Numerical Methods That (Usually) Work

A book that inspired me early in my career is Numerical Methods That Work by Forman S. Acton, published in 1970 by Harper and Row. Acton, a professor in the electrical engineering department at Princeton University, had a deep understanding of numerical computation and the book captures his many years of experience of practical problem solving using a combination of hand computations and early computers.

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Although written in the 1960s, Acton’s book is more about the 1950s world of computation; it makes only brief mention of the QR algorithm for eigenvalues and does not cover the singular value decomposition or variable step size ODE solvers. Moreover, the author has an aversion to library routines and to rigorous error bounds. Acton states that the students who have attended his numerical methods course have mostly “been Engineers and Scientists. (Mathematicians at Princeton are proudly Pure while most Computer Scientists find an obligatory decimal point to be slightly demeaning.)”. What, then, is special about this book from an applied mathematics point of view?

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(c) Princeton University Press Office of Communications

The book promotes timeless principles that are taught less and less nowadays. A general theme is to analyze a problem and exploit its structure, before applying the simplest suitable numerical method. One example that has stuck with me is the idea of trying to treat a given equation as a perturbation of an easier equation. For example, a quadratic equation \epsilon a x^2 + bx + c = 0 with small |\epsilon| can be thought of as a small perturbation of the linear equation bx + c = 0. Then simple fixed point iteration can be used to solve the quadratic with -c/b as a (good) starting value.

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The book is particularly strong on estimation or evaluation of integrals, dealing with singularities in functions, solving scalar nonlinear equations, exploiting asymptotic series, and avoiding instabilities. Several of these issues arise in the “railroad rail problem” presented at the start of the book, which every serious user of numerical methods should have a go at solving.

The pièce de résistance of the book is undoubtedly the 13-page “Interlude: What Not to Compute”. Described as a “cathartic essay” by James Daniel in SIAM Review in 1971, this essay is as relevant as ever, though Acton’s professed dislike of recursive calculations seems dated now that most programming languages fully support recursion.

Contemporary reviewers all note the practical slant of the book. I particularly like H. F. Trotter’s comment that “this reviewer, for one, would find it easier to supply theoretical discussion to supplement this text than to supply the lively practicality that is not always present in other books on this subject” (American Scientist, 59 (4), 1971). As this comment indicates, not only is the book full of excellent advice, but it is written in a distinctive and highly entertaining style. Here are a few examples:

  • “Newton’s predilection for wandering off to East Limbo on encountering a minimum” (On Newton’s method for solving nonlinear equations.)
  • “Only a socially irresponsible man would ignore such computational savings.” (On methods with operation counts proportional to n^2 versus n^3, respectively.)
  • “Many theorems are available for your pleasure.” (About positive definite matrices.)

The typesetting is excellent. One could hardly do better in \LaTeX. Moreover the diagrams are a paragon of good, minimal design and would not be easy to equal with today’s drawing packages.

In the original book the title on the cover is embossed in silver and the word “Usually” has been inserted, unembossed, just before “Work”. In the 1990 reprint by the Mathematical Association of America the “Usually” is in feint grey text. The reprint includes an extra “Preface-90″, an “Afterthoughts” (the quote in the first paragraph is taken from the latter), and some extra problems. The reprint is available on Google Books

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In 1996 Acton, by then an emeritus professor of computer science, published a second book Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations with Princeton University Press. It contains similar material on a smaller range of topics, and didn’t have the same impact on me as Numerical Methods that Work. Indeed, being published 26 years later it feels much more out of date. Unlike the first book, this one does mention Gaussian quadrature, but only to advise against its use. This book is now out of print at PUP but is available from Dover and at Google Books.

Acton died in 2014. Some brief biographical information can be found at a Wikipedia page, a Princeton University obituary, and a tribute from a former student.

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Michael J. D. Powell (1936–2015)

I would like to share a couple of photos of Mike Powell, FRS, who passed away last month. The photos are from early and late in his career.

The first is one of a set of contact prints from a role of Kodak black and white film that I came across in a collection of photos belonging to Gene Golub, which I was able to look through after Gene’s death in 2007. It is clear from the complete set of images that they were taken in or around the Courant Institute. The photos are undated, but Olof Widlund (who appears in some of them) tells me that the photos are most likely from 1965-1966.

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The second image is from June 2013 and was taken at the banquet at the Biennial Conference in Numerical Analysis at the University of Strathclyde. Mike is flanked on his right by Iain Duff and on his left by Juan Meza. Mike was a regular attendee at this conference and starting at next month’s conference there will be a regular Fletcher-Powell Invited Lecture, honouring Roger Fletcher and Mike Powell’s contributions to numerical analysis and, particularly, nonlinear optimization.

