Computing Bessel functions of the second kind in extreme parameter regimes

A useful method of computing the integral order Bessel functions of the second kind Y_n(x+iy) when either, the absolute value of the real part, or the imaginary part of the argument z=x+iy is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when x∼0 (or y∼0) and is useful because a number existing algorithms and methods fail to give correct results for small x or small y. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.     

Analytical and numerical results onM-variable generalized Bessel functions

Recently, some multivariable special functions have been obtained by generalizing functions of Bessel type. Here, we continue the treatment of these functions starting fromJ _n (x, y; i), which is of noticeable practical interest. Finally, we consider the cases of functionsJ _n(x₁, x₂,..., x_M) and the related modified version,I _n(x₁, x₂,..., x_M), with two significant physical applications. Calculations of multivariable generalized Bessel functions are discussed and numerical results are given forJ _n(x₁,x₂;i), withn=0, 1, in a region of interest.     

On the numerical evaluation of an oscillating infinite series III

An oscillating infinite series involving product of Bessel function J _o(x) and an oscillating infinite series involving trigonometric function sin(x) were evaluated and computed numerically in [1] and [2] respectively. In this paper, an oscillating infinite series involving product of exponential, Bessel and trigonometric functions is evaluated. The series is transformed first into the sum of two infinite integrals by using contour integration and then the infinite integral with oscillating integrand is transformed through some identities into a finite integral containing modified Bessel function K ₁(x). Finally, theset two integrals are evaluated numerically without any computational difficulties at all.     

Algorithms for regular and irregular coulomb and bessel functions

Algorithms for computing Coulomb-Bessel functions are considered, with emphasis on obtaining accurate values when the argument x is inside the classical turning point x_λ. Algorithms of Barnett et al. for the generalized Coulomb functions and their derivatives are discussed in the context of the phase integral formalism. Modified or alternative algorithms are considered that are designed to be valid for all values of argument x and index λ for the functions F_λ(x), G_λ(x). An algorithm for a ccelerating convergence of a power series by conversion to a continued fraction is presented, and is applied to the evaluation of spherical Bessel functions. An explicit formula for the integrand of the phase integral is presented for spherical Bessel functions. The methods considered need to be augmented by an efficient algorithm for computing the logarithmic derivative of G₀ + iF₀ for Coulomb functions when x is smaller than the charge parameter η.     

Abstracts to Forthcoming Papers

The problem is to find a point z such that y(z) = M where M is a given real number and y(x) is the unique solution of a Volterra integral equation of the first kind. A fourth order method illustrates the algorithm to be described.     

A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations

In this paper, to find an approximate solution of general linear Fredholm integro-differential–difference equations (FIDDEs) under the initial-boundary conditions in terms of the Bessel polynomials, a practical matrix method is presented. The idea behind the method is that it converts FIDDEs to a matrix equation which corresponds to a system of linear algebraic equations and is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. The solutions are obtained as the truncated Bessel series in terms of the Bessel polynomials J _n (x) of the first kind defined in the interval [0, ∞). The error analysis and the numerical examples are included to demonstrate the validity and applicability of the technique.     

Efficient algorithms to compute Hankel transforms using wavelets

The aim of the paper is to propose two efficient algorithms for the numerical evaluation of Hankel transform of order ν, ν>−1 using Legendre and rationalized Haar (RH) wavelets. The philosophy behind the algorithms is to replace the part xf(x) of the integrand by its wavelet decomposition obtained by using Legendre wavelets for the first algorithm and RH wavelets for the second one, thus representing Fν(y) as a Fourier-Bessel series with coefficients depending strongly on the input function xf(x) in both the cases. Numerical evaluations of test functions with known analytical Hankel transforms illustrate the proposed algorithms.     

Panconnectivity and edge-pancyclicity of 3-ary <Emphasis Type="Italic">N</Emphasis>-cubes

We study two topological properties of the 3-ary n-cube Q _n ³. Given two arbitrary distinct nodes x and y in Q _n ³, we prove that there exists an x–y path of every length ranging from d(x,y) to 3 ⁿ −1, where d(x,y) is the length of a shortest path between x and y. Based on this result, we prove that Q _n ³ is edge-pancyclic by showing that every edge in Q _n ³ lies on a cycle of every length ranging from 3 to 3 ⁿ .

