The Ten Most-Viewed Posts in 2021

Image credit: Stuart Miles.

According to the WordPress statistics, this website received over 138,000 visitors and 205,000 views in 2021.

Here are the ten blog posts (published at any time) that received the most views during the year.

  1. Better LaTeX Tables with Booktabs (2019)
  2. What Is a Symmetric Positive Definite Matrix? (2020)
  3. What Is a Condition Number? (2020)
  4. Can We Solve Linear Algebra Problems at Extreme Scale and Low Precisions? (2021)
  5. What Is a Householder Matrix? (2020)
  6. What Is the Cayley–Hamilton Theorem? (2020)
  7. What Is an LU Factorization? (2021)
  8. Five Examples of Proofreading (2017)
  9. What is Numerical Stability? (2020)
  10. What Is the Log-Sum-Exp Function? (2021)

Seven of the posts are from the What Is series, which passed fifty posts in April. The series is ongoing and I have several posts in preparation for early 2022.

Top Ten Posts of 2020


According to the WordPress statistics, this blog received over 64,000 visitors and 106,000 views in 2020.

Here are the ten most-viewed posts published during the year. All are in the What Is series that I started in March 2020, and which will continue in 2021.

  1. What is Numerical Stability?
  2. What Is a Condition Number?
  3. What Is a Symmetric Positive Definite Matrix?
  4. What Is the Sylvester Equation?
  5. What Is Backward Error?
  6. What Is a Matrix Square Root?
  7. What Is a Fréchet Derivative?
  8. What Is the Complex Step Approximation?
  9. What Is a Cholesky Factorization?
  10. What Is the Hilbert Matrix?

Top Five Posts of 2019

According to the WordPress statistics, this blog received over 39,000 visitors and 65,000 views in 2019. These are the five most-viewed posts published during the year.

Image courtesy of Stuart Miles at
  1. Who Invented the Matrix Condition Number?
  2. Numerical Algorithms for High-Performance Computational Science: Highlights of the Meeting
  3. Better LaTeX Tables with Booktabs
  4. Advances in Numerical Linear Algebra Conference and James Hardy Wilkinson Centenary
  5. The Argonne Tapes

Reflections on a SIAM Presidency

The Drexel Dragon, on the Drexel University campus a couple of blocks from SIAM.
Face Fragment (1975) sculpture, a block from SIAM.

My two-year term as SIAM President ended on December 31, 2018. It’s been an exciting and enjoyable two years, not least because of the excellent SIAM staff, leadership and other volunteers I’ve worked with.

My blog post Taking Up the SIAM Presidency and my SIAM News article Evolving and Innovating set out some ideas that I wanted to pursue during my two years as president. I will not attempt to review these here, but just list five highlights from the last two years.

  • We held the SIAM ADVANCE in April 2018: a two-day strategic planning workshop attended by 25 officers, staff, and other members of the SIAM community. The many ideas that emerged from the event are summarized in an 80-page report provided to the SIAM Council and Board of Trustees. Many of these have already been acted upon, others are in progress, and yet more will be considered in the future. My SIAM News article Advancing SIAM gives more details of the workshop.
  • A new journal SIAM Journal on Mathematics of Data Science was created. The first issue will be published in the first few months of 2019.
  • A new SIAM book series Data Science was created.
  • A new SIAG, the SIAM Activity Group on Applied and Computational Discrete Algorithms was approved and will begin operation in 2019.
  • The new SIAM website was launched (in June 2018).
The location of the SIAM office: 3600 Market Street, Philadelphia.

Here is a summary of my presidency in numbers:

  • 12 trips to the USA (with 0 upgrades from economy class to business class).
  • 8 visits to SIAM headquarters and 1 SIAM staff meeting attended.
  • 20 “From the SIAM President” columns written for SIAM News: they are listed here.
  • 2 SIAM Council Meetings chaired and 4 SIAM Board meetings attended.
  • 1 ICIAM board meeting attended and 1 ICIAM board meeting and workshop hosted by SIAM in Philadelphia.
  • 2 meetings of the Joint Policy Board for Mathematics in Washington chaired and 2 attended.
  • Over 230 appointments made to committees and of candidates for elections (with the advice of various SIAM committees).

Top Ten Posts of 2018


According to the WordPress statistics, this blog received over 42,000 visitors and 73,000 views in 2018. These are the ten most-viewed posts published during the year.