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A memorial web site site for Mike has been set up.

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Publication Peculiarities: Titles

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In this post in my series on publication peculiarities I look at titles (the first post was about papers as a whole). The paper titles below all have something interesting about them.

I have tried to give links to all the papers mentioned. Where one is not given the paper is (to my knowledge) not officially available on the web, although it may be nevertheless findable using Google Scholar.

Shortest Title

Here are some contenders for the shortest title (the first and third of these are taken from page 146 of my Handbook of Writing for the Mathematical Sciences, SIAM, 1998, second edition). Note that the following titles are all clickable, even though they are not underlined.

Charles McCarthy, c_p, Israel J. Math. 5(4), 249-271, 1967.

F. Brezzi, L. P. Franca, T. J. R. Hughes and A. Russo, b = \int g, Comput. Methods Appl. Mech. Engrg. 145(3-4), 329-339, 1997.

Norman Meyers and James Serrin, H=W, Proc. Natl. Acad. Sciences USA 51, 1055-1056, 1964.

Mitchell Feigenbaum and David Mermin, E=mc^2, American Journal of Physics 56, 18-21, 1988.

The last three titles have the virtue of forming a complete sentence with subject, verb and object.

Title Inspired by Film

Olivier Ledoit and Michael Wolf, Honey, I Shrunk the Sample Covariance Matrix, J. Portfolio Management 30(4), 110-119, 2004.

The next paper, referring to The Blair Witch Project (1999), uses a photo of Tony Blair to compare different colour maps:

Bernice E. Rogowitz and Alan D. Kalvin, The “Which Blair Project”: A Quick Visual Method for Evaluating Perceptual Color Maps, 183-556, in Proceedings of IEEE Visualization 2001, 2001.

Typo in Title

It is quite rare for a title to contain a typo. Here is an example

S. W. Ellacott and E. B. Saff, On Clenshaws’s Method and a Generalisation to Faber Series, Numer. Math., 52, 499-509, 1988.

In the body of the paper (in particular on the last line of the first page) the correct usage “Clenshaw’s” appears.

In the next example “Lambert W” is correctly spelled in the body of the paper, though not in the title:

M. M. Sherrif, N. S. Ravindran and P. Krishnapriya, Stability Analysis For Tumour Growth Model Through the Lambertz W Function, Journal of Advances in Mathematics 7(1), 1140-1146, 2014.

This paper has an incorrect spelling of Riccati:

George E. Trapp, Jr., The Ricatti Equation and the Geometric Mean, pages 437-445, in B. N. Datta, e.d., Linear Algebra and Its Role in Systems Theory, Contemporary Math., 47, 1985.

The next paper has a common grammatical error in the title:

A. A. Gurjar, S. Ladhake and A. Thakare, Analysis Of Acoustic of “OM” Chant To Study It’s Effect on Nervous System, International Journal of Computer Science and Network Security 9, 363-367, 2009.

The original 1942 edition of

Paul Halmos, Finite-Dimensional Vector Spaces, viii+200, Springer-Verlag, 1958.

was titled Finite Dimensional Vector Spaces, without the hyphen. Halmos explained that he became convinced of the need for the hyphen by the time of 1958 edition.

Finally, one should be aware that there can be errors in metadata if not in a paper itself. The title of

B. Wie, H. Weiss and A. Arapostathis, Quaternion Feedback Regulator for Spacecraft Eigenaxis Rotations, Journal of Guidance, Control, and Dynamics 12, 375-380, 1989.

has no typos, but the journal’s metadata at the above link, including the title displayed on the web page, has the typo “quarternion”.

Counting

Authors like enumerating things. Here we count to ten in paper titles.

H. F. Baker, The Reciprocation of One Quadric into Another, Proc. Cambridge Philos. Soc. 23, 22-27, 1925.

Donald Knuth, Two Notes on Notation, Amer. Math. Monthly, 99, 403-422, 1992.

Thomas Quinn, Scott Tremaine and Martin Duncan, A Three Million Year Integration of the Earth’s Orbit, Astron. J. 101, 2287-2305, 1991.

K. Appel and W. Haken, Every Planar Map is Four Colorable. Part I: Discharging, Illinois J. Math. 21, 429-490, 1977.

David Anderson, Tom Kilburn: A Tale of Five Computers, Comm. CACM, 57, 35-38, 2014.

