Showing papers on "Inverse trigonometric functions published in 1981"

High bandwidth evaluation of elementary functions

Abstract: Among the requirements currently being imposed on high-performance digital computers to an increasing extent are the high-bandwidth computations of elementary functions, which are relatively time-consuming procedures when conducted in software. In this paper, we elaborate on a technique for computing piecewise quadratric approximations to many elementary functions. This method permits the effective use of large RAMs or ROMs and parallel multipliers for rapidly generating single-precision floating-point function values (e.g., 30–45 bits of fraction, with current RAM and ROM technology). The technique, based on the use of Taylor series, may be readily pipelined. Its use for calculating values for floating-point reciprocal, square root, sine, cosine, arctangent, logarithm, exponential and error functions is discussed.

Hardware evaluation of mathematical functions

Abstract: The paper presents a family of algorithms for evaluating the elementary mathematical functions including division, sine, cosine, tangent, arctangent, logarithm, exponential and square root. The algorithms are based on continued products and continued sums, and generate the sum on a 2-bit-at-a-time basis. The only operations required are shifting, adding, subtracting and recall of prestored constants. Consequently, the method is suitable for hardware implementation, and two designs are presented. The faster design can generate a 64-bit result in under 1 μs for most of the functions, and under 2 μS for the remainder, when using ECL technology of the F100K type. The other design is based on a pipelined structure and is approximately 40% slower and cheaper. The performance of both designs compares favourably with that of a hardwired polynomial evaluation based on Homer's scheme when using a 300 ns 64- by 64-bit multiplier. Such a design is between 35% and 130% slower than the pipelined version depending upon the function evaluated, yet only 30% cheaper.

Differential and Integral Calculus

Abstract: Part One. Differential Calculus: Limits as $n=\infty$ Logarithms, powers, and roots Functions and continuity Limits as $x=\xi$ Definition of the derivative General theorems on the formation of the derivative Increase, decrease, maximum, minimum General properties of continuous functions on closed intervals Rolle's theorem and the theorem of the mean Derivatives of higher order Taylor's theorem "0/0" and similar matters Infinite series Uniform convergence Power series Exponential series and binomial series The trigonometric functions Functions of two variables and partial derivatives Inverse functions and implicit functions The inverse trigonometric functions Some necessary algebraic theorems Part Two. Integral Calculus: Definition of the integral Basic formulas of the integral calculus The integration of rational functions The integration of certain non-rational functions Concept of the definite integral Theorems on the definite integral The integration of infinite series The improper integral The integral with infinite limits The gamma function Fourier series Index of definitions Subject index.

Simple, wide-range approximations to trigonometric and inverse trigonometric functions useful in real-time signal processing

Abstract: By means of straightforward methods, simple approximations are derived for the cosine, sine and tangent functions as well as for their inverse functions. The approximations are simple in the sense that they contain few terms and have numerical coefficients with, at most, three significant figures. The trigonometric approximations presented are valid in all four quadrants, including the singularity points in the case of the tangent. The inverse trigonometric approximations are valid over the full natural argument intervals. The absolute approximation errors are shown to be near-minimax with peak values all less than 0.25% of maximum. The results are expected to be useful in real-time signal processing circuits and systems where speed and simplicity rather than extreme accuracy are the dominating factors. A translinear circuit realisation example is given in the Appendix.

The trigonometric and hyperbolic functions

Abstract: This chapter discusses calculus to the trigonometric functions. It discusses differentiation of trigonometric functions and presents the calculation of the six trigonometric functions: sin x , cos x , tan x , sec x , csc x , and cot x . The chapter discusses the rules for differentiating trigonometric functions. Periodic motion is a very common occurrence in physical or biological settings. A simple example is given by the back and forth motion of a pendulum. Another is given by the oscillating size of the population of an animal species. Yet another is given by the rise and fall of average daily temperature in a fixed location over a time span of several years. It often happens that the equations describing periodic or harmonic motion involve the sine and cosine functions. In many physical applications especially those involving differential equations, functions arise that are combinations of e x and e − x . This happens so often that certain of the combinations that occur most frequently have been given special names. These functions are named hyperbolic functions.

On integrals and summable trigonometric series

Abstract: Abstract In considering a problem on certain summable (C, k) trigonometric series, R. D. James [13] used a symmetric pk+2- integral defined earlier to recapture the coefficients of the series from the sum function. James' formulas for the coefficients are more complicated than the usual Euler-Fourier form since the pk + 2 - integral is of order k + 2. It is shown that a generalized integral of order one for each non-negative integer k can be suitably defined to reduce James' formulas to the usual form.