Showing papers on "Elementary function published in 1999"

Approximating elementary functions with symmetric bipartite tables

Abstract: This paper presents a high-speed method for function approximation that employs symmetric bipartite tables. This method performs two parallel table lookups to obtain a carry-save (borrow-save) function approximation, which is either converted to a two's complement number or is Booth encoded. Compared to previous methods for bipartite table approximations, this method uses less memory by taking advantage of symmetry and leading zeros in one of the two tables. It also has a closed-form solution for the table entries, provides tight bounds on the maximum absolute error, and can be applied to a wide range of functions. A variation of this method provides accurate initial approximations that are useful in multiplicative divide and square root algorithms.

Efficient calculation of lattice sums for free-space periodic Green's function

Abstract: An efficient method to calculate the lattice sums is presented for a one-dimensional (1-D) periodic array of line sources. The method is based on the recurrence relations for Hankel functions and the Fourier integral representation of the zeroth-order Hankel function. The lattice sums of arbitrary high order are then expressed by an integral of elementary functions, which is easily computed using a simple scheme of numerical integration. The calculated lattice sums are used to evaluate the free-space periodic Green's function. The numerical results show that the proposed method provides a highly accurate evaluation of the Green's function with far less computation time, even when the observation point is located near the plane of the array.

Non-grey benchmark results for two temperature non-equilibrium radiative transfer.

Abstract: Benchmark solutions to time-dependent radiative transfer problems involving non-equilibrium coupling to the material temperature field are crucial for validating time-dependent radiation transport codes. Previous efforts on generating analytical solutions to non-equilibrium radiative transfer problems were all restricted to the one-group grey model. In this paper, a non-grey model, namely the picket-fence model, is considered for a two temperature non-equilibrium radiative transfer problem in an infinite medium. The analytical solutions, as functions of space and time, are constructed in the form of infinite integrals for both the diffusion description and transport description. These expressions are evaluated numerically and the benchmark results are generated. The asymptotic solutions for large and small times are also derived in terms of elementary functions and are compared with the exact results. Comparisons are given between the transport and diffusion solutions and between the grey and non-grey solutions.

An Algorithm that Computes a Lower Bound on the Distance Between a Segment and ℤ2.

Abstract: We give a fast algorithm for computing a lower bound on the distance between a straight line and the points of a bounded regular grid. This algorithm is used to find worst cases when trying to round the elementary functions correctly in floating-point arithmetic. These worst cases are useful to design algorithms that guarantee the exact rounding of the elementary functions.

Numerical Integration with Exact Real Arithmetic

Abstract: We show that the classical techniques in numerical integration (namely the Darboux sums method, the compound trapezoidal and Simpson's rules and the Gauss-Legendre formulae) can be implemented in an exact real arithmetic framework in which the numerical value of an integral of an elementary function is obtained up to any desired accuracy without any round-off errors. Any exact framework which provides a library of algorithms for computing elementary functions with an arbitrary accuracy is suitable for such an implementation; we have used an exact real arithmetic framework based on linear fractional transformations and have thereby implemented these numerical integration techniques. We also show that Euler's and Runge-Kutta methods for solving the initial value problem of an ordinary differential equation can be implemented using an exact framework which will guarantee the convergence of the approximation to the actual solution of the differential equation as the step size in the partition of the interval in question tends to zero.

A convolutive source separation method with self-optimizing non-linearities

Abstract: This paper deals with the separation of two convolutively mixed signals. The proposed approach uses a recurrent structure adapted by a generic rule involving arbitrary separating functions. These functions should ideally be set so as to minimize the asymptotic error variance of the structure. However, these optimal functions are often unknown in practice. The proposed alternative is based on a self-adaptive (sub-)optimization of the separating functions, performed by estimating the projection of the optimal functions on a predefined set of elementary functions. The equilibrium and stability conditions of this rule and its asymptotic error variance are studied. Simulations are performed for real mixtures of speech signals. They show that the proposed approach yields much better performance than classical rules.

Open Least Element Principle and Bounded Query Computation

Abstract: We show that elementary arithmetic formulated in the language with a free function symbol f and the least element principle for open formulas (where we assume that the symbols for all elementary functions are included in the language) does not prove the least element principle for bounded formulas in the same language. A related result is that composition and anyn umber of unnested applications of bounded minimum operator are, in general, insufficient to generate the elementary closure of a function, even if all elementaryfunctions are available. Thus, unnested bounded minimum operator is weaker than unnested bounded recursion.

