Bessel function of the first kind with complex argument

A new method of computing integral order Bessel functions of the first kind J_n(z) 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 from asymptotic expressions when x∼ 0 (or y ∼ 0). These expansions are derived from the integral definition of Bessel functions. This method is necessary because some existing algorithms and methods fail to give correct results for small x small y. In addition, our overall method of computing Bessel functions of any order and argument is discussed and the logarithmic derivative is used in computing these functions. The starting point of the backward recurrence relations needed to evaluate the Bessel function and their logarithmic derivatives are investigated in order to obtain accurate numerical results. Our numerical method, together with established techniques of computing the Bessel functions, is easy to implement, efficient, and produces reliable results for all z.     

Optimized implementations of rational approximations—a case study on the Voigt and complex error function

Rational functions are frequently used as efficient yet accurate numerical approximations for real and complex valued special functions. For the complex error function w(x+iy), whose real part is the Voigt function K(x,y), the rational approximation developed by Hui, Armstrong, and Wray [Rapid computation of the Voigt and complex error functions, J. Quant. Spectrosc. Radiat. Transfer 19 (1978) 509-516] is investigated. Various optimizations for the algorithm are discussed. In many applications, where these functions have to be calculated for a large x grid with constant y, an implementation using real arithmetic and factorization of invariant terms is especially efficient.     

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.     

A novel approach to the fast computation of Zernike moments

This paper presents a novel approach to the fast computation of Zernike moments from a digital image. Most existing fast methods for computing Zernike moments have focused on the reduction of the computational complexity of the Zernike 1-D radial polynomials by introducing their recurrence relations. Instead, in our proposed method, we focus on the reduction of the complexity of the computation of the 2-D Zernike basis functions. As Zernike basis functions have specific symmetry or anti-symmetry about the x-axis, the y-axis, the origin, and the straight line y=x, we can generate the Zernike basis functions by only computing one of their octants. As a result, the proposed method makes the computation time eight times faster than existing methods. The proposed method is applicable to the computation of an individual Zernike moment as well as a set of Zernike moments. In addition, when computing a series of Zernike moments, the proposed method can be used with one of the existing fast methods for computing Zernike radial polynomials. This paper also presents an accurate form of Zernike moments for a discrete image function. In the experiments, results show the accuracy of the form for computing discrete Zernike moments and confirm that the proposed method for the fast computation of Zernike moments is much more efficient than existing fast methods in most cases.     

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.     

Quadrature rules for the integration of the product of two oscillatory functions with different frequencies

We construct quadrature rules for the efficient computation of the integral of a product of two oscillatory functions y₁(x) and y₂(x), where , and the functions f_i,j(x) are smooth. The weights are evaluated by the exponential fitting technique of Ixaru [Comput. Phys. Comm. 105 (1997) 1-19], which is now extended to cover the case of two frequencies. We give a numerical illustration on how the new rules compare for accuracy with the one-frequency dependent rules and with the classical ones.     

A combined symbolic and numerical algorithm for the computation of zeros of orthogonal polynomials and special functions

A Maple algorithm for the computation of the zeros of orthogonal polynomials (OPs) and special functions (SFs) in a given interval [x₁,x₂] is presented. The program combines symbolic and numerical calculations and it is based on fixed point iterations. The program uses as inputs the analytic expressions for the coefficients of the three-term recurrence relation and a difference-differential relation satisfied by the set of OPs or SFs. The performance of the method is illustrated with several examples: Hermite, Chebyshev, Legendre, Jacobi and Gegenbauer polynomials, Bessel, Coulomb and Conical functions.     

Fast error estimates for indirect measurements: applications to pavement engineering

In many real-life situations, we have the following problem: we want to know the value of some characteristicy that is difficult to measure directly (e.g., lifetime of a pavement, efficiency of an engine, etc.). To estimatey, we must know the relationship betweeny and some directly measurable physical quantitiesx ₁,...,x _n . From this relationship, we extract an algorithmf that allows us, givenx _i , to computey: y=f(x ₁, ...,x _n ). So, we measurex _i , apply an algorithmf, and get the desired estimate. Existing algorithms for error estimate (interval mathematics, Monte-Carlo methods, numerical differentiation, etc.) require computation time that is several times larger than the time necessary to computey=f(x ₁, ...,x _n ). So, if an algorithmf is already time-consuming, error estimates will take too long. In many cases, this algorithmf consists of two parts: first, we usex _i to determine the parametersz _k of a model that describes the measured object, and second, we use these parameters to estimatey. The most time-consuming part is findingz _k ; this is done by solving a system of non-linear equations; usually least squares method is used. We show that for suchf, one can estimate errors repeating this time-consuming part off only once. So, we can compute bothy and an error estimate fory with practically no increase in total computation time. As an example of this methodology, we give pavement lifetime estimates.     

