European Mathematical Society - David H. Bailey
https://euro-math-soc.eu/author/david-h-bailey
enPi: The Next Generation
https://euro-math-soc.eu/review/pi-next-generation
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In <em>Pi. A source book</em> the editors L. Berggren, J. Borwein and P. Borwein, assembled a number of reprints that sketch the history of pi, its mathematical importance and the broad interest that it has received through the centuries from the Rhind papyrus till modern times. The last edition (3rd edition, 2004, to which I will refer as SB3) added several papers that related to the computation of the digits of pi by digital computers. Rather than extending this with more recent developments (SB3 was already some 800 pages), it was decided to collect this computational aspect in a new volume. This "<em>The next generation</em>" volume got the rightful subtitle "<em>a source book on the recent history of pi and its computation</em>". Because it extends the papers on digital computation that were added in SB3, the trailing papers of SB3 are reprised here. The papers are ordered chronologically, so of the first 14 papers in this book, 12 were already at the end of SB3.</p>
<p>
It starts with the agm (algebraic-geometric mean) iteration attributed to Salamin and Brent who both published their papers in 1976. It generates two sequences of numbers by iteratively extending the sequences respectively with the algebraic and the geometric mean of the previous numbers. Given appropriate initial conditions, both sequences converge to a common limit related to pi. This method is widely used since these publications of 1976, but the agm idea was actually used already by Gauss and others although not in connection with computing pi. The Borwein brothers discuss a quartically convergent method based on it (1984) and Bailey and Kanada used it to compute millions of decimals of pi (1988). The number of digits computed today has exceeded these computer experiments by many orders of magnitudes and several papers in this book survey the history, and the diversity of formulas and methods and the successive records reached.</p>
<p>
There are, besides the classical methods to compute pi, also several computational methods to generate the expansion of pi. For example, a completely different spigot algorithm computes the decimals of pi one by one but using only integer arithmetic (originally from 1995 and extended in 2006). In a more classical vein is the BBP algorithm (named after the authors Bailey, Borwein and Plouffe) which allows to compute a set of binary (or hexadecimal) digits of pi without the need to compute all the previous ones (1997). This is of course a great help when computing trillions of digits. Of course there are a a number of papers devoted to Ramanujan's notebooks with formulas to compute pi.</p>
<p>
There are also some papers on the proof of irrationality of pi, and of related numbers such as its roots, ζ(2), ζ(3); (i.e. Apéry's constant), Catalan's constant etc. The investigation of the properties of the digits of pi, in particular the normality of pi (still unproved) is discussed and computationally tested. The tests can be nicely visualised using random walks and color coding. Normality means that every possible sequence of <em>m</em> successive digits is equally probable for any basis and for any <em>m</em>.</p>
<p>
The papers are reprinted in their original format, thus with different fonts, lay-out, etc. It happens that the end of a previous chapter or article is still on the first page of the reprint or the start of the next one is on the last page. Even some totally unrelated announcement that appeared at the end of the original journal paper, it is reprinted here unaltered. Just as one would in a pre-digital age collect photocopies of the papers. Nevertheless, the book has an overall name and subject index, which is not obvious in this case. Since the papers come from many different journals (and even some chapters of a book) not all of these papers may be readily available or even known to an interested researcher, or in this case, it may even be a lay person who is interested. Many of the papers have authors that are the main players in the field: David Bailey, Bruce Berndt, and Jonathan and Peter Borwein. As this book was being printed one of its editors, Jonathan (Jon) Borwein, passed away on 2 August 2016. So it was probably too late to add a dedication or a note in this book. This collection he helped to compile and containing several papers that he coauthored, can be considered one of his last gift to the scientific community. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of reprints of 25 papers discussing pi. Mostly about the computations of its digits and checking the normality. They are ordered chronologically from 1976 to 2015. This is an alternative for yet a fourth edition of <em>Pi. A Source Book</em> by Berggren and the Borwein brothers, the third edition of which appeared in 2004. The computational papers of that 3rd edition are reprised as the initial papers of this volume.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-h-bailey" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David H. Bailey</a></li><li class="vocabulary-links field-item odd"><a href="/author/jonathan-m-borwein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jonathan M. Borwein</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-verlag-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9783319323756 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">74.19 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">521</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</a></li><li class="vocabulary-links field-item even"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319323756" title="Link to web page">http://www.springer.com/gp/book/9783319323756</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11-04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-04</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11y16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11Y16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/68q25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68q25</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11k16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11K16</a></li></ul></span>Tue, 23 Aug 2016 07:39:47 +0000Adhemar Bultheel47119 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/pi-next-generation#comments