Numerical evaluation of the Hankel transform by using linear Legendre multi-wavelets

An efficient algorithm for evaluating the Hankel transform Fn(p) of order n of a function f(r) is given. As the continuous Legendre multi-wavelets forms an orthonormal basis for L²(R); we expand the part rf(r) of the integrand in its wavelet series reducing the Hankel transform integral as a series of Bessel functions multiplied by the wavelet coefficients of the input function. Numerical examples are given to illustrate the efficiency of the proposed method.     

A general method to derive implicit equations of curves and surfaces using interlineation and interflation of functions

A general method is proposed to derive equations of irregular curves O _D (x, y) = 0 and of irregular surfaces O _G (x, y, z) =0 in implicit form, where the functions O _D (x, y) and O _G (x, y, z) belong to a prescribed differentiability class. The method essentially involves interlineation and interflation of functions. An example is considered.     

On the computation of spherical Bessel functions of complex arguments

In this short paper, problems with existing algorithms for computing spherical Bessel functions of complex arguments are reported. As a remedy, a revised algorithm based on the recurrence is proposed. The upper and lower limits for the starting order that can be used in the backward recurrence to reach a desired accuracy are given. The proposed algorithm is stable and is capable of computing a wide range of arguments approaching machine accuracy.     

Linear Algebra Considerations for the Multi-Threaded Simulation of Mechanical Systems

A solution method suitable for the multi-threaded simulation ofmechanical systems represented in Cartesian coordinates isproposed and analyzed. In a state-space framework for thesolution of the Differential Algebraic Equations (DAE) ofMultibody Dynamics, the position/velocity stabilization and theacceleration computation are based on iterative solvers applied toequivalent reduced problems. The most in-depth computationalaspect analyzed is the preconditioning, i.e., the direct solutionof the reduced systems. Provided a topology index reduction is first applied to the model, the effort for the direct solution of the reduced systems is shown to be of order O(N _J ), where N _J is the number of joints in the model. The recurring theme of thepaper is the central role that the topology of the mechanicalsystem plays in the overall performance of the numericalsimulation. Based on the topology of the model, parallelcomputational threads can be established to start in the equationformulation and continue through the iterative numericalalgorithms employed for the numerical solution. Task schedulingthese parallel threads is expected to redeem real-time performancefor certain classes of complex applications.     

The evaluation of transition matrix elements for atomic excitations by a screened Coulomb potential

We have developed a FORTRAN code for calculating screened potential matrix elements for transitions between atomic bound states. In the applied procedure, the target wave functions can be arbitrarily chosen, and a wide class of screening potentials can be applied. The program includes subroutines for the calculation of the modified Bessel functions of fractional order and their derivatives for a broad range of argument and order.     

A nonholonomic control method for stabilizing an X4-AUV

A nonholonomic control method is considered for stabilizing all attitudes and positions (x, y, or z) of an underactuated X4 autonomous underwater vehicle (AUV) with four thrusters and six degrees of freedom (DOF), in which the positions are stabilized according to the Lyapunov stability theory. A dynamic model is first derived, and then a sequential nonlinear control strategy is implemented for the X4-AUV which is composed of translational and rotational subsystems. A controller for the translational subsystem stabilizes one position out of the x-, y-, and z-coordinates, whereas controllers for the rotational subsystems generate the desired roll, pitch, and yaw angles. Thus, the rotational controllers stabilize all the attitudes of the X4-AUV at the desired (x-, y-, or z-) position of the vehicle. Some numerical simulations are conducted to demonstrate the effectiveness of the proposed controllers.     

An equivalence between rational H2 and Hankel-norm approximations

Let H(z) be a given function in H₂ A classical problem in engineering analysis is to find a rational function G (z) ε H₂ degree M say, which is closest to H(z) in 2-norm. This problem is typically approached using the cost function |H(z) − G(z)|₂, in which G(z) is allowed to vary over the set of Mth-order rational functions in H₂ and for which stationary points are sought. We show that each stationary point of degree M of this functional coincides with a weighted Hankel-norm approximant to H(z). The weighting function derives from the outer factor of the error function H(z) − G(z) stationary point of the rational H₂ approximation problem.     

