Showing papers on "Elementary function published in 2000"

Arbitrary Load Distribution on a Layered Half Space

Abstract: This paper develops a method which allows one to calculate the complete elastic field (stress field unit displacements) of layered materials of transverse and complete isotropy under given load conditions. It is assumed that the layered body consists of at infinite half-space and various infinite planes which are all ideally bonded to each other. Thus, the interfaces are parallel to the surface of the resulting coated half space. The approach is based on the method of images in classical electrostatics. The final solution for an arbitrary load problem can be presented as a series of potential functions, where corresponding functions may be interpreted as image loads the analogous to image charges. The solution for the elastic field for any arbitrary stress distribution on the surface of the coated half space can be obtained in a relatively straightforward manner by using the method described here as long as the corresponding solution for the homogeneous half space is known. Further, if this solution of the homogeneous case may be expressed in terms of elementary functions, then the solution for the coated half space is elementary, too, Explicit formulas for the stress fields for some particular examples are given.

On the General Solution for Piezothermoelasticity for Transverse Isotropy With Application

Abstract: This paper derives a general solution of the three-dimensional equations of transversely isotropic piezothermoelastic materials (crystal class, 6 mm). Two displacement functions are first introduced to simplify the basic equations and a general solution is then derived using the operator theory. For the static case, the proposed general solution is very simple in form and can be used easily in certain boundary value problems. An illustrative example is given in the paper by considering the symmetric crack problem of an arbitrary temperature applied over the faces of a flat crack in an infinite space. The governing integro-differential equations of the problem are derived. It is found that exact expressions for the piezothermoelastic field for a penny-shaped crack subject to a uniform temperature can be obtained in terms of elementary functions.

Elementary function generators for neural-network emulators

Abstract: Piecewise first- and second-order approximations are employed to design commonly used elementary function generators for neural-network emulators Three novel schemes are proposed for the first-order approximations The first scheme requires one multiplication, one addition, and a 28-byte lookup table The second scheme requires one addition, a 14-byte lookup table, and no multiplication The third scheme needs a 16-byte lookup table, no multiplication, and no addition A second-order approximation approach provides better function precision; it requires more hardware and involves the computation of one multiplication and two additions and access to a 28-byte lookup table We consider bit serial implementations of the schemes to reduce the hardware cost The maximum delay for the four schemes ranges from 24- to 32-bit serial machine cycles; the second-order approximation approach has the largest delay The proposed approach can be applied to compute other elementary function with proper considerations

Numerical and asymptotic aspects of parabolic cylinder functions

Abstract: Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.

Numerical and asymptotic aspects of parabolic cylinder functions

Abstract: Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.

Optimal) duplication is not elementary recursive

Abstract: In 1998, Asperti and Mairson proved that the cost of reducing a λ-term using an optimal λ-reducer (a la Levy) cannot be bound by any elementary function in the number of shared-beta steps. We prove in this paper that an analogous result holds for Lamping's abstract algorithm. That is, there is no elementary function in the number of shared beta steps bounding the number of duplication steps of the optimal reducer. This theorem vindicates the oracle of Lamping's algorithm as the culprit for the negative result of Asperti and Mairson. The result is obtained using as a technical tool Elementary Affine Logic.

Reasoning about the Elementary Functions of Complex Analysis

Abstract: There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make major errors, or not to simplify enough. In this paper we outline the "unwinding number" approach to such problems, and show how it can be used to prevent errors and to systematise such simplification, even though we have not yet reduced the simplification process to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complex-variable analysis.

