A Quartically Convergent Square Root Algorithm

Abstract

In this presentation I will summarize some interesting research Jon Borwein and I have done lately on India mathematics.

As it is well known, in 1984 Jon and Peter found a quartically convergent algorithm for pi (i.e., each iteration approximately quadruples the number of correct digits). Along this line, I have found a quartically convergent algorithm for the square root, certainly not due to me but instead appearing in an ancient Indian manuscript that dates to 200 CE or so! What's more, the scheme may well predate that manuscript, perhaps by several hundred years, to 400 BCE or so. There are some other interesting items on cube roots and the origin of positional decimal arithmetic as well.

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