Unrestricted Algorithms for Reciprocals and Square Roots

摘要

In previous works the authors introduced a new form of computer arithmetic, called level-index arithmetic, the main purpose of which is to eradicate the problems of overflow and underflow associated with the floating-point and fixed-point systems. In order to implement the new arithmetic in multiple precision, unrestricted algorithms are needed to evaluate exponentials and logarithms. By an unrestricted algorithm we mean one in which the user may demand any accuracy in the function values for arguments of any magnitude. An unrestricted algorithm for the exponential function was described previously. This paper we constructs an unrestricted algorithm for computing square roots. This is a necessary preliminary in the construction of an unrestricted algorithm for the logarithmic function. At the same time this paper a quite similar algorithm for forming reciprocals: the direct division process that is used for low-precision work is inefficient for multiple-precision calculations.