Showing papers on "Elementary function published in 1994"
TL;DR: In this article, a generalization of the Goulian-Li continuation in the power of the 2D cosmological term is proposed to construct the two-and three-point correlation functions for Liouville exponentials with generic real coefficients.
593 citations
TL;DR: In this article, the precision results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta, and the imaginary parts of these integrals are expressed as complete elliptic integrals.
Abstract: Motivated by the precision results in the electroweak theory studies of two-loopFeynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop self-energy diagrams in arbitrary and especially four dimensions.This is done in three respects firstly results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta. Secondly the imaginary parts of these integrals are expressed as complete elliptic integrals.Finally one-dimensional integral representations with elementary functions are derived.They are very well suited for the numerical evaluations.
135 citations
TL;DR: In this paper, expressions for the elements of the dielectric tensor for linear waves propagating at an arbitrary angle to a uniform magnetic field in a fully hot plasma whose constituent particle species σ are modeled by generalized Lorentzian distribution functions are derived.
Abstract: Expressions are derived for the elements of the dielectric tensor for linear waves propagating at an arbitrary angle to a uniform magnetic field in a fully hot plasma whose constituent particle species σ are modeled by generalized Lorentzian distribution functions. The expressions involve readily computable single integrals whose integrands involve only elementary functions, Bessel functions, and modified plasma dispersion functions, the latter being available in the form of finite algebraic series. Analytical forms for the integrals are derived in the limits λ→0 and λ→∞, where λ=(k⊥ρLσ)2/2, with k⊥ the component of wave vector perpendicular to the ambient magnetic field, and ρLσ the Larmor radius for the particle species σ. Consideration is given to the important limits of wave propagation parallel and perpendicular to the ambient magnetic field, and also to the cold plasma limit. Since most space plasmas are well modeled by generalized Lorentzian particle distribution functions, the results obtained in this paper provide a powerful tool for analyzing kinetic (micro‐) instabilities in space plasmas in a very general context, limited only by the assumptions of linear plasma theory.
122 citations
TL;DR: It is proved that the functions computable in on-line by a finite automaton are piecewise affine functions whose coefficients are rational numbers.
Abstract: After a short introduction to on-line computing, we prove that the functions computable in on-line by a finite automaton are piecewise affine functions whose coefficients are rational numbers (i.e., the functions f(x)=ax+b, or f(x,y)=ax+by+c where a, b, and c are rational). A consequence of this study is that multiplication, division and elementary functions of operands of arbitrarily long length cannot be performed using bounded-size operators. >
42 citations
TL;DR: A new algorithm for computing the complex logarithm and exponential functions is proposed, based on shift-and-add elementary steps, and it generalizes some algorithms by Briggs and De Lugish (1970), as well as the CORDIC algorithm.
Abstract: A new algorithm for computing the complex logarithm and exponential functions is proposed. This algorithm is based on shift-and-add elementary steps, and it generalizes some algorithms by Briggs and De Lugish (1970), as well as the CORDIC algorithm. It can easily be used to compute the classical real elementary functions (sin, cos, arctan, ln, exp). This algorithm is more suitable for computations in a redundant number system than the CORDIC algorithm, since there is no scaling factor when computing trigonometric functions. >
37 citations
TL;DR: In this article, a new trial function which extends the generalized Gaussian method for the study of weakly guiding single-mode fibers is presented, which uses only simple elementary functions to approximate the fundamental modal fields.
Abstract: A new trial function which extends the generalized Gaussian method for the study of weakly guiding single-mode fibers is presented. This function uses only simple elementary functions to approximate the fundamental modal fields. In many cases of study, we show that the present trial function gives a significant improvement over the existing trial functions, and it is very useful in the analysis of fibers and related devices. >
24 citations
TL;DR: This work describes a floating-point arithmetic unit based on the CORDIC algorithm that computes a full set of high level arithmetic and elementary functions: multiplication, division, (co)sine, hyperbolic, square root, natural logarithm, inverse (hyperbolic) tangent, vector norm, and phase.
Abstract: This work describes a floating-point arithmetic unit based on the CORDIC algorithm. The unit computes a full set of high level arithmetic and elementary functions: multiplication, division, (co)sine, hyperbolic (co)sine, square root, natural logarithm, inverse (hyperbolic) tangent, vector norm, and phase. The chip has been integrated in 1.6 /spl mu/m double-metal n-well CMOS technology and achieves a normalized peak performance of 220 MFLOPS. >
24 citations
TL;DR: In this article, the fundamental matrix of the system of partial differential operators that governs the diffusion of heat and the strains in elastic media is represented by simple definite integrals and power series, and the constant and the linear term in the Taylor series about 0 with respect to the coupling constant are expressed explicitly by error functions, elementary functions, and delta functions.
Abstract: The fundamental matrix of the system of partial differential operators that governs the diffusion of heat and the strains in elastic media is represented by simple definite integrals and power series. For the elements of this matrix, the constant and the linear term in the Taylor series about 0 with respect to the coupling constant are expressed explicitly by error functions, elementary functions, and delta functions.
