the trapezoid rule on a Talbot-type contour
FIESTA 2: parallelizeable multiloop numerical calculations
The program FIESTA has been completely rewritten. Now it can be used not only
as a tool to evaluate Feynman integrals numerically, but also to expand Feynman
integrals automatically in limits of momenta and masses with the use of sector
decompositions and Mellin-Barnes representations. Other important improvements
to the code are complete parallelization (even to multiple computers),
high-precision arithmetics (allowing to calculate integrals which were undoable
before), new integrators and Speer sectors as a strategy, the possibility to
evaluate more general parametric integrals.Comment: 31 pages, 5 figure
Efficient approximation of functions of some large matrices by partial fraction expansions
Some important applicative problems require the evaluation of functions
of large and sparse and/or \emph{localized} matrices . Popular and
interesting techniques for computing and , where
is a vector, are based on partial fraction expansions. However,
some of these techniques require solving several linear systems whose matrices
differ from by a complex multiple of the identity matrix for computing
or require inverting sequences of matrices with the same
characteristics for computing . Here we study the use and the
convergence of a recent technique for generating sequences of incomplete
factorizations of matrices in order to face with both these issues. The
solution of the sequences of linear systems and approximate matrix inversions
above can be computed efficiently provided that shows certain decay
properties. These strategies have good parallel potentialities. Our claims are
confirmed by numerical tests
Algorithms to Evaluate Multiple Sums for Loop Computations
We present algorithms to evaluate two types of multiple sums, which appear in
higher-order loop computations. We consider expansions of a generalized
hypergeometric-type sums, \sum_{n_1,...,n_N} [Gamma(a1.n+c1) Gamma(a2.n}+c2)
... Gamma(aM.n+cM)] / [Gamma(b1.n+d1) Gamma(b2.n+d2) ... Gamma(bM.n+dM)]
x1^n1...xN^nN with , etc., in a small parameter
epsilon around rational values of ci,di's. Type I sum corresponds to the case
where, in the limit epsilon -> 0, the summand reduces to a rational function of
nj's times x1^n1...xN^nN; ci,di's can depend on an external integer index. Type
II sum is a double sum (N=2), where ci,di's are half-integers or integers as
epsilon -> 0 and xi=1; we consider some specific cases where at most six Gamma
functions remain in the limit epsilon -> 0. The algorithms enable evaluations
of arbitrary expansion coefficients in epsilon in terms of Z-sums and multiple
polylogarithms (generalized multiple zeta values). We also present applications
of these algorithms. In particular, Type I sums can be used to generate a new
class of relations among generalized multiple zeta values. We provide a
Mathematica package, in which these algorithms are implemented.Comment: 30 pages, 2 figures; address of Mathematica package in Sec.6; version
to appear in J.Math.Phy
Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED
We compute the two-loop QED helicity amplitudes for the scattering of a
lepton pair with an off-shell and an on-shell photon,
, using the approximation of massless leptons.
We express all master integrals relevant for the scattering of four massless
particles with a single external off-shell leg up to two loops in a basis of
algebraically independent multiple polylogarithms, which guarantees an
efficient numerical evaluation and compact analytic representations of the
amplitudes. Analytic forms of the amplitudes are reconstructed from numerical
evaluations over finite fields. Our results complete the amplitude-level
ingredients contributing to the NLO predictions of electron-muon scattering
, which are required to meet the precision goal of the future
MUonE experiment.Comment: 16 + 13 pages, 2 figures, 2 table
Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED
We compute the two-loop QED helicity amplitudes for the scattering of a lepton pair with an off-shell and an on-shell photon, 0 → ℓℓ¯γγ*, using the approximation of massless leptons. We express all master integrals relevant for the scattering of four massless particles with a single external off-shell leg up to two loops in a basis of algebraically independent multiple polylogarithm, which guarantees an efficient numerical evaluation and compact analytic representations of the amplitudes. Analytic forms of the amplitudes are reconstructed from numerical evaluations over finite fields. Our results complete the amplitude-level ingredients contributing to the N3LO predictions of electron-muon scattering eμ → eμ, which are required to meet the precision goal of the future MUonE experiment
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