Ergin Elmacioglu and Dongwon Lee, On Six Degrees of Separation in DBLP-DB and More, SIGMOD Rec. 34, 33-40, 2005.

Brian McCartin, Seven Deadly Sins of Numerical Computation, Amer. Math. Monthly, 105, 929-941, 1998.

E. C. Berkeley, Eight Hundred People Interested in Mechanical Brains, Amer. Statist., 4, 11-12, 1950.

Desmond Higham, Nine Ways to Implement the Binomial Method for Option Valuation in MATLAB, SIAM Review, 44, 661-677, 2002.

Gian-Carlo Rota, Ten Lessons I Wish I Had Been Taught, Notices Amer. Math. Soc., 44, 22-25, 1997.

As for continuing beyond 10, I will just pick out two specific cases.

Joseph Lee Rodgers and Alan Nicewander, Thirteen Ways to Look at the Correlation Coefficient, Amer. Statist., 42, 59-66, 1988.

Cleve B. Moler and Charles F. Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five years Later, SIAM Review, 45, 3-49, 2003.

Meta-Titles

This category refers to titles that are looking at a topic from a higher perspective.

Michael Berry, Why Are Special Functions Special?, Physics Today 54, 11-12, 2001.

Barbara Kitchenham, Pearl Brereton, David Budgen, Mark Turner, John Bailey and Stephen Linkman, Systematic Literature Reviews in Software Engineering—A Systematic Literature Review, Information and Software Technology 51, 7-15, 2009.

The final article is somewhat controversial: see The Tears of Donald Knuth.

Martin Campbell-Kelly, The History of the History of Software, IEEE Annals of the History of Computing 29, 40-51, 2007.

Shakespearean Titles

Now we are looking for titles that make reference to a Shakespearean play.

R. W. Bemer, Towards Standards for Handwritten Zero and Oh: Much Ado About Nothing (and a Letter), or a Partial Dossier on Distinguishing Between Handwritten Zero and Oh, Comm. ACM 10, 513-518, 1967.

William Kahan, Branch Cuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit, in A. Iserles and M. J. D. Powell, eds, The State of the Art in Numerical Analysis, Oxford University Press, 1987, pages 165-211.

Peter Hall, A Comedy of Errors: The Canonical Form for a Stable Characteristic Function, Bull. London Math. Soc. 13, 23-27, 1981

Joel David Hamkins, Destruction or Preservation as You Like It, Annals of Pure and Applied Logic 91, 191-229, 1998

Miscellany

Someone whose titles I have often admired is William (Velvel) Kahan (he has already appeared in the previous section). I particular like his alliterative

William Kahan, Conserving Confluence Curbs Ill-Condition, Technical Report number 6, Computer Science Department, University of California, Berkeley, 1972.

and the pun in

William Kahan and Beresford N. Parlett, Can You Count on Your Calculator?, Memorandum No. UCB/ERL M77/21, Electronics Research Laboratory, College of Engineering, University of California, 1977.

If you know of further interesting titles please add them to the comments below.

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Publication Peculiarities: Papers

I read a lot of papers. I also glance at many papers in journal contents pages and Google search results. From time to time I notice a paper that has an interesting title, author list, abstract, or some other notable feature. This post is the first in a series collecting such publication peculiarities. It concerns papers with striking features other than the title, author list, or abstract. The rules of the game are that I prefer examples from mathematics and related areas and that I must be able to provide a link to the article in question.

If you know of other good examples, please add them in the comments box at the end of this post.

The Letter W

The paper

Brian Hayes, Why W?, American Scientist 93, 104-108, 2005,

which is about the Lambert W function, has the remarkable feature that every sentence contains at least one instance of the letter “w” (as the author admits in the final section). There does not appear to be a word for the result of this constrained writing, but it is a kind of opposite of a lipogram: a text in which a certain letter is avoided entirely.

A Computer Program

Charles Lindsey was a Senior Lecturer in the Department of Computer Science at the University of Manchester and was one of the designers of the language Algol 68. I took a course on programming languages from him when I was a student. His paper

Charles H. Lindsey, ALGOL 68 with fewer tears, Comput. J. 15 (2), 176-188, 1972

is a syntactically valid Algol 68 program. Nowadays we would call this literate programming!

