Pi wars: dueling with supercomputers.

Pi wars: Dueling supercomputers

The relentless pursuit of pi ( ) hasnow pushed computation of that elusive number's decimal expansion beyond 134 million digits. This recent effort by Yasumasa Kanada of the University of Tokyo and his colleagues, on an NEC SX-2 supercomputer, eclipses the record set last year by David H. Bailey using a Cray-2 supercomputer at the NASA Ames Research Center at Moffett Field, Calif. (SN: 2/8/86, p.91).

"The story of computing digits of pi isno longer a story of great practicality,' says mathematician Peter B. Borwein of Dalhousie University in Halifax, Nova Scotia. "It hasn't been a story of great practicality since maybe the 16th century . . ., but it is a problem that has captured many, many people's imaginations.' Borwein this week discussed the latest achievements at the annual meeting in Chicago of the American Association for the Advancement of Science.

Archimedes started the chase morethan 2,000 years ago when he developed a method for approximating pi, the ratio of a circle's circumference to its diameter, by nesting a circle between a pair of polygons whose perimeters were easy to calculate. In the 17th century, Isaac Newton, using his own method, calculated at least 15 digits of pi. But, in a letter, he sheepishly admitted: "I am ashamed to tell you to how many figures I have carried these computations, having no other business at the time.'

In 1949, a primitive computer pushedthe computation to 2,037 digits. In recent years, the computation of pi has become an appealing, though not particularly useful or revealing, way to demonstrate publicly the capabilities of rival supercomputers.

The latest computation, which wasdone twice using two computer algorithms to check the result, took about two days each time on the SX-2. Borwein estimates that reciting the number's 134,217,700 digits, one digit every second, would take about four years.

"A reasonable question is, why doesone compute pi and not compute something else to 134 million digits?' says Borwein. "Part of the reason is that pi is the most naturally occurring of the nonalgebraic numbers [the next level of complexity of numbers]. And it's a number we know a little bit about but not a great deal about.'

Mathematicians, for instance, provedlong ago that pi is an irrational number. This means that it takes a never-ending string of digits to express pi as a decimal number. However, no one knows whether all of the digits from 0 to 9 appear infinitely often in this expansion or whether one-tenth of the digits are ones, and so on. Tests show that the first 30 million or so digits do behave regularly as expected.

Peter Borwein and his brother Jonathandeveloped the equations and the improved computer algorithms used for the last few record-setting computations of pi. A comparison with earlier methods shows how much these techniques have advanced. To get half a billion digits, Archimedes's method would have to be applied more than a billion times. "The current method, says Peter Borwein, "reduces that to 12 iterations.' In the Borwein method, he says, "each time you take the next step, you get four times as many correct digits as you had before.' Combining this with a fast way to multiply leads to a remarkably efficient procedure for computing pi.

The Borwein algorithm is close to thetheoretically best possible algorithm for computing pi. "There's a very small gap between what is known and what is possible,' says Borwein. On that basis, he conjectures that no one will ever know the 10(1,000)th digit of pi. Assuming that all of the preceding digits must be computed to arrive at this particular digit, even the age of the universe would allow too little time for the computation.