  1. How to Program log z
  2. Tricks and Tips in Numerical Computing
  3. Conversations with Gil Strang
  4. What’s New in MATLAB R2018a?
  5. Numerical Linear Algebra Group 2017
  6. Bohemian Matrices in Manchester
  7. Half Precision Arithmetic: fp16 Versus bfloat16
  8. Palomino Blackwing Pencil Tribute to Ada Lovelace
  9. Joan E. Walsh (1932–2017)
  10. Lectures on Multiprecision Algorithms in Kácov

In addition, the article Differentiation With(out) a Difference in my SIAM News “From the SIAM President” column was the most viewed SIAM News article of 2018.

Taking Up the SIAM Presidency

I am honored to be taking over the reins from Pam Cook as president of the Society for Industrial and Applied Mathematics (SIAM) for the next two years, starting January 1, 2017. Pam remains as past-president during 2017. I look forward to helping to address the challenges facing SIAM and to working with the excellent SIAM officers and staff.

SIAM headquarters, Philadelphia.

Eighteen months ago I wrote a “candidate statement” for the fall 2015 SIAM elections. The comments I made then remain valid and so I thought it would be worth reproducing the statement here.

The January/February 2017 issue of SIAM News will contain my first From the SIAM President column, in which I give further thoughts on SIAM’s future.

I am happy to receive comments from SIAM members or potential members, either in the box below or by email.

Candidate Statement: SIAM is the leading international organization for applied mathematics and has been an important part of my professional life since I joined as a PhD student, 31 years ago. SIAM is the first place that many people turn to for publications, conferences, and news about applied mathematics and it represents the profession nationally and internationally.

I have been fortunate to be involved in the leadership for many years, having spent six years on the Council, eight years on the Board, and having recently served two terms as Vice President At Large (2010-2013).

SIAM faces a number of challenges that, if elected as President, I relish helping to address, working with SIAM members, SIAM officers, and the excellent SIAM staff.

SIAM’s publications remain strong, but are vulnerable to changes in the way scholarly journals operate (open access, article processing charges, etc.). SIAM needs to monitor the situation and respond appropriately, while striving to provide an even greater service to authors, referees and editors, for example by better use of web tools.

SIAM’s membership is also healthy, but SIAM must continue to enhance membership benefits and work hard to attract and retain student members, who are the future of the society, and to provide value for its members in industry.

Book sales are declining globally and in academic publishing it is becoming harder to find authors with the time to write a book. Nevertheless, the SIAM book program is in a strong position and the 2015 review of the program that I chaired has produced a list of recommendations that should help it to thrive.

SIAM conferences are a terrific place to learn about the latest developments in the subject, meet SIAM staff, browse SIAM books, and attend a business meeting. Attendances continue to grow (the SIAM CSE meeting in Salt Lake City last March was the largest ever SIAM meeting, with over 1700 attendees), but in any given year, the majority of SIAM’s 14,000 members do not attend a SIAM conference. Audio and slide captures of selected lectures are made available on SIAM Presents, but we need to do more to help members engage in virtual participation.

The SIAM web site has provided sterling service for a number of years, but is in need of a major redesign, which is underway. This is an excellent opportunity to integrate better the many services (conferences, journals, books, membership, activity groups, chapters, sections, etc.) in a responsive design. Beyond the core website, SIAM has a strong social media presence, posts a wide variety of videos on its YouTube channel, hosts SIAM Blogs (which I was involved in setting up in 2013), has recently made SIAM News available online, and has SIAM Connect and SIAM Unwrapped as further outlets. Optimizing the use of all these communication tools will be an ongoing effort.

These are just some of the challenges facing SIAM in the future as it continues to play a global leadership role for applied mathematics.

July 2015

Mathematical Word Processing: Historical Snippets

Matthew G. Kirschenbaum’s recent book Track Changes: A Literary History of Word Processing contains a lot of interesting detail about the early days of word processing, covering the period 1964 to 1984. Most of the book concerns non-scientific writing, though \TeX gets a brief mention.

Inspired by the book, I thought it might be useful to collect some information on early mathematical wordprocessing. Little information of this kind seems to be available online.