Latin

The paper

Clifford Truesdell, Solutio Generalis et Accurata Problematum Quamplurimorum de Motu Corporum Elasticorum incomprimibilium in Deformationibus valde Magnis, Arch. Rational Mech. Anal. 11, 106-113, 1962

has been described by Ball and James (in The Scientific Life and Influence of Clifford Ambrose Truesdell III) as “perhaps the only serious scientific paper published in Latin in the 20th century”.

Shortest Paper

A contender for shortest paper is

L. J. Lander and T. R. Parkin, Counterexample to Euler’s Conjecture on Sums of Like Powers, Bull. Amer. Math. Soc. 72, 1079, 1966,

which consists of just two sentences. However, brevity is taken to extremes in the next paper, for which writer’s block led to an empty body:

Dennis Upper, The Unsuccessful Self-Treatment of a Case of “Writer’s Block”, Journal of Applied Behavior Analysis 7, 497, 1974.

This experiment has been successfully replicated:

Geoffrey Molloy, The Unsuccessful Self-Treatment of a Case of “Writer’s Block”: A Replication, Perceptual and Motor Skills 57, 566, 1983,

Robert Didden, Jeff Sigafoos, Mark O’Reilly, Giulio Lancioni and Peter Sturmey, A Multisite Cross-Cultural Replication of Upper’s (1974) Unsuccessful Self-Treatment of Writer’s Block, J. Appl. Behav. Anal. 40, 773, 2007.

Order of Authors

Most fields have conventions about the order in which author names appear. The authors of the paper

M. P. Hassell and R. M. May, Aggregation of Predators and Insect Parasites and its Effect on Stability, Journal of Animal Ecology 43, 567-594, 1974

state that “The order of authorship was determined from a twenty-five-game croquet series held at Imperial College Field Station during summer 1973.”

First Word

The first word of the first article in the journal Nature was, appropriately, “Nature”:

T. H. Huxley, Nature: Aphorims by Goethe, Nature 1(1), 9-11, 1869.

Remnant

Occasionally, a paper contains something the authors meant to remove before publication. The originally published version of the paper

Zachary W. Culumber, Christian E. Bautista-Hernández, Scott Monks, Lenin Arias-Rodriguez and Michael Tobler, Variation in Melanism and Female Preference in Proximate but Ecologically Distinct Environments, Ethology 120, 1090-1100, 2014

contained the sentence

Although association preferences documented in our study theoretically could be a consequence of either mating or shoaling preferences in the different female groups investigated (should we cite the crappy Gabor paper here?),

Some time after the paper was published it was updated, with the parenthetical phrase replaced by “(Gabor 1999)”.

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Numerical Linear Algebra Group 2014

The Manchester Numerical Linear Algebra group was very active in 2014. This post summarizes what we got up to. Publications are not included here, but many of them can be found on MIMS EPrints under the category Numerical Analysis.

A new venture for several of us this year was to make our software available on GitHub: Deadman, Higham, Relton, Sego, Zhang.

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PhD Students

Three students successfully defended their theses:

Sam and Leo are now postdoctoral Research Associates in the group. Ramaseshan is a Senior Engineer at Arup, working in the Manchester office.

Weijian Zhang joined the group in September as a PhD student working with Nick Higham.

Sam Relton and Mary Aprahamian served as President and Treasurer, respectively, of the Manchester SIAM Student Chapter.

Postdoctoral Research Associates (PDRAs)

Jennifer Pestana joined the group in January, from Oxford University, to work with Françoise Tisseur.

Javier Perez joined the group in September, to work with Françoise Tisseur.

Tim Butters joined the group in January to work as a postdoctoral Knowledge Transfer Associate with Stefan Guettel and Nick Higham, in a project with Sabisu.

Lijing Lin (2007-2014) left the group in September to take up a Research Associate position in the Faculty of Life Sciences at The University of Manchester.

Amal Khabou (2013-2014) left in September to take up a Maître de conférences position at Université Paris Sud.

Meisam Sharify (2013-2014) left in May to take up a post as Assistant Professor in the Department of Computer Science, Shahid Beheshti University, G.C. Tehran, Iran.

Presentations

Members of the group gave presentations at the the following conferences and workshops.

IMA Conference on the Mathematical Challenges of Big Data, London, December 16-17, 2014 (Higham).

Structured Numerical Linear and Multilinear Algebra: Analysis, Algorithms and Applications, Kalamata, Greece, September 8-12 2014 (Noferini).

Spectral Theory Workshop to celebrate the 70th birthday of Brian Davies, King’s College London, 30 October 2014 (Tisseur).

