On-Line Algorithms for Division and Multiplication
IEEE TRANSACTIONS ON COMPUTERS, VOL. C-26, NO. 7, JULY 1977 681 On-Line Algorithms for Division and Multiplication KISHOR S. TRIVEDI AND MILO0 D. ERCEGOVAC Abstract-In this paper, on-line algorithms for division and subtraction operations always satisfy this property. multiplication are developed. It is assumed that the operands as well Atrubin [3] has developed a right-to-left on-line algorithm as the result flow through the arithmetic unit in a digit-by-digit, most significant digit first fashion. The use of a redundant digit set, for multiplication. In this paper, we will retain our earlier at least for the digits of the result, is required. definition of the on-line property where all of the operands as well as the result digits flow through the arithmetic unit Index Terms-Computer arithmetic, division, multiplication, in a left-to-right digit-by-digit fashion. on-line algorithms, pipelining, radix, redundancy. Consider an mr-digit radix r number N = 1=l nir-i. In the conventional representation, each digit ni can take any I. INTRODUCTION value from the digit set 0,1, - - ,r - 1}. Such representa- IN THIS PAPER we consider problems of divi- tions, which allow only r values in the digit set, are non- sion and multiplication in a computational environ- redundant since there is a unique representation for each ment in which all basic arithmetic algorithms satisfy the (representable) number. By contrast, number systems that "on-line" property. This implies that to generate the jth allow more than r values in the digit set are redundant.
[Show full text]