Generalized Gaussian quadrature rules over regions with parabolic edges

This paper presents a generalized Gaussian quadrature method for numerical integration over regions with parabolic edges. Any region represented by R ₁={(x, y)| a≤x≤b, f(x)≤y≤g(x)} or R ₂={(x, y)| a≤y≤b, f(y)≤x≤g(y)}, where f(x), g(x), f(y) and g(y) are quadratic functions, is a region bounded by two parabolic arcs or a triangular or a rectangular region with two parabolic edges. Using transformation of variables, a general formula for integration over the above-mentioned regions is provided. A numerical method is also illustrated to show how to apply this formula for other regions with more number of linear and parabolic sides. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear and parabolic edges. Finally, the computational efficiency of the derived formulae is demonstrated through several numerical examples.     

Three-dimensional analysis of Marangoni flow and radial segregation in Ge<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Si<Subscript>1-<Emphasis Type="Italic">x</Emphasis></Subscript> melt with Czochralski configuration

In order to examine the flow field and the radial segregation of silicon (Si) in a Ge _x Si_1-x melt with an idealized Czochralski (Cz) configuration, we conducted a series of unsteady three-dimensional (3-D) numerical simulations under zero-gravity conditions. The effect of convection driven by surface tension on the free surface of the melt was included in the model, by considering thermal, as well as solutal Marangoni convection. The concentration and flow fields at several stages during crystal growth are presented for several temperature differences, driving the Marangoni convection. The simulation results indicate that the flow and concentration fields are axisymmetric for Ma _T < 625 and become oscillatory and 3-D for higher values. It was found that the maximum Si concentration difference at the growth interface decreases as thermal Marangoni number increases due to higher flow velocities in the vicinity of the interface. However, temporal fluctuations of Si concentration at the interface increase at higher thermal Marangoni numbers. The effects of aspect ratio (A _r) were also considered in the model. It was found that the aspect ratio of the melt in the crucible has a prominent influence on the flow pattern in the melt which, in turn, effects the Si concentration at the growth interface.     

Application of the modified Taylor approximation

In this paper we introduce and describe a new scheme for the numerical integration of smooth functions. The scheme is based on the modified Taylor expansion and is suitable for functions that exhibit near-sinusoidal or repetitive behaviour. We discuss the method and its rate of convergence, then implement it for the approximation of certain integrals. Examples include integrands involving Airy wave, Bessel, Gamma, and elliptic functions. The results, from the data in the tables, demonstrate that the method converges rapidly and approximates the integral as well as some well-known numerical integration methods used with sufficiently small step sizes.     

An algorithm for regular and irregular Coulomb and Bessel functions of real order to machine accuracy

We describe an algorithm to evaluate a wide class of functions and their derivatives, to extreme precision (25–30S) if required, which does not use any function calls other than square root. The functions are the Coulomb functions of positive argument (F_λ(x, η), G_λ(x, η), x > 0, η, λ real) and hence, as special cases with η = 0, the cylindrical Bessel functions (J_μ(x), Y_μ(x), x > 0, μ real), the spherical Bessel functions (i_λ(x), y_λ(x), x > 0, λ real), Airy functions of negative argument Ai(-x), Bi(-x) and others. The present method has a number of attractive features: both the regular and irregular solution are calculated, all others of the functions can be produced from a specified minimum (not necessarily zero) to a specified maximum, functions of a single order can be found without all of the orders from zero, the derivatives of the functions arise naturally in the solution and are readily available, the results are available to different precisions from the same subroutine (in contrast to rational approximation techniques) and the methods can be used for estimating final accuracies. In addition, the sole constant required in the algorithm is π, no precalculated arrays of coefficients are needed, and the final accuracy is not dependent on that of other subroutines. The method works most efficiently in the region x ≈ 0.5 to x ≈ 1000 but outside this region the results are still reliable, even though the number of iterations within the subroutine rises. Even in these more asymptotic regions the unchanged algorithm can be used with known accuracy to test other specific subroutines more appropriate to these regions. The algorithm uses the recursion relations satisfied by the Coulomb functions and contains a significant advance over Miller's method for evaluating the ratio of successive minimal solutions (F_λ+1/F_λ). It relies on the evaluation of two continued fractions and no infinite series is required for normalisation: instead the Wronskian is used.