4th IMA Conference on Numerical Linear Algebra and Optimisation, Birmingham, September 3-5, 2014 (Aprahamian, Berljafa, Khabou, Pestana, Relton, Strabić, Tisseur).

11th World Congress on Computational Mechanics, Barcelona, 20-25 July 2014 (Kannan).

International Workshop on Operator Theory and Applications, Amsterdam, July 14-18, 2014 (Tisseur).

SIAM Annual Meeting, Chicago, July 7-11, 2014 (Aprahamian, Deadman, Guettel, Lotz, Zhang).

First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI, Bilbao, June 30-July 4, 2014 (Noferini).

Householder Symposium XIX on Numerical Linear Algebra, Spa, Belgium, June 8-13, 2014 (Deadman, Guettel, Higham, Khabou, Lin, Noferini, Pestana, Taslaman, Tisseur).

10th International Workshop on Accurate Solution of Eigenvalue Problems (IWASEP10), Dubrovnik, Croatia, June 2-5 (Sego, Strabić).

Structured Matrix Days, XLIM, Université de Limoges, France, May 26-27, 2014 (Noferini).

Advances in Numerical Algorithms and High Performance Computing, University College London, April 14-15, 2014 (Deadman, Higham, Relton).

Napier 400th Anniversary Celebrations: Computation in Mathematics Workshop, ICMS, Edinburgh, April 2, 2014 (Higham: Video podcast).

Workshop on Nonlinear Eigenvalue Problems, The University of Manchester, April 23-25, 2014 (organized by Tisseur, Pestana, Kressner and Michiels and attended by the whole group).

Annual Meeting of the International Association of Applied Mathematics and Mechanics, Erlangen, March 10-14, 2014 (Guettel).

International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing March 7-9, 2014 Tsukuba, Japan (Guettel, Tisseur).

2014 SET for BRITAIN exhibition at the House of Commons (Mary Aprahamian presented a poster).

Conference and Workshop Organization

The group organized three events in Manchester.

Several minisymposia were organized:

Visitors

Vedran Sego visited the group throughout 2014.

Zhi-Gang Jia from Jiangsu Normal University, China, visited the group August 2013-July 2014.

Philip Gill (UC San Diego) and Margaret Wright (New York University) both visited for two weeks in March.

Jan Papež, a PhD student at the Academy of Sciences of the Czech Republic, Prague, visited for a week in November.

Marcel Schweitzer, a PhD student at the Universit of Wuppertal, visited the group for one week in September 2014.

Vladimir Druskin (Schlumberger-Doll Research Center, Cambridge, USA) visited the group for one week in June 2014.

Daniel Ruprecht (Università della Svizzera Italiana) and Robert Speck (Jülich Supercomputing Centre) visited the group for one week in February 2014.

Massimiliano Fasi, an M.Sc. student at the University of Bologna, Italy, visited for a week in November. He will join the group as a PhD student in September 2015, funded by a University of Manchester President’s Doctoral Scholarship Award.

Caterina Fenu, a PhD student at the University of Cagliari, Italy, visited for a week in November.

Knowledge Transfer

In The Knowledge Transfer Partnership with Sabisu, involving KTP Associate Tim Butters, Stefan Guettel, Nick Higham, and Jon Shapiro (School of Computer Science), an alarm management system has been developed and launched as a product.

Recognition and Service

Françoise Tisseur was awarded a Royal Society Wolfson Research Merit Award (2014-2019) to support her work on the numerical solution of nonlinear eigenvalue problems.

Former PDRA Yuji Nakatsukasa (2011-2013) was awarded the Householder Prize 2014 for his PhD thesis “Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition” (2011), written at the University of California, Davis.

Nick Higham served on Subpanel 10, Mathematical Sciences, in the Research Excellence Framework (REF) 2014.

Françoise Tisseur served as

  • Vice-President of the UK & Republic of Ireland SIAM Section,
  • Program Director of the SIAM Activity Group on Linear Algebra,
  • Member of the Householder Prize Committee 2014.
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More Tips on Book and Thesis Writing

Following my earlier post Top Five Tips on Book Writing, here are seven more tips. These apply equally well to writing a thesis.

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Book sculpture at Fudan University, Shanghai.

1. Signpost Citations

In academic writing we inevitably include a fair number of citations to entries in the bibliography. In a book, even more so than in a paper, we do not want the reader to have to turn to the bibliography every time a citation is reached in order to understand what is being cited. So a sentence such as

The matrix logarithm appears in a wide variety of applications
[2], [8], [14].

is better phrased as the more informative

The matrix logarithm appears in a wide variety of applications,
such as reduced-order models [2], image registration [8],
and computer animations [14].