17 citations
03 Aug 1994
TL;DR: A CORDIC algorithm with a critically damped convergence process by changing the restriction to either positive or negative "pseudo rotations" so that the scale factor is a constant.
Abstract: The COordinate Rotation DIgital Computer (CORDIC) algorithm has been widely used in evaluating elementary functions. It employs only basic arithmetic operations such as additions and shifts rather than multiplications. This attractive hardware property is achieved by restricting the operation at each iteration step to either positive or negative "pseudo rotations," so that the scale factor is a constant. However, because of this restriction the angular error does not converge directly towards zero. Instead, it asymptotically oscillates about zero like an underdamped second-order control system responding to a step input. Multiple overshots may also be observed during the convergence. According to control theory, this kind of convergence is not fast. In this paper, we develop a CORDIC algorithm with a critically damped convergence process by changing the restriction to either.
15 citations
TL;DR: A new theorem is reported that generalizes Liouville's theorem on integration in finite terms and allows dilogarithms to occur in the integral in addition to elementary functions.
14 citations
01 Aug 1994
TL;DR: A solution for a version of the identify problem is proposed for a class of functions including the elementary functions, and a semi algorithm is given to solve the elementary constant problem.
Abstract: A solution for a version of the identify problem is proposed for a class of functions including the elementary functions. Given f(x), g(x), defined at some point b we decide whether or not f(x) = g(x) in some neighbourhood of b. This problem is first reduced to a problem about zero equivalence of elementary constants. Then a semi algorithm is given to solve the elementary constant problem. This semi algorithm is guaranteed to give the correct answer whenever it terminates, and it terminates unless the problem being considered contains a counterexample to Schanuel's conjecture.
01 Aug 1994
TL;DR: Methods for constructing signature functions for expressions containing algebraic numbers are presented and compared and some experimental results are given.
Abstract: In 1980 Schwartz gave a fast probabilistic method which tests if a matrix of polynomials over Z is singular or not. The method is based on the idea of signature functions which are mappings of mathematical expressions into finite rings. In Schwartz's paper, they were polynomials over Z into GF(p). Because computation in GF(p) is very fast compared with computing with polynomials, Schwartz's method yields an enormous speedup both in theory and in practice. Therefore it is desirable to extend the class of expressions for which we can find effective signature functions. In the mid 80's Gonnet extended the class of expressions, for which signature functions could be found, to include a restricted class of elementary functions and integer roots. In this paper we present and compare methods for constructing signature functions for expressions containing algebraic numbers. Some experimental results are given.
01 Jan 1994
TL;DR: This paper proposes a method called ATA-M (Add-Table Lookup-Add with Multiplication) for evaluating polynomials with the aid of tables for computing the elementary functions.
Abstract: One of the most spectacular development in computer technology is the growth in memory density and speed. It is with this development in mind that we intend to tackle the old problem of computing the elementary functions. Since the dawn of computing, the fast and accurate computation of the elementary functions has been a constant concern of numerical computing. It now seems possible to use tables of sizes in the range of megabits to aid in such computation. To this end, in this paper, we propose a method called ATA-M (Add-Table Lookup-Add with Multiplication) for evaluating polynomials with the aid of tables. When applied to the elementary functions, we obtained a set of algorithms which computes the reciprocal, square root, exponential, sine, cosine, logarithm, are tangent and the hyperbolic functions in about 3 to 4 double precision floating point multiplication time and utilizing about 2 Mbyte of tables. >
TL;DR: In this paper, the transfer function associated with the nonlinear Fabry-Perot resonator was derived in terms of simple approximate expressions of various orders of accuracy, and the validity criteria of validity for the approximate formulae were derived.
Abstract: In this paper we present an exact calculation of the transfer function associated with the nonlinear Fabry-Perot resonator While our exact result cannot be evaluated in terms of elementary functions, it does permit us to obtain a number of simple approximate expressions of various orders of accuracy In addition, our derivation yields criteria of validity for the approximate formulae Our approach is to be compared with others in which approximations are introduced in the model itself, either through the equations or through the boundary conditions Our lowest order approximate formula turns out to be identical, interestingly, with the result obtained from the slowly varying envelope approximation (SVEA) Thus, our validity criteria apply to the SVEA result, and predict well its domain of validity and its breakdown for short wavelengths and for very high intensities and nonlinearities The simple higher order formulae we present provide improved estimations in such regimes
01 Aug 1994
TL;DR: A conjecture that gives the form of an integral if it can be expressed using elementary functions and polylogarithms is reported, which consists of a generalization of Liouville's theorem on integration in finite terms with elementary functions.
