Fast spherical Bessel transform via fast Fourier transform and recurrence formula

JRAF: A Julia package for computation of relativistic molecular auxiliary functions

In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations.

In this paper, we present p+q+m and p+q+2m parametric S-functions, exploring the relationships between these two types of functions. We derive their fundamental properties, including generating functions, recurrence relations, differential formulas and integral representations. From these results, we establish several notablecorollaries. Furthermore, we present a distribution function involving the S-function and compute the Laplace transform of its density function. Additionally, we explore the connections between the S-function and other significant transcendental functions.

Spherical area light sources are widely used in synthetic rendering. However, traditional Monte Carlo methods can require an excessive number of samples for sufficient accuracy. We propose a Spherical Harmonics (SH) based method to provide a trade-off between performance and accuracy. Our key idea is an analytical integration of SH over spherical caps. The SH integration is first decomposed into a weighted sum of Zonal Harmonics (ZH) integration, which could be evaluated using recurrence formulae. The resulting integration could then be used for rendering spherical area lights efficiently, saving 50% light samples at best while maintaining competitive accuracy. Our method can easily fit into an existing SH based rendering framework to support near-field sphere lighting.

For any vector r = r1 + r2 an expansion is derived for the product of a power rN of its magnitude and a surface spherical harmonic YLM(ϑ, φ) of its polar angles in terms of spherical harmonics of the angles (ϑ1, φ1) and (ϑ2, φ2). The radial factors satisfy simple differential equations; their solutions can be expressed in terms of hypergeometric functions of the variable (r</r>)2, and the leading coefficients by means of Gaunt's coefficients or 3j symbols. A number of linear transformations and three-term recurrence relations between the radial function are derived; but in contrast to the case L = 0, no generally valid expressions symmetric in r1 and r2 could be found. By interpreting the terms operationally, an expansion is derived for the product of YLM(ϑ, φ) and an arbitrary function f(r). The radial factors are expansions in derivatives of f(r>); for spherical waves, they factorize into Bessel functions of r1 and r2 in agreement with the expansion by Friedman and Russek. The 3j symbols are briefly discussed in an unnormalized form; the new coefficients are integers, satisfying a simple recurrence relation through which they can be arranged on a five-dimensional generalization of Pascal's triangle.

The objective of present research is to introduce and analyze a new special function that occurs as a part of the kernel of the integral representation of the λ-generalized Hurwitz-Lerch zeta functions. One important aspect of the analysis of special functions is to study their properties. It is found that λ-generalized gamma function satisfies the log convex and derivative properties. Recurrence relation and reflection formula are achieved that are always important to study the behavior of new functions. As an application, a new series representation of the λ-generalized Hurwitz-Lerch zeta functions is established. These new results are validated by analyzing their important cases that agree with the known results. It is interesting to note that the coefficients in the series representation of the family of zeta functions are generalized from “1” to “gamma function” and then to “generalized gamma” and “λ-generalized gamma functions” in a simple and natural way.

A method is presented for the accurate and efficient computation of the forces and their first derivatives arising from any number of zonal and tesseral terms in the Earth's gravitational potential. The basic formulae are recurrence relations between some solid spherical harmonics,V n,m, associated with the standard polynomial ones.

The Parlange hysteresis model is reformulated as a pair of recurrence relations to provide relationships between wetting and drying phases to any order. The model is applied to the classical van Genuchten model for soil water retention used as the main wetting curve. The nonphysical behaviour of these retention curves is related to the existence of a point of inflection in the van Genuchten model when it is used for the wetting boundary. Where the van Genuchten form is used as the main drying curve, the Parlange hysteresis model provides an ordinary differential equation describing the main wetting curve. A number of simple analytical solutions, relating to particular values of the parameters of the van Genuchten model, then provide forms for the main wetting curve. In general, a numerical integration is required to generate the main wetting curve, for general values of the parameters of the van Genuchten model. The recurrence relations for the hysteresis cycling are still applicable, even when the main wetting curve is only known numerically. The new main wetting curves do not have inflection points and there is no nonphysical behaviour. The model is then applied to the experimental data of Viaene et al. (1994)

