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PSLQ: An algorithm to discover mathematical identities

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Speaker: Dr. David H. Bailey, Lawrence Berkeley Lab and U.C. Davis
Location: 3106 MSB
Start time: Tue, Apr 16 2013, 11:30AM

Discovered by mathematician-sculptor Helaman Ferguson in the early 1990s, the ``PSLQ'' algorithm, given an input vector of real or complex numbers $(x_1, x_2, \cdots, x_n)$, attempts to find nontrivial integers $(a_1, a_2, \cdots, a_n)$ such that the inner product $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = 0$, to within available precision, or else finds bounds within which no such set of integers exist. Implementations of the PSLQ algorithm must employ very high precision arithmetic (typically hundreds of digits), or else the true relation will be lost in numerical noise. The first ``big'' discovery found using PSLQ is the ``BBP'' formula for $\pi$, namely \begin{eqnarray*} \pi &=& \sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{4}{8k+1} - \frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6}\right) \end{eqnarray*} This formula has the remarkable property that it permits one to directly calculate base-16 (or binary) digits of $\pi$ beginning at an arbitrary starting position $n$, without needing to compute any of the previous $n-1$ digits. Such formulas are now known for many other basic constants of mathematics, and these formulas have deep connections to the fundamental question of whether (and why) the digits of these constants (including $\pi$) appear to be ``random.''

Contact Jesus De Loera if you wish to meet the speaker.