Rigorous analysis of scattering by a periodic array of cylindrical objects

Abstract: Periodic arrays of cylindrical objects are widely used as the wavelength selective or polarization selective components in microwave and optical wave regions. Various theoretical approaches have been developed to solve the periodic boundary-value problems. The generalized recursive algorithm (T-matrix approach) can be applied to analyze a periodic array. In this case, the scattered fields are calculated using the T-matrix of the isolated group of scatterers within the unit cell of an array and the lattice sums characterizing the periodic configuration. Recently, an efficient and accurate method to calculate the lattice sums was presented. According to this method, the lattice sums are evaluated using an integration of elementary functions that give a possibility to reduce the computation time substantially. In this paper, we present an algorithm connecting the T-matrix representation for group scatterers and the lattice sums, that is attractive from the practical point of view for obtaining the desired frequency or polarization selective characteristics. We applied this algorithm to study the scattering properties of various arrays composed of periodic cells, containing two circular cylindrical objects, which can be dielectric, metallic or a mixture of both and different dimensions.

Effect of the instrumental function on measurements of phase functions of scattering by large particles at small scattering angles

Abstract: Within the framework of the small-angle approximation, a rigorous analytic solution to the problem of determining the instrumental function has been obtained for measurements of the scattering phase functions considering finite angular sizes of a radiation source and receiver. It has been shown that the instrumental function is described by an integral of a product of the Bessel functions. The instrumental function is expressed in terms of elementary functions. The effect of the instrumental function on the accuracy of measuring small-angle scattering-phase functions for different particle sizes is discussed.© (1999) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Device for computing elementary functions by table-and-algorithm method

Abstract: FIELD: computer engineering. SUBSTANCE: device is designed to compute functions by setting reference values of functions in interpolation blocks. Function increments are interpolated between interpolation blocks in function increment unit. Errors in reproduction of functions at device result output caused by digital data presentation in interpolation blocks and in function increment unit are relatively compensated for with finite number of bits and reduced impact of error in setting rate of function variation between interpolation blocks by first-order polynomial on resultant computing error. Four structures are proposed for implementing function increment unit depending on bit number n of code being converted and read-only memory capacity. Bit number n is between 11 and 32. Function computation error corresponds to 0.5-0.75 units of less significant bit of result. EFFECT: improved precision of elementary function computation. 6 dwg, 1 tbl

Estimating some integral parameters of phase function of large particles to analytically compute light propagation through a medium

Abstract: Analytical formulas are derived for integrals of phase function of large particles, namely for light fluxes scattered singly within an arbitrary angular range and mean scattering angle squared. The relations of the first kind are obtained via direct integration, by the scattering angle, of the Fresnel's reflectivities weighted with some angular function. The Fraunhofer's diffraction and geometrical optics parts are taken into account. As a result, the light flux is expressed as a sum of elementary functions. The formulas can be obviously converted to the known relations for the single-scattering albedo and mean cosine of the phase function for a particular case of the integration over full range from) to 180 degrees. The mean scattering angle squared is used, for example, by the small- angle diffusion approximation to compute light propagation. The corresponding formula is derived by comparing the solutions to the radiative transfer equation with the said approximation and with the small-angle one. The mean scattering angle squared is particularly shown to be inversely proportional to the effective size parameter squared of particles. The proportionality coefficient is found.

College Mathematics With Technology: For Business, Economics, Life and Social Sciences

Abstract: I. A LIBRARY OF ELEMENTARY FUNCTIONS. 1. A Beginning Library of Elementary Functions. 2. Additional Elementary Functions. II. FINITE MATHEMATICS. 3. Mathematics of Finance. 4. Systems of Linear Equations Matrices. 5. Linear Inequalities and Linear Programming. 6. Probability. 7. Markov Chains. 8. The Derivative. III. CALCULUS. 9. Graphing and Optimization. 10. Additional Derivative Topics. 11. Integration. 12. Additional Integration Topics. 13. Multivariable Calculus. 14. Differential Equations. Appendix A: Tables. I. Basic Geometric Formulas. II. Integration Formulas.

