Showing papers on "Elementary function published in 1986"

New scalar and vector elementary functions for the IBM system/370

Abstract: Algorithms have been developed to compute short- and long-precision elementary functions: SIN, COS, TAN, COTAN, LOG, LOG10, EXP, POWER, SQRT, ATAN, ASIN, ACOS, ATAN2, and CABS, in scalar (28 functions) and vector (22 functions) mode. They have been implemented as part of the new VS FORTRAN library recently announced along with the IBM 3090 Vector Facility. These algorithms are essentially table-based algorithms. An important feature of these algorithms is that they produce bitwise-identical results on scalar and vector System/370 machines. Of these, for five functions the computed value result is always the correctly rounded value of the infinite-precision result. For the rest of the functions, the value returned is one of the two floating-point neighbors bordering the infinite-precision result, which implies exact results if they are machine-representable. For the five correctly rounded elementary functions, scalar and vector algorithms are designed independently so as to optimize performance in each case. For other functions, the bitwise-identical constraint leads to algorithms which compromise between scalar and vector performance. The authors have been able to design algorithms where this compromise is minimal and thus achieve very good performance on both scalar and vector implementations. For test measurements on high-end System/370 machines, scalar functions aremore » always faster (sometimes by as much as a factor of 2.5) as compared to the old VS FORTRAN library. The vector functions are usually two to three times faster than scalar functions.« less

A new class of analytic solutions of the two-state problem

Abstract: A new class of solutions of the time-dependent Schrodinger equation is found for the two-state problem often encountered in quantum optics, magnetic resonance and atomic collisions. The authors use the Riemann-Papperitz differential equation to find exact solutions in terms of hypergeometric functions. They consider only cases in which the final occupation probabilities are elementary functions of the parameters of the model.

Integration in finite terms with special functions: the logarithmic integral

Abstract: Since R. Risch published an algorithm for calculating symbolic integrals of elementary functions in 1969 (Traps. Amer. Math. Soc., 139 (1969), pp. 167–189), there has been an interest in extending his methods to include nonelementary functions. In this paper, we use the framework of differential algebra to make precise the notion of integration in terms of elementary functions and logarithmic integrals. Basing our work on a recent extension of Liouville’s theorem on integration in finite terms, we then describe a decision procedure for determining if a given element in a transcendental elementary field has an integral which can be written in terms of elementary functions and logarithmic integrals. This algorithm first examines the structure of the integrand in order to limit the logarithmic integrals which could appear in the integral to a finite number. This allows us to write a general expression for the integral and then use techniques similar to those employed by Risch to calculate the undetermined parts.

The Matrix Exponential Approach To Elementary Operations

Abstract: In 1971, J.S. Walther generalized and unified J.E. Volder's coordinate rotation (CORDIC) algorithms. Using Walther's algorithms a few commonly used functions such as divide, multiply-and-accumulate, arctan, plane rotation, arctanh, hyperbolic rotation can be implemented on the same simple hardware (shifters and adders, elementary controller) and computed in approximately the same time. Can other useful functions be computed on the same hardware by further generalizing these algorithms? Our positive answer lies in a deeper understanding of Walther's unification: the key to the CORDIC algorithms is that all of them effect the multiplication of a vector by the exponential of a 2 X 2 matrix. The importance of this observation is readily demonstrated as it easily yields the convergence conditions for the CORDIC algorithms and an efficient way of extending the domain of convergence for the hyperbolic functions. A correspondence may be established between elementary functions such as square-root, √(x2+y) , inverse square-root or cubic root and exponentials of simple matrices. Whenever such a correspondence is found, a CORDIC-like algorithm for computing the function can be synthesized in a very straightforward manner. The algorithms thus derived have a simple structure and exhibit uniform convergence inside an adjustable, precisely defined, domain.

Integration of Liouvillian functions with special functions

Abstract: In this paper, we discuss a decision procedure for the indefinite integration of transcendental Liouvillian functions in terms of elementary functions and logarithmic integrals. We also discuss a decision procedure for the integration of a large class of transcendental Liouvillian functions in terms of elementary functions and error-functions.

APL, the language and its actuarial applications

Abstract: Introduction. Elementary Operations. Scalars, Vectors, Matrices and Arrays. Basic Monadic and Dyadic APL Functions: Elementary Functions. Relational and Logical Functions. Circular Functions. Further Operations. Reduction, Scan, Compression, Expansion. Functions. The Outer and Inner Product. Further Operations: The Reversal, Rotate, Transposition. Decode and Encode. Format and Execute. Matrix and Matrix Inverse. System Commands, System Functions and System Variables. Applications: Loss Reserving Methods. Credibility Theory. Probability Functions. Elements of Numerical Analysis. Forecasting. Appendix. References. Index.

Symbolic integration in terms of error-functions and logarithmic integrals (liouvillian, decision procedure, differential field, elementary extension, computer algebra)

Abstract: We deal with the problem of symbolic integration of transcendental liouvillian functions in terms of logarithmic integrals or error-functions. We give an algorithm to decide if such a function has an integral which can be expressed in terms of the elementary functions and logarithmic integrals, and, if so, find it. We give a similar algorithm involving error-functions, but have a restriction, in this case, on the function to be integrated. We give several examples illustrating each algorithm. We also give an example demonstrating the difficulty in removing the restriction in the error-function algorithm.

Exponentials and logarithms

Abstract: This chapter discusses two of the most important functions of mathematics: the exponential and logarithmic functions. It discusses why they are important by giving a large variety of applications in physics, biology, and economics. There are two very different ways to define the exponential and logarithmic functions. The first of these starts with the basic rules of exponents and the second way starts off with a special integral. In both cases, one is led to the same functions. Both ways of defining these functions can give insight into how to work with them. It is useful to see how two very different approaches lead to the same central idea. The chapter discusses the relationship between the exponential and logarithmic functions and the differentiation and integration of more general exponential and logarithmic functions.

Taylor’s Formula

Abstract: We finally come to the point where we develop a method which allows us to compute the values of the elementary functions like sine, exp, and log The method is to approximate these functions by polynomials, with an error term which is easily estimated This error term will be given by an integral, and our first task is to estimate integrals We then go through the elementary functions systematically, and derive the approximating polynomials