A Closed-Form Solution for the Irradiance due to Linearly-Varying Luminaires

Abstract: We present a closed-form expression for the irradiance at a point on a surface due to an arbitrary polygonal Lambertian luminaire with linearly-varying radiant exitance. The solution consists of elementary functions and a single well-behaved special function that can be either approximated directly or computed exactly in terms of classical special functions such as Clausen’s integral or the closely related dilogarithm. We first provide a general boundary integral that applies to all planar luminaires and then derive the closed-form expression that applies to arbitrary polygons, which is the result most relevant for global illumination. Our approach is to express the problem as an integral of a simple class of rational functions over regions of the sphere, and to convert the surface integral to a boundary integral using a generalization of irradiance tensors. The result extends the class of available closed-form expressions for computing direct radiative transfer from finite areas to differential areas. We provide an outline of the derivation, a detailed proof of the resulting formula, and complete pseudo-code of the resulting algorithm. Finally, we demonstrate the validity of our algorithm by comparison with Monte Carlo. While there are direct applications of this work, it is primarily of theoretical interest as it introduces much of the machinery needed to derive closed-form solutions for the general case of luminaires with radiance distributions that vary polynomially in both position and direction.

Systematic study of elementary models of analytical signals in the form of peaks and waves

Abstract: An analytical signal represented as a symmetrical peak or a corresponding integral curve (wave) was described using three elementary functions: Gaussian function, derivative of a logistic function, and Cauchy function. The shape and geometric properties of such an analytic peak were characterized by a triangular frame formed by the tangents at the inflection points and the asymptotes to peak branches. In the case of a wave, a frame formed by the tangent at the inflection point of the wave and the asymptotes to its lower and upper branches was used for the same purpose. The use of the shape of differential curves as increments for physicochemical calculations was discussed.

A Radix-10 BKM Algorithm for Computing Transcendentals on Pocket Computers

Abstract: We present a radix-10 variant of the BKM algorithm. It is ashift-and-add, CORDIC-like algorithm that allows fast computation of complex exponentials and logarithms. It can easily be used to compute the classical real elementary functions (sin, cos, arctan, ln, exp). This radix-10 version is suitable for implementationin a pocket computer.

Upper and Lower Bounds on Continuous-Time Computation

Abstract: We consider various extensions and modifications of Shannon’s General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive functions, the elementary functions, the levels of the Grzegorczyk hierarchy, and the arithmetical and analytical hierarchies.

When is the Fourier transform of an elementary function elementary

Abstract: Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the form f(x)/t, where f is a cubic form in many variables (the simplest case after quadratic forms). It turns out that this classification is closely related to Zak's classification of Severi varieties. The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify combinations of multiplicative characters that can arise. The answer (for real and complex fields) is given in terms of exact covering systems. We also describe examples related to prehomogeneous vector spaces. Finally, we consider examples over p-adic fields, and in particular give a local proof of an integral formula of D.K. that could previously be proved only by a global method.

Parallel computation of complex elementary functions using quaternary signed-digit arithmetic

Abstract: A variety of algorithms for computing complex elementary functions based on the quaternary signed-digit (QSD) number system are proposed. An arithmetic unit that performs parallel one-step addition (subtraction), multiplication, and division is proposed to perform the computations of elementary functions such as square root, logarithmic, exponential, and other related functions. An optoelectronic-correlator-based architecture is suggested for implementing the proposed QSD elementary function algorithms. We used the symbolic substitution technique to reduce the number of the computation rules involved.

An Analog Characterization of the Subrecursive Functions

Abstract: We study a restricted version of Shannon's General Purpose Analog Computer in which we only allow the machine to solve linear differential equations. This corresponds to only allowing local feedback in the machine's variables. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions. Furthermore, we show that if the machine has access to an oracle which computes a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n-3 non-linear differential equations of a certain kind. Therefore, we claim that there is a close connection between analog complexity classes, and the dynamical systems that compute them, and classical sets of subrecursive functions.

Design of a unified arithmetic processor based on redundant constant-factor CORDIC with merged scaling operation

Abstract: An arithmetic processor is designed based on redundant constant-factor implementation of the coordinate rotation digital computer (CORDIC) algorithm with three different modes: circular, hyperbolic and linear. Both CORDIC types (angle calculation and vector rotation) are considered in this unified processor that is capable of computing a wide variety of arithmetic and elementary functions including: multiplication, division, some common trigonometric functions, natural logarithms, square roots, vector norm and phase. Furthermore, by merging the scaling operation with the regular CORDIC iterations, the processor based on folded (iterative) CORDIC architecture reduces by about 1/4 the total number of iterations in one complete CORDIC operation.

