Monday, October 18, 2010

Chakravala - Decision Analytics in Ancient India

More than two thousand years ago, many priests in India had to double up as mathematicians. They practiced the Sanathana Dharma, 'the eternal way of ethical living' (or Hinduism as it is popularly known today). Hinduism is richly influenced by nature, as well as the earthly and celestial elements, and fire rituals were quite important in those times. Careful attention was paid to the geometry of the altar, since different shapes were required depending on the objective of the ritual. This naturally gave rise to analytics, and a 'textbook' in those days (800-200 BCE) was the 'Sulba Sutras' to help figure out the correct angles and lengths to optimally design them (that lead to the discovery of Pythagorean triples and trigonometry, among other things). Inevitably, the beautiful natural patterns inherent in numbers awoke the inner geek in some of these priests.

Among the many famous mathematicians who carried forward this rich Vedic tradition was an astronomer named Brahmagupta (~ 600 CE). He extensively explored solutions to linear Diophantine equations that are central to integer programming today. Of course, his bigger distinction is for 'much ado about nothing'. He is known to be the first human being to clearly define, publish, and use zero as a number! He also explored Diophantine equations of the second degree, and came up with ideas that led to a recursive, iterative solution method for such equations; again something that is very useful in modern numerical optimization. He generalized an idea discovered by Diophantus and used this to achieve some success in finding solutions to Pell's equation. This was generalized to the Chakravala (Sanskrit for 'cyclic') algorithm by Jayadeva (950CE) and Bhaskara II (1100CE). A key subroutine at an iteration involves a neat rational scaling operation, followed by the solving of a simple discrete optimization problem that finds an integer m, such that it minimizes |m2N|/k, where N and k are input parameters. Furthermore, they recognize that such problems have degenerate solutions, and in some cases, find minimal integer feasible solutions using this approach. This attention to detail toward handling numerical issues stands out. Recognizing and tackling degeneracy is at the very heart of modern decision-analytics practice!

The Chakravala turns out to be an easy-to-use iterative method to find good approximations for square roots of integers. Indeed, this approach has been recognized for its ingenuity and 'careful simplicity' that allows us to work with well-conditioned real numbers; ideas that we in the OR community know are critical to matrix refactoring within a successful dual simplex implementation, for example.

1. Shiva,

a bunch of links on Indian math. Pearce's overview is very helpful, in particular.

Although Brahmagupta formalized Zero as a number and wrote about it, Aryabhatta may have pioneered it before him. I also think that there is a very good case for calling Aryabhatta the "Father of Algebra," at least in the currently extant civilization; he should almost certainly get precedence over Al-Khwarzmi (Persian mathematician from the 9th century, who seems to have picked up Indian mathematics from translations of Brahmagupta's Siddhantas, and advanced Indian numerals and arithmetic/algorithms to work with them. Persian and Arabic works of Indian math eventually reached Europe via Spain) for that distinction.

-- Panini Bio: http://www-history.mcs.st-andrews.ac.uk/Biographies/Panini.html

-- Indian Mathematics: Redressing the balance. by Ian G Pearce
http://www-history.mcs.st-andrews.ac.uk/Projects/Pearce/index.html

-- 8 II. Aryabhata and his commentators
http://www-history.mcs.st-andrews.ac.uk/Projects/Pearce/Chapters/Ch8_2.html

-- 8 III. Brahmagupta, and the influence on Arabia
http://www-history.mcs.st-andrews.ac.uk/Projects/Pearce/Chapters/Ch8_3.html

-- 8 V. Bhaskaracharya II
http://www-history.mcs.st-andrews.ac.uk/Projects/Pearce/Chapters/Ch8_5.html

-- 9 III. Madhava of Sangamagramma
http://www-history.mcs.st-andrews.ac.uk/Projects/Pearce/Chapters/Ch9_3.html

-- Prof. David Mumford's Review of Kim Plofker's book on Indian Mathematics:
www.ams.org/notices/201003/rtx100300385p.pdf

-- Mathematics in India, Kim Plofker
http://www.amazon.com/Mathematics-India-Kim-Plofker/dp/0691120676

-- A Cross-Cultural History of Mathematics. By Profs. Robin Hartshorne and David Mumford
(syllabus gives a neat chronological order)
http://www.dam.brown.edu/people/mumford/Math191/

-- Some notes from Dr. Mumford's class on Math History (see Chaps. 3 & 4 on Ind. Math.)
http://www.dam.brown.edu/people/mumford/Math191/lectures/

-- Indian numerals
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_numerals.html

-- A history of Zero
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html

-- History Topics: Index of Ancient Indian mathematics
http://www-history.mcs.st-andrews.ac.uk/Indexes/Indians.html

-- History Topics Index
http://www-history.mcs.st-andrews.ac.uk/Indexes/HistoryTopics.html

-- Mathematicians born in India
http://www-history.mcs.st-andrews.ac.uk/Countries/India.html

-- Srinivasa Ramanujan Bio
http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Ramanujan.html

Finally, from an Islamic and Islam-skeptic angles, the following are closely related to Indian mathematics:

-- al-Khwarizmi (Persian) Bio
http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Al-Khwarizmi.html

-- al-Kindi (Arab) Bio
http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Al-Kindi.html

-- al-Biruni (Central Asian who came to India with Ghazni) Bio
http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Al-Biruni.html

-- A Brief History of Zero and Indian Numerals. By Fjordman
http://www.brusselsjournal.com/node/4107

-- Medieval Muslim Scholars -- Their Contributions and Shortcomings. By Fjordman