6,263 research outputs found
Design of quadrature rules for MĆ¼ntz and MĆ¼ntz-logarithmic polynomials using monomial transformation
A method for constructing the exact quadratures for MĆ¼ntz and MĆ¼ntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of MĆ¼ntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for MĆ¼ntz polynomial
Contour integral method for obtaining the self-energy matrices of electrodes in electron transport calculations
We propose an efficient computational method for evaluating the self-energy
matrices of electrodes to study ballistic electron transport properties in
nanoscale systems. To reduce the high computational cost incurred in large
systems, a contour integral eigensolver based on the Sakurai-Sugiura method
combined with the shifted biconjugate gradient method is developed to solve
exponential-type eigenvalue problem for complex wave vectors. A remarkable
feature of the proposed algorithm is that the numerical procedure is very
similar to that of conventional band structure calculations. We implement the
developed method in the framework of the real-space higher-order finite
difference scheme with nonlocal pseudopotentials. Numerical tests for a wide
variety of materials validate the robustness, accuracy, and efficiency of the
proposed method. As an illustration of the method, we present the electron
transport property of the free-standing silicene with the line defect
originating from the reversed buckled phases.Comment: 36 pages, 13 figures, 2 table
Solving integral equations in
A dispersive analysis of decays has been performed in the past
by many authors. The numerical analysis of the pertinent integral equations is
hampered by two technical difficulties: i) The angular averages of the
amplitudes need to be performed along a complicated path in the complex plane.
ii) The averaged amplitudes develop singularities along the path of integration
in the dispersive representation of the full amplitudes. It is a delicate
affair to handle these singularities properly, and independent checks of the
obtained solutions are demanding and time consuming. In the present article, we
propose a solution method that avoids these difficulties. It is based on a
simple deformation of the path of integration in the dispersive representation
(not in the angular average). Numerical solutions are then obtained rather
straightforwardly. We expect that the method also works for .Comment: 11 pages, 10 Figures. Version accepted for publication in EPJC. The
ancillary files contain an updated set of fundamental solutions. The
numerical differences to the former set are tiny, see the READMEv2 file for
detail
The Kink Phenomenon in FejƩr and Clenshaw-Curtis Quadrature
The FejƩr and Clenshaw-Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N, (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like , where is a constant greater than 1. For these values of N the accuracy of both the FejƩr and Clenshaw-Curtis rules is almost indistinguishable from that of the more celebrated Gauss-Legendre quadrature rule. For larger N, however, the error decreases at the rate , i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.\ud
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This work was supported by the Royal Society of the UK and the National Research Foundation of South Africa under the South Africa-UK Science Network Scheme. The first author also acknowledges grant FA2005032300018 of the NRF
On the computation of Gaussian quadrature rules for Chebyshev sets of linearly independent functions
We consider the computation of quadrature rules that are exact for a
Chebyshev set of linearly independent functions on an interval . A
general theory of Chebyshev sets guarantees the existence of rules with a
Gaussian property, in the sense that basis functions can be integrated
exactly with just points and weights. Moreover, all weights are positive
and the points lie inside the interval . However, the points are not the
roots of an orthogonal polynomial or any other known special function as in the
case of regular Gaussian quadrature. The rules are characterized by a nonlinear
system of equations, and earlier numerical methods have mostly focused on
finding suitable starting values for a Newton iteration to solve this system.
In this paper we describe an alternative scheme that is robust and generally
applicable for so-called complete Chebyshev sets. These are ordered Chebyshev
sets where the first elements also form a Chebyshev set for each . The
points of the quadrature rule are computed one by one, increasing exactness of
the rule in each step. Each step reduces to finding the unique root of a
univariate and monotonic function. As such, the scheme of this paper is
guaranteed to succeed. The quadrature rules are of interest for integrals with
non-smooth integrands that are not well approximated by polynomials
Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems
On computers, discrete problems are solved instead of continuous ones. One
must be sure that the solutions of the former problems, obtained in real time
(i.e., when the stepsize h is not infinitesimal) are good approximations of the
solutions of the latter ones. However, since the discrete world is much richer
than the continuous one (the latter being a limit case of the former), the
classical definitions and techniques, devised to analyze the behaviors of
continuous problems, are often insufficient to handle the discrete case, and
new specific tools are needed. Often, the insistence in following a path
already traced in the continuous setting, has caused waste of time and efforts,
whereas new specific tools have solved the problems both more easily and
elegantly. In this paper we survey three of the main difficulties encountered
in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed
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