Likewise, instead of

Versions of the algorithm have been developed by several authors
[1], [3], [7].

I would write

Versions of the algorithm have been developed by Chester [1], 
Hughes [3, Sec. 2], and Smith and Jones [7].

Even that example lacks information about the date of publication. In my books I have used my own version of the \LaTeX \cite macro that allows me to include the year:

Versions of the algorithm have been developed by Baker and
Chester [1, 2006], Hughes [3, 2001, Sec. 2], 
and Smith and Jones [7, 2004].

The macro is

\def\ycite[#1#2#3#4#5]#6{\cite[$\mit{#1#2#3#4}$#5]{#6}}

(which puts the year in the distinctive math italic font) and the first two citations in the previous sentence would be typed as \ycite[2006]{bach06} and \ycite[2001, Sec.~2]{hugh01}.

2. Produce a Good Index

A good index is essential, since it is the main way that readers can find content. The vast majority of books that I read have an inadequate index, as I have noted in my post A Call for Better Indexes at SIAM Blogs. Usually the index is too small. Occasionally the index is of about right length but is flawed. The main problems are

  • Items that should be indexed are absent from the index.
  • An index entry does not point to all (significant) occurrences of the term.
  • Related entries are not grouped properly.

Advice on producing an index can be found in Section 13.4 of my Handbook of Writing for the Mathematical Sciences and various other sources (try a Google search), and I intend to wrote a post on indexing soon.

\LaTeX, through its \index command, used in conjunction with the MakeIndex program, provides an excellent way to produce an index.

3. Use the Backref \LaTeX package

Backref.sty is a \LaTeX package that adds to each bibliography entry the text “cited on pages” and then lists the pages on which that item was cited. It costs nothing to use it, but it adds great value to the bibliography, which then functions as a separate index into the book. I started using backref with my book MATLAB Guide (2005). To a large extent it removes the need for an author index, and if I do a third edition of Accuracy and Stability of Numerical Algorithms I will probably use backref and drop the author index.

The backref package is not widely used, though a number of SIAM books have made use of it.

4. Use Hyperlinks

For a book provided in PDF form, hyperlinks from an equation reference to the equation, a citation to the bibliography entry, a URL to the web page, and so on, are a great aid to the reader. In \LaTeX obtaining the hyperlinks is usually just a matter of adding \usepackage{hyperref} in the pre-amble.

5. Make Figures Readable and Consistent

It’s very easy nowadays to produce figures containing plots of functions or computational results. But it’s much harder to produce a set of figures that

  • are clearly legible,
  • have labels, legends, and annotations that are of similar size to the main text,
  • are consistent in format (axes, line thicknesses etc.)

All too often I see figures in which the text is so small that I cannot read it at a normal reading distance. My experience (which is mainly with MATLAB, and with the \LaTeX packages TikZ and PGFplots) is that it is a time-consuming process to produce high quality plots. But it is worth the effort.

6. Use Short Captions in the List of Figures/Tables

The general form of the \LaTeX caption command is \caption[short caption]{long caption}. The short caption is what is printed in the List of Figures or List of Tables at the front of the book, if you are printing those lists. The short caption will be read in isolation from the figure or table so it should omit all unnecessary detail, such as explaining line or marker types. All too often, the short and long captions are the same, resulting in unnecessarily long and detailed lists of figures or tables.

Here is an example (simplified, with other macros removed) of the caption from a figure in my book Functions of Matrices:

\caption[Illustration of condition (b) of Theorem~11.4.]%
        {Illustration of condition (b) of Theorem~11.4,
         which requires every eigenvalue of $B$ to lie in the
         open half-plane (shaded) of the corresponding eigenvalue
         of $A^{1/2}$.}

7. Make the Header Contain the Section and Chapter Number and Title

I like to know where I am when I am reading a book, so I expect the page headers to tell me the section number and chapter number, and preferably their titles as well. I cannot understand why some books omit this information. Without it, phrases such as “as discussed in the previous chapter” become harder to follow up, and searching for a particular section is more difficult.

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Emacs and Org Mode: What People are Saying

For a couple of years I’ve been collecting tweets about Emacs and Org mode. With the Twitter app’s new ability to provide code to embed tweets I decided to create a post listing the collection. If you are not an existing user of Emacs or Org mode these tweets should give you a feel for whether you might want to explore further. If you are already a convert then many of the sentiments expressed here will be familiar. Note that the links and hashtags below are clickable. (Like all of this Blog, this post was written in Org mode.)

