Peter Borwein
The Notices of the American Mathematical Society has just published a memorial tribute, written by the present author, that summarizes Peter’s life and career. Here are a few highlights:
Peter Borwein is perhaps best known for discovering (often but not always with his brother Jonathan) new formulas and algorithms for $\pi$ and other mathematical constants. One of these algorithms is the following: Set $a_0 = 6 – 4 \sqrt{2}$ and $y_0 = \sqrt{2} – 1$. Then iterate, for $k \ge 0$,
\begin{align}
y_{k+1} &= \frac{1 – (1 – y_k^4)^{1/4}}{1 + (1 – y_k^4)^{1/4}}, \nonumber \\
a_{k+1} &= a_k (1 + y_{k+1})^4 – 2^{2k+3} y_{k+1} (1 + y_{k+1} + y_{k+1}^2). \label{form:q4}
\end{align}
Then $1/a_k$ converges quartically to $\pi$: each iteration approximately quadruples the number of correct digits (provided that each iteration is performed with at least the numeric precision required for the final result). This algorithm, together with a quadratically convergent algorithm independently discovered by Brent and Salamin, has been employed in several large computations of $\pi$.
In 1995, Peter posed the question to some students and post-docs of whether there was any economical way to calculate digits in some base of a mathematical constant such as $\pi$, beginning at a given digit position, without needing to calculate the preceding digits. Peter and Simon Plouffe subsequently found the following surprisingly simple scheme for binary digits of $\log 2$, based on the formula $\log 2 = \sum_{k \ge 1} 1/(k 2^k)$, due to Euler. First note that binary digits of $\log 2$ starting at position $d + 1$ can be written ${\rm frac} \, (2^d \log 2)$, where ${\rm frac} \, (x) = x – \{x\}$ is the fractional part. Then
\begin{align}
{\rm frac} \, (2^d \log 2) &= {\rm frac} \, \left(\sum_{k=1}^\infty \frac{2^d}{k 2^k} \right)
= {\rm frac} \left( \sum_{k=1}^d \frac{2^{d-k}}{k} + \sum_{k=d+1}^\infty \frac{2^{d-k}}{k} \right) \nonumber \\
&= {\rm frac} \left(\sum_{k=1}^d \frac{2^{d-k} \bmod k}{k} \right) + {\rm frac} \, \left(\sum_{k=d+1}^\infty \frac{2^{d-k}}{k} \right), \label{form:bor4}
\end{align}
where $\bmod \, k$ has been added to the numerator of the first term, since we are only interested in the fractional part after division by $k$. The key point here is that the numerator expression, namely $2^{d-k} \bmod k$, can be computed very rapidly by the binary algorithm for exponentiation mod $k$, without any need for extra-high numeric precision, even when the position $d$ is very large, say one billion or one trillion. The second sum can be evaluated as written, again using standard double-precision or quad-precision floating-point arithmetic. The final result, expressed as a binary floating-point value, gives a string of binary digits of $\log 2$ beginning at position $d+1$.
In the wake of this observation, Peter and others searched the literature for a formula for $\pi$, analogous to Euler’s formula for $\log 2$, but none was known at the time. Finally, a computer search conducted by Simon Plouffe numerically discovered this formula, now known as the BBP formula for $\pi$:
\begin{align}
\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). \label{form:bbp}
\end{align}
Indeed, this formula permits one to efficiently calculate a string of base-16 digits (and hence base-2 digits) of $\pi$, beginning at an arbitrary starting point, by means of a relatively simple algorithm similar to that described above for $\log 2$. Nicholas Sze used a variation of this scheme to calculate binary digits of $\pi$ starting at position two quadrillion.
No account of Peter Borwein’s career would be complete without mentioning the remarkable grace with which he faced his condition of multiple sclerosis. Initially diagnosed prior to the year 2000, the disease eventually left him confined to a wheelchair, increasingly dependent on family and caregivers, and, sadly, increasingly unable to pursue research or to effectively collaborate with colleagues. The present author recalls visiting Peter in January 2019 at his home in Burnaby, British Columbia. In spite of his paralysis and infirmity, Peter’s pleasant demeanor and humor were on full display. Would that we could all bear our misfortunes with such equanimity!
Full details of the above and other highlights of Peter Borwein’s remarkable life are in the AMS Notices article.