Upper and Lower Bounds on Continuous-Time Computation

Abstract: We consider various extensions and modifications of Shannon's General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive functions, the elementary functions, the levels of the Grzegorczyk hierarchy, and the arithmetical and analytical hierarchies.

Orthogonal Polynomials in Analytical Method of solving Differential Efquations Describing Dynamics of Multilevel Systems

Abstract: An effective method to obtain exact analysis solutions of equations describing the coherent dynamics of multilevel systems is presented.The method is based on the use of orthogonal polynominals,integral transform and discrete analogues.All the obtained solutions are expressed in terms of the special or elementary functions.

Elementary and integral-elementary functions

Abstract: By an integral-elementary function we mean any real function that can be obtained from the constants, sin $x$, $e^{x}$, $\log x$, and arcsin $x$ (defined on ($-1,1$)) using the basic algebraic operations, composition and integration. The rank of an integral-elementary function $f$ is the depth of the formula defining $f$. The integral-elementary functions of rank $\leq n$ are real-analytic and satisfy a common algebraic differential equation $P_{n} (f,f',\ldots,f^(k))=0$ with integer coefficients. We prove that every continuous function $f:\mathbf{R} \rightarrow \mathbf{R}$ can be approximated uniformly by integral-elementary functions of bounded rank. Consequently, there exists an algebraic differential equation with integer coefficients such that its everywhere analytic solutions approximate every continuous function uniformly. This solves a problem posed by L. A. Rubel. Using the same basic functions as above, but allowing only the basic algebraic operations and compositions, we obtain the class of elementary functions. We show that every differentiable function with a derivative not exceeding an iterated exponential can be uniformly approximated by elementary functions of bounded rank. If we include the function arcsin $x$ defined on $[-1,1]$, then the resulting class of naive-elementary functions will approximate every continuous function uniformly. We also show that every sequence can be uniformly approximated by elementary functions, and that every integer sequence can be represented .in the form $f(n)$, where $f$ is naive-elementary.

Accelerated Shift-and-Add Algorithms

Abstract: The problem addressed in this paper is the computation of elementary functions (exponential, logarithm, trigonometric functions, hyperbolic functions and their reciprocals) in fixed precision, typically the computer single or double precision. The method proposed here combines Shift-and-Add algorithms and classical methods for the numerical integration of ODEs: it consists in performing the Shift-and-Add iteration until a point close enough to the argument is reached, thus only one step of Euler method or Runge-Kutta method is performed. This speeds up the computation while ensuring the desired accuracy is preserved. Time estimations on various processors are presented which illustrate the advantage of this hybrid method.

Schrodinger's Cataplex

Abstract: We discuss elementary entwiners that cross-weave the variables of certain integrable models: Liouville, sine-Gordon, and sinh-Gordon field theories in two-dimensional spacetime, and their quantum mechanical reductions. First we define a complex time parameter that varies from one energy-shell to another. Then we explain how field propagators can be simply expressed in terms of elementary functions through the combination of an evolution in this complex time and a duality transformation.

A novel model for optical functions of GaSb

Abstract: In this paper we propose an analytical expression for the complex dielectric function of semiconductors which includes both discrete and continuum exciton effects. We start from the unbroadened expression for the dielectric function based on Elliott's work [R. J. Elliott, Phys. Rev. 108 (1957) 1384], and after the introduction of broadening we obtain the expression for the complex dielectric function. The proposed analytical model accurately takes into account the excitonic effects, while it satisfies the requirements that the imaginary part of the dielectric function is an odd function of energy, and the real part of the dielectric function is an even function. We show that accurate description of broadening leads to equations for the dielectric function containing only elementary functions, with terms dependent on the exciton order m describing discrete exciton states. The proposed model has been applied to model the experimental data for the absorption edge of GaSb. We have obtained good agreement with the experiment. The agreement with experimental data can be improved further if adjustable broadening function is considered instead of a simple Lorentzian one.