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Hans Schneider (1927–2014)

I first met Hans in 1984 at the Gatlinburg meeting IX in Waterloo, Canada, at which time I was a PhD student. When I discussed my work on matrix square roots with him he recalled a 1966 paper by Culver “On the Existence and Uniqueness of the Real Logarithm of a Matrix”, of which I was unaware. By the time I returned to Manchester, after visiting Stanford for a few weeks, a copy of the paper was waiting for me, with an explanation of how the results of that paper could be adapted to analyze real square roots of a real matrix.

As chair of the 2002 Householder symposium XV in Peebles, Scotland, I was delighted to invite Hans to deliver the after-dinner speech. (The Gatlinburg meeting was renamed the Householder symposium in 1990, in honour of Alston Householder, who organized the early meetings.) Having Hans speak was particularly appropriate as he had studied at the nearby University of Edinburgh. I believe this was the last Householder Symposium that Hans attended.

I kept a copy of my introduction of Hans at the banquet. It seems appropriate to reproduce it here.

Ladies and gentlemen, our after-dinner speaker this evening is Hans Schneider, who is James Joseph Sylvester Emeritus Professor of Mathematics at the University of Wisconsin.

There’s an old definition that an intellectual is somebody who can hear the William Tell overture and not think of the Lone Ranger. I don’t think there are many people who can hear the term “linear algebra and its applications” and not think of Hans Schneider. After all, Hans has been Editor-in-Chief of the journal of that name since 1972, and developed it into a major mathematics journal. Hans was also instrumental in the foundation of the International Linear Algebra Society, of which he served as President from 1987 to 1996.

Some of you may be surprised to know that Hans has a strong connection with Scotland. He studied here and received his Ph.D. at Edinburgh University in 1952 under the famous Alexander Craig Aitken. I understand that Aitken gave him two words of advice: “Read Frobenius!”.

Well, it’s a real pleasure to introduce Hans and to ask him to speak on “The Debt Linear Algebra Owes Helmut Wielandt”.

The reference to Frobenius is apposite, given my original conversation with Hans since, as I have only recently discovered, Frobenius gave one of the earliest proofs of the existence of matrix square roots in 1896. That result, and much more about Frobenius’s wide range of contributions to mathematics is discussed in a 2013 book by Thomas Hawkins, The Mathematics of Frobenius in Context. A Journey Through 18th to 20th Century Mathematics (of which my copy has the rare error of having the odd pages on the left, rather than the right, of each two-page spread).

The photo below was taken during Hans’s after-dinner speech (more photos from the meeting are available in this gallery).

020619-2159-45.jpg

Links:

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Top Five Tips on Book Writing

Snoopy writing

I’ve written four books, and am currently writing and editing a fifth (The Princeton Companion to Applied Mathematics). I am also an editor of two SIAM book series and chair the SIAM Book Committee. Based on this experience here are my top five tips about writing an (academic) book. These cover high level issues. In a subsequent post I will give some more specific tips relating to writing and typesetting a book or thesis.

1. Identify Your Audience

Book publishers ask prospective authors to complete a proposal form, one part of which asks who is the audience for the book. This is a crucial question that should be answered before a book is written, as the answer will influence the book in many ways.

As an example, you might be contemplating writing a book about the numerical solution of a certain class of equations and intend to include computer code. Your audience might be

  • readers in mathematics or a related subject who wish to learn about numerical methods for solving the equations and are most concerned with the theory or algorithms,
  • readers whose primary interest is in solving the equations and who wish to have lots of sample code that they can run,
  • readers in the previous class who also need to learn the language in which the examples are written.

The choice of content, and how the book is presented, will depend very much on which audience you are writing for.

2. Revise, Revise, Revise

Just like a paper, a book draft needs to go through multiple revisions, and you must not be afraid to make major changes at any stage. You may receive constructive criticisms from reviewers of your book proposal, but reviewers may not have time to read the complete manuscript carefully and you should not assume that they have found all errors, typos, and areas for improvement.

3. Take Time to Choose Your Publisher

Given the huge effort that goes into writing a book you should take the time to find the right publisher. Discuss your book with several publishers and compare what they can offer in the way of

  • format (hardback, paperback, electronic) and, if more than one format, the timescale in which each is made available,
  • if the publisher has branches in more than one country, how price and publication schedule will differ between the countries,
  • whether you are allowed to make a PDF version of the book freely available on your website, if this interests you,
  • willingness to allow you to choose the book design (page size, font, cover, etc.),
  • use of colour (which increases the cost),
  • royalties (including a possible advance),
  • pricing,
  • the publisher’s policy on translations,
  • copy editing (see the next section),
  • time from delivering a completed manuscript to publication,
  • marketing (will the book be advertised at all, and if so how?), and
  • how long your book is guaranteed to stay in print.