Chapter 14 - Series Solutions: Important ODEs

A family of multivariate orthogonal polynomials generalizing the standard (univariate) Charlier polynomials is shown to arise in the matrix elements of the unitary representation of the Euclidean group E(d) on oscillator states. These polynomials in d discrete variables are orthogonal on the product of d Poisson distributions. The accent is put on the d = 2 case and the group theoretical setting is used to obtain the main properties of the polynomials: orthogonality and recurrence relations, difference equation, raising/lowering relations, generating function, hypergeometric and integral representations and explicit expression in terms of standard Charlier and Krawtchouk polynomials. The approach is seen to extend straightforwardly to an arbitrary number of variables. The contraction of SO(3) to E(2) is used to show that the bivariate Charlier polynomials correspond to a limit of the bivariate Krawtchouk polynomials.

: The solution to the problem of scattering by an arbitrary homogeneous sphere (Mie theory) yields an infinite set of coefficients. These scattering coefficients depend on various Bessel functions and their derivatives. The usual approach to computing scattering coefficients is to compute the Bessel functions by recurrence. The Bessel functions in the expressions for the scattering coefficients satisfy recurrence relations. It has been shown that the scattering coefficients themselves satisfy recurrence relations. Whether or not the recurrence relations can simplify scattering calculations must be determined by extensive calculations. It is not known if the recurrence relations are stable, either upward or downward. These matters are for further investigation.

We present an approach to solve numerical integration. The theorem to solve numerical integration is proven. This paper also gives the examples of numerical integration. The results show that numerical integration method presented in this paper has a high accuracy, hence, its application values is significant in the engineering practice.

The aim of the present paper is to deliver a method for exact calculation of the probability of photon migration in an infinite and homogeneous medium scattering photons anisotropically. The phase function is represented as an expansion over spherical harmonics. The probability of photon migration is obtained as an expansion with respect to the number of scatterings with the coefficients depending on the distance-to-time ratio and can be easily calculated from recurrence relations. Up to 30 scatterings are taken into account when computing the migration probabilities, the number of effectively contributing scatterings being strongly dependent on the distance-to-time ratio decreasing when one approaches the propagation front. The important property of the method is its capability to describe exactly migration from the early arriving photons up to the scattering ones. The limits of the approximate description of anisotropic scattering as isotropic with an effective value of the scattering coefficient is analyzed by calculating the best-fit value of the scattering coefficient.

In this chapter we derive semi closed-form solutions of European interest-rate derivatives in terms of their transformed payoff functions, for all contracts given in Chapter 3. Equipped with this frequency representation of the payoff function, the contract can be priced with the general valuation formula according to equation (4.21). This procedure, combined with a standardized numerical integration routine, can then be used to compute the desired quantities. Apart from the generality of this method, we observe that all call and put option contracts exhibit identical payoff representations in Fourier space. The difference between them are the different strips in the imaginary plane, parallel to the real axis, on which the transform operation is valid for the particular contract.KeywordsOption PriceCall OptionShort RatePrice FormulaOption ContractThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

<abstract> <p>Numerical integration plays an important role in solving various engineering and scientific problems and is often learnt in applied calculus commonly through the trapezium and Simpson's methods (or rules). A common misconception for some students is that Simpson's method is the default choice for numerical integration due to its higher accuracy in approximation over the trapezium method by overlooking the requirement for using Simpson's method. As learning progressed to other numerical methods scheduled later in advanced mathematics, such as interpolations and computational modelling using computing tools like MATLAB, there is a lack of articulation among these numerical methods for students to solve problems solvable only by combining two or more approaches. This classroom note shares a few teaching and learning practices the author experienced in lectures, tutorials, and formal assessments on comparing or combining different numerical methods for numerical integration for engineering students in applied calculus and advanced mathematics over the past decade at Central Queensland University (CQU), a regional university in Australia. Each case represents a common concern raised or a mistake made by some students in different times. These efforts helped not only correct the misconception on the use of Simpson's method by some students, but also develop students' strategic thinking in problem solving, particularly involving decision-making for choosing the best possible method to produce a more appropriate solution to a problem that does not have an analytical solution.</p> </abstract>