Exact Solutions of a (2 + 1)-Dimensional Nonlinear Klein-Gordon Equation

Abstract: The purpose of this paper is to present a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations of second order and their solutions by a singularity analysis are classified. In particular, it has been shown that whenever they have the Painleve property, they can be transformed to standard forms by Moebius transformations of dependent variable and arbitrary smooth transformations of independent variable whose solutions, depending on the values of parameters, are expressible in terms of either elementary functions or Jacobi elliptic functions.

Correctly rounded functions for better arithmetic

Abstract: The IEEE 754 standard for floating-point arithmetic requires that the four arithmetic operations and the square root should be correctly rounded. This has improved the accuracy, reliability and portability of numerical software. Unfortunately, such a requirement does not exist for the elementary functions. As a consequence, some libraries that are used everyday are rather poor. The problem of correctly rounding these functions is called the "Table Maker's Dilemma". After a survey on this problem, we give some new results. They show that correctly rounded functions could be provided at a reasonable cost.

Closed-Form Expressions for Irradiance from Non-Uniform Lambertian Luminaires Part II: Polynomially-Varying Radiant Exitance

Abstract: We present a closed-form expression for the irradiance at a point on a surface due to an arbitrary polygonal Lambertian lurninaire with linearly-varying radiant exitance. The solution consists of elementary functions and a single well-behaved special function that can be either approximated directly or computed exactly in terms of classical special functions such as Clausen's integral or the closely related dilogarithm. We first provide a general boundary integral that applies to all planar luminaires and then derive the closed-form expression that applies to arbitrary polygons, which is the result most relevant for global illumination. Our approach is to express the problem as an integral of a simple class of rational functions over regions of the sphere, and to convert the surface integral to a boundary integral using a generalization of irradiance tensors. The result extends the class of available closed-form expressions for computing direct radiative transfer from finite areas to differential areas. We provide an outline of the derivation, a detailed proof of the resulting formula, and complete pseudo-code of the resulting algorithm. Finally, we demonstrate the validity of our algorithm by comparison with Monte Carlo. While there are direct applications of this work, it is primarily of theoretical interest as it introduces much of the machinery needed to derive closed-form solutions for the general case of luminaires with radiance distributions that vary polynomially in both position and direction.

An apparatus for determining the values of elementary, trigonometric and hyperbolic functions and process for making same

Abstract: An electronic device is operated and controlled by a microprogram to provide as improved elementary function floating-point processor. The microprogram includes a set of routines which arc based on a modified and expanded COordinate Rotation DIgital Computer (CORDIC) algorithm for calculating a number of elementary functions including trigonometric and inverse trigonometric functions, hyperbolic and inverse hyperbolic functions, exponential, logarithm, square root, multiply, divide and rotation of vectors. The device is capable of handling a plural number of consecutive bits from associated data storage registers.

Implementation of the Prelle-Singer Method for 1ODEs

Abstract: A set of MapleV R5 software routines for solving first order ordinary differential equations (1ODEs) is presented. The package implements the Prelle-Singer Method in its original form plus its extension to include elementary functions (ELFs) on the integrating factor . The package also presents a theoretical extension to deal with all 1ODEs presenting liouvillian solutions (LIS).

Fourier transform over finite field and identities between Gauss sums

Abstract: This is a sequel to math.AG/0003009. Here we study identities for the Fourier transform of "elementary functions" over finite field containing "exponents" of monomial rational functions. It turns out that these identities are governed by monomial identities between Gauss sums. We show that similar to the case of complex numbers such identities correspond to linear relations between certain divisors on the space of multiplicative characters.