Transformed Bakshali Square-Root Algorithm with Higher Orders of Convergence

The Bakshali formula for square root extraction is an amazing achievement of ancient Indian mathematics. Understanding the logic behind the Bakshali formula as a cycle of approximation, error determination, correction, and new approximation, a Bakshali algorithm for square root extraction in two variables (the approximation X_n and its correction Y_n ) is presented. The recursive algorithm extendable to any depth n leads to Bakshali-type formulas of high orders of convergence that goes as 2ⁿ, where n=1,2,3, …. . The Bakshali algorithm is streamlined by recasting it into a transformed form with three variables (y_n, t_n, z_n). The transformed Bakshali algorithm runs by generating integer sequences with no division in the stages and a single division at the end. It obeys the exact invariant z_n² − N t_n² = y_n², which places every iterate on a Pell conic and gives closed forms for the iterate and its error, and the order of convergence 2ⁿ. The iterates coincide with those of Newton’s method in rational form, but the route does not. The correction term of Bakshali algorihm is such the new residual is exactly the square of the correction. This exact error transfer builds stages that no longer involve N and save at least half a squaring each. For an integer radicand and an integer seed every stage is exact. Three consequences follow: the number of correct digits after each stage is known in advance; the integers never exceed the accuracy they deliver by more than one digit, so no precision schedule is needed; and a rigorous enclosure of the error follows from the algorithm’s own variables alone, at no extra cost. Nested versions with 8-fold and 16-fold convergence are demonstrated. In five 16-fold iterations, square root of a 5 digit number is obtained to more than 2.8 million correct digits, and in seven 8-fold iterations 5.6 million correct digits are obtained.