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 $100$ the asymptotic methods are enough for a double precision accuracy computation ($15$-$16$ digits) of the nodes and weights of the Gauss--Hermite and Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic

Computation of the Marcum Q-function

Methods and an algorithm for computing the generalized Marcum $Q-$function ($Q_{\mu}(x,y)$) and the complementary function ($P_{\mu}(x,y)$) are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics and probability theory, where they are called the non-central chi-square or the non central gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to $10^{-12}$ can be obtained in the parameter region $(x,y,\mu) \in [0,\,A]\times [0,\,A]\times [1,\,A]$, $A=200$, while for larger parameters the accuracy decreases (close to $10^{-11}$ for $A=1000$ and close to $5\times 10^{-11}$ for $A=10000$).Comment: Accepted for publication in ACM Trans. Math. Soft

Stochastic Matrix Product States

The concept of stochastic matrix product states is introduced and a natural form for the states is derived. This allows to define the analogue of Schmidt coefficients for steady states of non-equilibrium stochastic processes. We discuss a new measure for correlations which is analogous to the entanglement entropy, the entropy cost $S_C$, and show that this measure quantifies the bond dimension needed to represent a steady state as a matrix product state. We illustrate these concepts on the hand of the asymmetric exclusion process

Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine for computing the function ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$, a real-valued numerically satisfactory companion of the function ${\rm P}^m_{-\tfrac12+i\tau}(x)$ for $x>1$. In this way, a natural basis for solving Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication

Non-equilibrium dynamics of a Bose-Einstein condensate in an optical lattice

The dynamical evolution of a Bose-Einstein condensate trapped in a one-dimensional lattice potential is investigated theoretically in the framework of the Bose-Hubbard model. The emphasis is set on the far-from-equilibrium evolution in a case where the gas is strongly interacting. This is realized by an appropriate choice of the parameters in the Hamiltonian, and by starting with an initial state, where one lattice well contains a Bose-Einstein condensate while all other wells are empty. Oscillations of the condensate as well as non-condensate fractions of the gas between the different sites of the lattice are found to be damped as a consequence of the collisional interactions between the atoms. Functional integral techniques involving self-consistently determined mean fields as well as two-point correlation functions are used to derive the two-particle-irreducible (2PI) effective action. The action is expanded in inverse powers of the number of field components N, and the dynamic equations are derived from it to next-to-leading order in this expansion. This approach reaches considerably beyond the Hartree-Fock-Bogoliubov mean-field theory, and its results are compared to the exact quantum dynamics obtained by A.M. Rey et al., Phys. Rev. A 69, 033610 (2004) for small atom numbers.Comment: 9 pages RevTeX, 3 figure

Asymptotics and zero distribution of Padé polynomials associated with the exponential function

AbstractThe polynomials Pn and Qm having degrees n and m, respectively, with Pn monic, that solve the approximation problem Pn(z)e−z + Qm(z) = O(zn+m+1) will be investigated for their asymptotic behavior, in particular in connection with the distribution of their zeros. The symbol O means that the left-hand side should vanish at the origin at least to the order n + m + 1. This problem is discussed in great detail in a series of papers by Saff and Varga. In the present paper, we show how their results can be obtained by using uniform expansions of integrals in which Airy functions are the main approximants. We give approximations of the zeros of Pn and Qm in terms of zeros of certain Airy functions, as well of those of the remainder defined by En,m(z) = Pn(z)e−z + Qm(z)

On Polynomials Related with Hermite–Padé Approximations to the Exponential Function

AbstractWe investigate the polynomialsPn,Qm, andRs, having degreesn,m, ands, respectively, withPnmonic, that solve the approximation problem [formula]We give a connection between the coefficients of each of the polynomialsPn,Qm, andRsand certain hypergeometric functions, which leads to a simple expression forQmin the casen=s. The approximate location of the zeros ofQm, whenn⪢mandnequals;s, are deduced from the zeros of the classical Hermite polynomial. Contour integral representations ofPn,Qm,Rs, andEnmsare given and, using saddle point methods, we derive the exact asymptotics ofPn,Qm, andRsasn,m, andstend to infinity through certain ray sequences. We also discuss aspects of the more complicated uniform asymptotic methods for obtaining insight into the zero distribution of the polynomials, and we give an example showing the zeros of the polynomialsPn,Qm, andRsfor the casen=s=40,m=45

Computer based human-centered display system

A human centered informational display is disclosed that can be used with vehicles (e.g. aircraft) and in other operational environments where rapid human centered comprehension of an operational environment is required. The informational display integrates all cockpit information into a single display in such a way that the pilot can clearly understand with a glance, his or her spatial orientation, flight performance, engine status and power management issues, radio aids, and the location of other air traffic, runways, weather, and terrain features. With OZ the information is presented as an integrated whole, the pilot instantaneously recognizes flight path deviations, and is instinctively drawn to the corrective maneuvers. Our laboratory studies indicate that OZ transfers to the pilot all of the integrated display information in less than 200 milliseconds. The reacquisition of scan can be accomplished just as quickly. Thus, the time constants for forming a mental model are near instantaneous. The pilot's ability to keep up with rapidly changing and threatening environments is tremendously enhanced. OZ is most easily compatible with aircraft that has flight path information coded electronically. With the correct sensors (which are currently available) OZ can be installed in essentially all current aircraft

Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders

We consider the problem of computing satisfactory pairs of solutions of the differential equation for Legendre functions of non-negative integer order $\mu$ and degree $-\frac12+i\tau$, where $\tau$ is a non-negative real parameter. Solutions of this equation are the conical functions ${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ and ${Q}^{\mu}_{-\frac12+i\tau}(x)$, $x>-1$. An algorithm for computing a numerically satisfactory pair of solutions is already available when $-1<x<1$ (see \cite{gil:2009:con}, \cite{gil:2012:cpc}).In this paper, we present a stable computational scheme for a real valued numerically satisfactory companion of the function ${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ for $x>1$, the function $\Re\left\{e^{-i\pi \mu} {{Q}}^{\mu}_{-\frac{1}{2}+i\tau}(x) \right\}$. The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.Comment: To be published in Numerical Algoritm