Showing papers on "Inverse trigonometric functions published in 1972"

An algorithm for computing logarithms and arctangents

Abstract: An iterative algorithm with fast convergence can be used to compute logarithms, inverse circular functions, or inverse hyperbolic functions according to the choice of initial conditions. Only rational operations and square roots are required. The method consists in adding an auxiliary recurrence relation to Borchardt's algorithm to speed the convergence. 1. Introduction. Logarithms, inverse circular functions, and inverse hyperbolic functions are computed in practice by a variety of methods, including infinite series, continued fractions, Chebyshev approximations, and rational approximations (4). A method known as Borchardt's algorithm (2, Eq. (2.3)), (3) employs very simple recurrence relations containing a square root but is seldom used, because the error is reduced by a factor of only four per cycle. Thus, lOD accuracy typically requires fifteen to twenty square roots. A related algorithm, due to Thacher (6), has a factor of sixteen per cycle but loses some significant figures through cancellation. In the present paper, we introduce an auxiliary recurrence relation to speed the convergence of Borchardt's algorithm so that only three or four square roots are required for 1OD accuracy. (We assume here that the range of the independent variable has been reduced by familiar devices such as arctan(l/x) = 7r/2 - arctan x. More cycles are required outside the reduced range.) The method amounts to repeated application of a well-known technique (5, pp. 86-87) for improving convergence by extrapolation. The recurrence relations are the same for all the functions con- sidered, there are no constants to be stored, and no serious cancellation occurs. The precision is limited only by the number of cycles performed, and the rate of convergence gradually accelerates. Although still not as fast as some other methods because of the square roots, the algorithm might well be preferable when economy of storage space is important. It might be useful also in verifying the accuracy of faster algorithms. The method is being extended to computation of elliptic integrals, but only elementary functions are discussed in the present paper. 2. Statement of the Algorithm. The same recurrence relations will be used for computing a logarithm, an inverse circular function, or an inverse hyperbolic function, but the initial values and the final step will depend on which function is being computed. For definiteness, we state first the complete algorithms for com- puting natural logarithms and arctangents and later give the changes necessary for computing other functions.