| |
Shortcuts in
Continued Fraction Method for Solving Brahmagupta-Bhaskara equation.
Dr. R. Rangarajan
Dept. of Mathematics, Kuvempu University,
Shankaraghatta,
Dist. Shimoga, Karnataka.
The Chakravala method of Bhaskara for solving
Brahmagupta-Bhaskara equation is a milestone in the glorious history of
ancient Indian Mathematics. Continued fraction algorithm is intimately
connected to Chakravala algorithm just like it is connected to Euclid's
algorithm for finding G.C.D. of two positive integers. Continued
fraction algorithm must have been known to Indian Mathematicians. Until
the discovery of Euclid's algorithm for computing convergents, continued
fraction method was not popular. It was first systematically stated and
proved the continued fraction method by Lagrange for Brahmagupta-Bhaskara
equation (which Euler mistakenly called Pell's equation). In the continued
fraction method, we compute the convergents of a periodic continued fraction
by setting up a sequence called PQ-sequence. If n is the length of period
of the continued fraction, then whenever n is even, we need to compute
nth convergents otherwise we need to compute 2nth convergent. If n is large,
the method will not be efficient. In the present paper a systematic study
of PQ-scquence is carried out to identify a type of symmetric pattern.
By using the symmetric pattern, short-cuts are obtained. In fact the information
upto n/2th element of the PQ sequence is sufficient to compute the solution.
|
|
home
membership