Nearest-Integer Continued Fractions in DÃkkaraõa

Abstract

The Karaõa texts of Indian astronomy give simplified expressions for the mean rates of motion of planets. The Kerala text Karaõapaddhati (c. 1532-1566 CE) expresses these rates which involve ratios of large numerators or multipliers (guõakras) and large demominators or divisors (hÀrakas), as ratios of smaller numbers using essentially the method of simple continued fraction expansion.A modified version of this method is described in a slightly later Malayalam text named DÃkkaraõa (c. 1608 CE), also. A very interesting feature of the DÃkkaraõa algorithm is that a nearest-integer continued fraction expansion with the minimal length is implicit in it. We discuss this algorithm in this paper.