Robust algorithms that locate local extrema of a function of one variable from interval measurement results: A remark

The problem of locating local maxima and minima of a function from approximate measurement results is vital for many physical applications: inspectral analysis, chemical species are identified by locating local maxima of the spectra; inradioastronomy, sources of celestial radio emission, and their subcomponents, are identified by locating local maxima of the measured brightness of the radio sky;elementary particles are identified by locating local maxima of the experimental curves. Since measurements are never absolutely precise, as a result of the measurements, we have aclass of possible functions. If we measuref(x _i ) with interval uncertainty, this class consists of all functionsf for whichf(x _i ) ε [y _i ??, y _i +?], wherey _i are the results of measuringf(x _i ), andε is the measurement accuracy. For this class, in [2], a linear-time algorithm was described. In real life, a measuring instrument can sometimes malfunction, leading to the so-calledoutliers, i.e., measurementsy _i that can be way offf(x _i ) (and thus do not restrict the actual valuesf(x _i ) at all). In this paper, we describerobust algorithms, i.e., algorithms that find the number of local extrema in the presence of possible outliers. These algorithms solve an important practical problem, but they are not based on any new mathematical results: they simply use algorithms from [2] and [3].     

Numerische Lösung von Anfangswertaufgaben gewöhnlicher Differentialgleichungen n-ter Ordnung

In the present paper a new method is given for the numerical treatment of the initial problemsy ⁽ⁿ⁾=f(x,y,y′, ...,y ^(n?1),y ⁽ⁱ⁾ (x _o )=y _o ⁽ⁱ⁾ , i=0, 1, ...,n?1. This method is an one-step process of order four. For a class of linear differential equations the exact solution is obtained. Moreover some numerical results are presented.     

Curve generation of implicit functions by incremental computers

The conventional numerical solution of an implicit function f(x, y) = 0 is substantially complicated for calculating by any computer. We propose a new method representing the argument of the implicit function as a unary function of a parameter, t, if the continuous and unique solution of f(x, y) = 0 exists. The total differential $\text{d}\text{f}\text{d}\text{t}$ constitutes simultaneous differential equations of which the solution about x and y is unique. The Newton-Raphson method must be used to calculate the values near singular points of an implicit function and then the sign of dt has to be decided according to four special cases. Incremental computers are suitable for curve generation of implicit functions by the new method, because the incremental computer can perform more complex algorithms than the analog computer and can calculate faster than the digital computer. This method is easily applicable to curve generation in three-dimensional space.     

In2O3 + xBaO (x = 0.5-5 at.%) - A novel material for trace level detection of NOx in the ambient

Indium oxide (In₂O₃) doped with 0.5-5 at.% of Ba was examined for their response towards trace levels of NO_x in the ambient. Crystallographic phase studies, electrical conductivity and sensor studies for NO_x with cross interference for hydrogen, petroleum gas (PG) and ammonia were carried out. Bulk compositions with x ≤ 1 at.% of Ba exhibited high response towards NO_x with extremely low cross interference for hydrogen, PG and ammonia, offering high selectivity. Thin films of 0.5 at.% Ba doped In₂O₃ were deposited using pulsed laser deposition technique using an excimer laser (KrF) operating at a wavelength of (λ) 248 nm with a fluence of ∼3 J/cm² and pulsed at 10 Hz. Thin film sensors exhibited better response towards 3 ppm NO_x quite reliably and reproducibly and offer the potential to develop NO_x sensors (Threshold limit value of NO₂ and NO is 3 and 25 ppm, respectively).     

Structure and humidity sensing properties of La1−xKxCo0.3Fe0.7O3−δ perovskite

The effects of K-substitution at La-site of La_1−xK_xCo_0.3Fe_0.7O_3-δ perovskite on its structure and humidity sensing properties were studied in detail. The XRD, SEM-EDS, N₂ adsorption-desorption measurements (BET), ICP-AES and XPS were used to characterize the microstructure of La_1−xK_xCo_0.3Fe_0.7O_3−δ perovskite. The results show that the partial substitution of K at La-site has no obvious effect on the crystal phase, morphology and surface area of samples, but leads to the increased oxygen vacancies and surface enriched K⁺. The sensitivity of humidity sensor based on all samples was evaluated by measuring the impedance response to the humidity changes. The partial substitution of K at La-site significantly enhanced the humidity sensitivity of La_1−xK_xCo_0.3Fe_0.7O_3−δ perovskite at low relative humidity (RH). By correlating the structure of material with its sensing properties, the probable reasons that lead to the remarkable sensitivity enhancement of the K-substitution samples compared with the unsubstituted sample (LaCo_0.3Fe_0.7O_3−δ) were given; moreover, the sensing mechanism was also discussed by the complex impedance spectra in detail.     