A Hardware Approach to Interval Arithmetic for Sine and Cosine Functions

Abstract: The existing software packages for interval arithmetic are often too slow for numerically intensive computation, whereas hardware solutions do not handle elementary functions directly. We propose some minimal modifications to the CORDIC architecture to efficiently support either interval or regular sine and cosine functions. The computational time for an interval operation is close to that of a regular operation for most cases and twice that of a regular operation for the remaining ones. The CORDIC algorithm is also slightly modified to guarantee correct choice of the bounds of the result, and a low worst case error is obtained.

Arithmetic unit for the computation of interval elementary functions

Abstract: Interval arithmetic for elementary functions has been tackled by different researchers from a software point of view. In this paper, we present a hardware solution to efficiently support the computation of some classic elementary functions: sin, sinh, cos, cosh and exp, using an interval as input argument. To perform this, we carry out a modification of the classic CORDIC architecture with a moderate hardware increase. The total computation time for the evaluation of these functions over an argument interval is, in most of the interesting cases, just a few cycle times greater than the computation time of the corresponding point functions.

Derivations of the Solid Angle Subtended at a Point by First- and Second-Order Surfaces and Volumes as a Function of Elliptic Integrals

Abstract: An analytical study of the solid angle subtended at a point by objects of first and second algebraic order has been made. It is shown that the derived solid angle for all such objects is in the form of a general elliptic integral, which can be written as a linear combination of elliptic integrals of the first and third kind and elementary functions. Many common surfaces and volumes have been investigated, including the conic sections and their volumes of revolution. The principal feature of the study is the manipulation of solid-angle equations into integral forms that can be matched with those found in handbook tables. These integrals are amenable to computer special function library routine analysis requiring no direct interaction with elliptic integrals by the user. The general case requires the solution of a fourth-order equation before specific solid-angle formulations can be made, but for many common geometric objects this equation can be solved by elementary means. Methods for the testing and application of solid-angle equations with Monte Carlo rejection and estimation techniques are presented. Approximate and degenerate forms of the equations are shown, and methods for the evaluation of the solid angle of a torus are outlined.

Stress state of a rock mass in the vicinity of seam containing a working. Three-dimensional problem

Abstract: Formulas are obtained to determine the stresses at any point of a rock mass. The formulas are derived through double integrals of an auxiliary function represented by a convergent Neumann series. Approximations of the kernels are presented by algebraic combinations of elementary functions.

Multiple precision computation of interval elementary functions

Abstract: The control of error during numerical evaluation of an expression is a crucial problem. A natural solution is to use interval arithmetic during calculations. For this reason, we need algorithms able to efficiently determine the image of an interval by the elementary functions. Adapting classical algorithms to interval arithmetic leads to a large overestimation of the result. In this paper, we propose a method based on Taylor approximation which evaluates functions on the bounds of the interval and deduces the resulting image. The computation of each bound in done with the Taylor approximation evaluated by the Smith scheme. Indeed, polynomial approximation seems to be the most adapted to our range of precision (several hundreds of digits). This approach is better than the classical algorithm using the Horner scheme and than the other polynomial approximations (e.g. minimax and Chebyshev polynomials) which are more precise but also more complicated. Furthermore, we present a rigorous evaluation of the error. Thus, the desired accuracy of the result is given, even on the numerically instable points (big number, (pi) /2 ...). For the computation of a point, our algorithm implemented with GMP is up to 2 times faster than the corresponding computation with softwares like MAPLE or MUPAD which don't even guarantee the relative precision. We show that the control of the error is not costly. And since the Taylor approximation is just a bit less precise but far more simple than the other approximations, it can be rapidly determined and, all in all, is the most efficient.© (1999) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Some Co-Circular Crack Problems with Cyclic Symmetry

Abstract: Some co-circular crack problems with cyclic symmetry in the infinite plane are considered. They are reduced to singular integral equations along a single crack, whose solutions may be expressed in terms of elementary functions with certain coefficients determined by some integrals.

Table methods for elementary functions

Abstract: We suggest a new table-based method for evaluating the exponential function in double precision arithmetic. This method can easily be extended to the base 2 exponential function.