3,074 research outputs found
Efficient computation of Laguerre polynomials
An efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials View the MathML source are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for n large and a small, are used depending on the parameter region.
Based on tests of contiguous relations in the parameter a and the degree n satisfied by the polynomials, we claim that a relative accuracy close to or better than 10-12 can be obtained using the module LaguerrePol for computing the functions View the MathML source in the parameter range z=0, -1<a=5, n=0.
Program summary
Program Title: Module LaguerrePol
Program Files doi:http://dx.doi.org/10.17632/3jkk659cn8.1
Licensing provisions: CC by 4.0
Programming language: Fortran 90
Nature of problem: Laguerre polynomials View the MathML source appear in a vast number of applications in physics, such as quantum mechanics, plasma physics, etc.
Solution method: The algorithm uses asymptotic expansions or recurrence relations for computing the function values depending on the range of parameters.
Restrictions: The admissible input parameter ranges for computing the Laguerre View the MathML source are z=0, -1<a=5, n=0.The authors acknowledge financial support from Ministerio de Ciencia e Innovación, projects MTM2012-34787, MTM2015-67142-
Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures
Asymptotic approximations to the zeros of Hermite and Laguerre polynomials
are given, together with methods for obtaining the coefficients in the
expansions. These approximations can be used as a standalone method of
computation of Gaussian quadratures for high enough degrees, with Gaussian
weights computed from asymptotic approximations for the orthogonal polynomials.
We provide numerical evidence showing that for degrees greater than the
asymptotic methods are enough for a double precision accuracy computation
(- digits) of the nodes and weights of the Gauss--Hermite and
Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic
Comparison of numerical solutions for Q^2 evolution equations
Q^2 evolution equations are important not only for describing hadron
reactions in accelerator experiments but also for investigating
ultrahigh-energy cosmic rays. The standard ones are called DGLAP evolution
equations, which are integrodifferential equations. There are methods for
solving the Q^2 evolution equations for parton-distribution and fragmentation
functions. Because the equations cannot be solved analytically, various methods
have been developed for the numerical solution. We compare brute-force,
Laguerre-polynomial, and Mellin-transformation methods particularly by focusing
on the numerical accuracy and computational efficiency. An efficient solution
could be used, for example, in the studies of a top-down scenario for the
ultrahigh-energy cosmic rays.Comment: 12 pages, LaTeX, 13 eps files, Journal of Computational Physics in
press, http://hs.phys.saga-u.ac.j
Numerical approximations for population growth model by Rational Chebyshev and Hermite Functions collocation approach: A comparison
This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF)
collocation approach to solve the Volterra's model for population growth of a
species within a closed system. This model is a nonlinear integro-differential
equation where the integral term represents the effect of toxin. This approach
is based on orthogonal functions which will be defined. The collocation method
reduces the solution of this problem to the solution of a system of algebraic
equations. We also compare these methods with some other numerical results and
show that the present approach is applicable for solving nonlinear
integro-differential equations.Comment: 18 pages, 5 figures; Published online in the journal of "Mathematical
Methods in the Applied Sciences
Efficient Algorithm for Two-Center Coulomb and Exchange Integrals of Electronic Prolate Spheroidal Orbitals
We present a fast algorithm to calculate Coulomb/exchange integrals of
prolate spheroidal electronic orbitals, which are the exact solutions of the
single-electron, two-center Schr\"odinger equation for diatomic molecules. Our
approach employs Neumann's expansion of the Coulomb repulsion 1/|x-y|, solves
the resulting integrals symbolically in closed form and subsequently performs a
numeric Taylor expansion for efficiency. Thanks to the general form of the
integrals, the obtained coefficients are independent of the particular
wavefunctions and can thus be reused later.
Key features of our algorithm include complete avoidance of numeric
integration, drafting of the individual steps as fast matrix operations and
high accuracy due to the exponential convergence of the expansions.
Application to the diatomic molecules O2 and CO exemplifies the developed
methods, which can be relevant for a quantitative understanding of chemical
bonds in general.Comment: 27 pages, 9 figure
- …