An extension of Kedlaya’s algorithm for hyperelliptic curves

In this paper we describe a generalisation and adaptation of Kedlaya’s algorithm for computing the zeta-function of a hyperelliptic curve over a finite field of odd characteristic that the author used for the implementation of the algorithm in the Magma library. We generalise the algorithm to the case of an even degree model. We also analyse the adaptation of working with the xⁱdx/y³ rather than the xⁱdx/y differential basis. This basis has the computational advantage of always leading to an integral transformation matrix whereas the latter fails to in small genus cases. There are some theoretical subtleties that arise in the even degree case where the two differential bases actually lead to different redundant eigenvalues that must be discarded.     

A class of hybrid collocation methods for third-order ordinary differential equations

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x _n+j , i = 0(1)k and at an off grid point x = x _n+u , where k is the step number of the method and u is an arbitrary rational number in (x _n , x _n+k ). A predictor of order 2k ? 1 is also proposed to cater for y _n+k in the main method. Taylor series expansion is employed for the calculation of y _n+1, y _n+2, y _n+u and their higher derivatives. Evaluation of the resulting method at x = x _n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.     

A parametric optimization approach to H∞ and H2 strong stabilization

This paper presents an approach for designing stable MIMO H_∞ and H₂ controllers by directly computing the norm-constrained stable transfer matrices Q in the H_∞ and H₂ suboptimal controller parameterizations. This is done by first converting the H₂ and H_∞ strong stabilization problems into some nonlinear unconstrained optimization problems through explicit parameterization of the norm-constrained Q's for any fixed order. Then, a two-stage numerical search is carried out by using a combination of a genetic algorithm and a quasi-Newton algorithm in order to reach an optimal solution. The effectiveness of the proposed algorithms is illustrated through some benchmark numerical examples.     

A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation

In this paper we present a new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. The scheme uses a symmetrical multi-point difference formula to represent the partial differentials of the two-dimensional variables, which can improve the accuracy of the numerical solutions to the order of Δx2Nq+2 when a (2Nq+1)-point formula is used for any positive integer Nq with Δx=Δy, while Nq=1 equivalent to the traditional scheme. On the other hand, the new scheme keeps the same form of the traditional matrix equation so that the standard algebraic eigenvalue algorithm with a real, symmetric, large sparse matrix is still applicable. Therefore, for the same dimension, only a little more CPU time than the traditional one should be used for diagonalizing the matrix. The numerical examples of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heiles potential demonstrate that by using the new method, the error in the numerical solutions can be reduced steadily and extensively through the increase of Nq, which is more efficient than the traditional methods through the decrease of the step size.     

On the moore test for coupled equations

In this paper an improved Moore test for the coupled system:f(x, y)=0,g(x, y)=0 is described: x⁺ is calculated from x and y in a forward-substep, and we use x⁺ and y to compute y⁺ in a backward-substep. It is shown that, if x⁺ ? x, y⁺ ? y, then a solution of the coupled system (x*,y*) ∈ (x⁺, y⁺) exists. On this foundation, we prove convergence of a point iterative algorithm for solving coupled systems.     

Effect of La2CuO4 electrode area on potentiometric NOx sensor response and its implications on sensing mechanism

The intent of this work is to look at the effects of varying the La₂CuO₄ electrode area and the asymmetry between the sensing and counter electrode in a solid state potentiometric sensor with respect to NO_x sensitivity. NO₂ sensitivity was observed at 500-600 °C with a maximum sensitivity of ∼22 mV/decade [NO₂] observed at 500 °C for the sensor with a La₂CuO₄ electrode area of ∼30 mm². The relationship between NO₂ sensitivity and area is nearly parabolic at 500 °C, decreases linearly with increasing electrode area at 600 °C, and was a mixture of parabolic and linear behavior 550 °C. NO sensitivity varied non-linearly with electrode area with a minima (maximum sensitivity) of ∼−22 mV/decade [NO] at 450 °C for the sensor with a La₂CuO₄ electrode area of 16 mm². The behavior at 400 °C was similar to that of 450 °C, but with smaller sensitivities due to a saturation effect. At 500 °C, NO sensitivity decreases linearly with area.We also used electrochemical impedance spectroscopy (EIS) to investigate the electrochemical processes that are affected when the sensing electrode area is changed. Changes in impedance with exposure to NO_x were attributed to either changes in La₂CuO₄ conductivity due to gas adsorption (high frequency impedance) or electrocatalysis occurring at the electrode/electrolyte interface (total electrode impedance). NO₂ caused a decrease in high frequency impedance while NO caused an increase. In contrast, NO₂ and NO both caused a decrease in the total electrode impedance. The effect of area on both the potentiometric and impedance responses show relationships that can be explained through the mechanistic contributions included in differential electrode equilibria.     

不确定的中心支持向量机

在一般支持向量机中,训练集的每个元素（x_i,y_i）,i=1,2,…,l,对输入x_i有确定的类别标号y_i 。而在许多实际问题中,x_i的类别往往是不确定的,常常是以概率z_i⁺属于正类的概率z_i^-属于负类。针对这种实际情况,改造训练集,把原来的y_i,用z_i⁺,z_i^-代替,然后建立分类最优先模型,构建不确定中心支持向量机。