It is perfectly acceptable to submit a proposal to several publishers and see what they are willing to offer. However, it is only fair and proper to make clear to a publisher that you are talking to other publishers and, once you have set the wheels of a publisher’s review process in motion, to wait for an offer before making a decision to go with another publisher.

I am always surprised when I hear of authors who approach only one publisher, or who go with the first publisher to express an interest in the book. As in many contexts, it is best to make an informed choice from among the available options.

4. Ensure Your Book is Copy Edited

If you are an inexperienced writer, or your first language is not English, the benefits of copy editing are obvious. But even an experienced author finds it virtually impossible to think about all the little details that a copy editor will check for, such as correctness and consistency of spelling, notation, punctuation (notably the serial comma), citations, and references. For example, I sometimes mix US and UK spellings and don’t want to have to worry about finding and correcting my occasional lapses. A good copy editor will also suggest minor improvements of the text that might escape even the best writers.

Unfortunately, not all publishers copy edit all books nowadays. Notable exceptions that always do copy edit (and, as I know from experience, work to the highest standards in every respect) are Princeton University Press and SIAM.

If your publisher has a Style Manual it obviously makes sense to follow its guidelines in order to minimize changes at the copy editing stage. Here is a link to the SIAM Style Manual.

5. Think Twice Before Co-Authoring a Book

It might seem an attractive proposition to share authorship of a book: surely having n co-authors reduces the work by a factor 1/n? Unfortunately it often does not work out like that, despite best intentions. In fact, n co-authors can easily take n times as long to write a book as any one of them would. One of the biggest difficulties is timescale: one author may be willing and able to finish a book in a year but another may need twice that period to make their contribution. Indeed it is rare for the co-authors to be matched in the amount of effort they can put into the book; this is clearly problematic if initial expectations are not realized. Other potential problems are potentially differing opinions on content, notation, level, length, and almost anything else associated with a book.

Successful authorship teams often have a track record of co-authoring papers together. Although it is no guarantee that a much larger book project will run smoothly, experience with writing papers together will at least have given a good indication of where disagreements are likely to lie.

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The Spotlight Factor

In my Handbook of Writing for the Mathematical Sciences I described the spotlight factor, originally introduced by Tompa in 1989. The spotlight factor is defined for the first author of a paper in which there are n authors listed alphabetically, and it is assumed that the paper is from a community where it is the custom to order authors alphabetically.

The spotlight factor is the probability that if n-1 coauthors are chosen independently at random they will all have surnames later in the alphabet than the first author. This definition is not precise, since it is not clear what is the sample space of all possible names, so it is better to regard the spotlight factor as being defined by the formula given by Tompa, which is implemented in the MATLAB function below.

The smallest spotlight factor I have found is the value 0.0244 for Zielinski, for the paper

Pawel Zielinski and Krystyna Zietak, The Polar Decomposition—Properties, Applications and Algorithms, Applied Mathematics, Ann. Pol. Math. Soc. 38, 23-49, 1995

This beats the best factor of 0.0251 reported by Tompa in a 1990 follow-up paper.

Can you do better?

Here is a MATLAB M-file to compute the spotlight factor, preceded by an example of its usage:

>> spotlight('zielinski',1)
ans =
   2.4414e-02
function s = spotlight(x, k)
%SPOTLIGHT   Tompa's spotlight factor of authorship.
%   SPOTLIGHT(X, K) is the spotlight factor for the author whose 
%   last name is specified in the string X, with K coauthors.
%   Mixed upper and lower case can be used.
%   Smaller spotlight factors correspond to rarer events.

%   Reference:
%   Martin Tompa, Figures of Merit, SIGACT News 20 (1), 62-71, 1989

if ~ischar(x), error('First argument must be a string.'), end
if nargin < 2, error('Must give two arguments.'), end

x = double(upper(x)) - double('A') + 1;
x( find(x < 0 | x > 26) ) = 0;  % Handle punctuation and spaces.

s = 0;

% Ideally use Horner's rule, but the following is clearer.

for i=1:length(x)
    t = x(i);
    s = s + t/27^i;
end

s = (1 - s)^k;
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