Abstract: In this abstract, we report on a conjecture that gives the form of an integral if it can be expressed using elementary functions and polylogarithms. The conjecture is proved by the author in the cases of the dilogarithm and the trilogarithm [3] and consists of a generalization of Liouville's theorem on integration in finite terms with elementary functions. Those last structure theorems, for the dilogarithm and the trilogarithm, are the first case of structure theorems where logarithms can appear with non-constant coefficients. In order to prove the conjecture for higher polylogarithms we need to find the functional identities, for the polylogarithms that we are using, that characterize all the possible algebraic relations among the considered polylogarithms of functions that are built up from the rational functions by taking the considered polylogarithms, exponentials, logarithms and algebraics. The task of finding those functional identities seems to be a difficult one and is an unsolved problem for the most part to this date.
01 Jan 1994
TL;DR: This study proposes approaches for the hardware design of the sigmoid and the logarithm function based on a hybrid approach, and shows that one of the proposed designs can accommodate all ten elementary functions without loss of performance.
Abstract: In this study we present a number of schemes for the generation of elementary functions for hardwired neural network emulators. We propose approaches for the hardware design of the sigmoid and the logarithm function based on a hybrid approach. The hybrid approach requires access to lookup tables and direct computations. Our proposed approaches outperform existing schemes in terms of speed and hardware requirements, while keeping an accuracy in the order of the IEEE double precision floating point format. Additionally, ten elementary functions that are commonly used in the neural network paradigm have been designed using first and second degree piece-wise approximations. These functions have a precision on the order of 2-$\sp{10}$ with inexpensive hardware. The elementary functions are: the sigmoid and its derivative, the logarithm, the sine and cosine trigonometric functions, the exponential, the hyper tangent, the square root, the inverse and inverse square. The proposed design approaches outperform existing schemes in terms of performance, hardware cost, and precision. It is shown that one of the proposed designs can accommodate all ten elementary functions without loss of performance. Furthermore, the function generator can be programmable; this in turn provides the capability of extending the computations to other elementary functions with no penalties in terms of performance, hardware cost, or additional design effort. Those features make our low precision schemes suitable for neural network emulators that require "moderate" precision for the computation of elementary functions. Their inclusion in the design allows those emulators to achieve high performance computations with low hardware cost.
Patent•
22 Jul 1994
TL;DR: In this paper, a function computing system consisting of a processor, a molecule orbit calculating means, and a data base was proposed to reduce the computing time and improve the operability of a function computation system by converting previously a multi-variable function into the product of the uni-variable functions.
Abstract: PURPOSE:To reduce the computing time and to improve the operability in a function computing system by converting previously a multi-variable function into the product of the uni-variable functions to store the function value in a prescribed array and then reading out the function value to compute the multi-variable function. CONSTITUTION:A computing means of a function computing system 21 consists of a processor 22, a molecule orbit calculating means 24, and a data base 25. Then the function value is acquired at each lattice point from the multi-variable function which defines the space coordinates as an independent function so that plural lattice points of the space coordinates in a three-dimensional space area are continuously shown on the screen of a display device 26 in a three-dimensional way. Under such conditions, the computing means previously replaces the multi-variable function with a prescribed number of uni-variable functions and computes each uni-variable function to store them in a storage in a prescribed array. Then the processor 22 reads the value of uni-variable functions out of the storage 23 and calculates the multi-variable function. Therefore the operation which calculates the function value at each lattice point of a three-dimensional space finally includes no elementary function. Thus the computing time can be reduced.
TL;DR: L'A. et al. as discussed by the authors etudie la prehistoire puis la chronologie de la theorie de l'integration en termes finis de Liouville par laquelle le mathematicien opere le passage de l"approche analytique des fonctions a l'approche algebrique
Abstract: L'A. etudie la prehistoire puis la chronologie de la theorie de l'integration en termes finis de Liouville par laquelle le mathematicien opere le passage de l'approche analytique des fonctions a l'approche algebrique
TL;DR: New algorithms using shifts, adds and “small multiplications” (i. e. multiplications by few-digit-numbers) are presented and the number of the required steps to compute functions with a given accuracy is reduced and the computation time is reduced.
Abstract: Many hardware-oriented algorithms computing the usual elementary functions (sine, cosine, exponential, logarithm, ...) only use shifts and additions. In this paper, we present new algorithms using shifts, adds and “small multiplications” (i. e. multiplications by few-digit-numbers). These CORDIC-like algorithms compute the elementary functions in radix 2 p (instead of the standard radix 2) and use table look-ups. The number of the required steps to compute functions with a given accuracy is reduced and since we use a quick “small multiplier”, the computation time is reduced.
01 Jan 1994
TL;DR: In this article, the use of Hermite functions in vision and image analysis has gained interest in the image processing community, and some results on texture analysis using Hermite function have been presented.
Abstract: Recently the use of Hermite functions in vision and image analysis has gained interest in the image processing community. Hermite functions are related to Gaussian functions through a differential operator. In the search for a suitable spatially localized, but not-redundant, alternative to the Gabor approach, Gaussian derivatives as basis functions were introduced. The literature reports that Hermite functions fit the profiles of many cells in the striate cortex very well. We will present some results on texture analysis using Hermite functions.