It is first worth noting that before the advent of word processors papers were typed on a typewriter and mathematical symbols were typically filled in by hand, as in this example from A Study of the Matrix Exponential (1975) by Charlie Van Loan: vanl75-eq-p15.jpg

Some institutions had the luxury of IBM Selectric typewriters, which had a “golf ball” that could be changed in order to type special characters. (See this page for informative videos about the Selectric.) Here is an example of output from the Selectric, taken from my MSc thesis: high83m-eq224.jpg This illustrates some characteristic weaknesses of typewriter output: superscripts, subscripts, and operators are of the same size as the main symbols and spacing between characters is fixed (as in the vertical bars making up the norm signs here).

In 1980s at the Department of Mathematics at the University of Manchester the word processor Vuwriter was used. It was produced by a spin-off company Vuman Computer Systems, which took its name from “Victoria University of Manchester”, the university’s full name. Vuwriter ran on the Apricot PC, produced by the British company Apricot computers. At least one of my first papers was typed on Vuwriter by the office staff and I still have the original 1984 technical report Computing the Polar Decomposition—with Applications, which I have scanned and is available here. A typical equation from the report is this one:


The article

Peter Dolton, Comparing Scientific Word Processor Packages: T^3 and Vuwriter, The Economic Journal 100, 311-315, 1990

reviews a version of Vuwriter that ran under MS DOS on IBM PC compatibles. Another review is

D. L. Mealand, Word Processing in Greek using Vuwriter Arts: A Test case for Foreign Language Word Processing, Literary and Linguistic Computing 2, 30-33, 1987

which describes a version of the program for use with foreign languages.

The Department of Mathematics at UMIST (which merged with the University of Manchester in 2004) used an MSDOS word processor called ChiWriter.

In the same period I also prepared manuscripts on my own microcomputers: first on a Commodore 64 and then on a Commodore 128 (essentially a Commodore 64 with a screen 80 characters wide rather than 40 characters wide), using a wordprocessor called Vizawrite. For output I used an Epson FX-80 dot matrix printer, and later an Epson LQ 850 (which produced high resolution output thanks to its 24 pin print head, as opposed to the 9 pin print head of the FX-80). Vizawrite was able to take advantage of the Epson printers’ ability to produce subscripts and superscripts, Greek characters, and mathematical symbols. An earlier post links to a scan of my 1985 article Matrix Computations in Basic on a Microcomputer produced in Vizawrite.

In the 1980s some colleagues wrote papers on a Mac. An example (preserved at the Cornell eCommons digital repository) is A Storage Efficient WY Representation for Products of Householder Transformations (1987) by Charlie Van Loan. I think that report was prepared in Mac Write. Here is a sample equation:


Also in the 1980s I built a database of papers that I had read, and wrote a program that could extract the items that I wanted to cite, format them, and print a sorted list. This was a big time-saver for producing reference lists, especially for my PhD thesis. The database was originally held in Superbase for the Commodore C128, with the program written in Superbase’s own language, and was later transferred to PC-File running on an IBM PC-clone with the program converted to GW-Basic. I was essentially building my own much simpler version of BibTeX, which did not exist when I started the database.

I am aware of two good sources of information about technical word processors for the IBM PC. The first is the article

P. K. Wong, Choices for Mathematical Wordprocessing Software, SIAM News 17(6), pp. 8-9, 1984

This article notes that

“There are over 120 wordprocessing programs for the IBM PC alone and the machine is not yet three years old! Of this large number, however, less than half a dozen can claim scientific wordprocessing capabilities, and these have only been available within the past six to nine months.”

The other source is two articles published in the Notices of the American Mathematical Society in the 1980s.

PC Technical Group of IBM PC Users Group of the Boston Computer Society, Technical Wordprocessors for the IBM PC and Compatibles, Notices Amer. Math. Soc. 33, 8-37, 1986

PC Technical Group of IBM PC Users Group of the Boston Computer Society, Technical Wordprocessors for the IBM PC and Compatibles, Part IIB: Reviews, Notices Amer. Math. Soc. 34, 462-491, 1987

These two articles do not appear to be available online. The first of them includes a set of benchmarks, consisting of extracts from technical journals and books, which were used to test the abilities of the packages. The authors make the interesting comment that

“Microsoft chose not to answer our review request for Word, and based on discussion with Word owners, Word is not set up for equations.”

Finally, Roger Horn told me that his book Topics in Matrix Analysis (CUP, 1991), co-authored with Charlie Johnson, was produced from the camera-ready output of the T^3 wordprocessing system (reviewed in this 1988 article and in the SIAM News article above). T^3 was chosen because \TeX was not available on PCs when work on the the book began. It must have been a huge effort to produce this 607-page book in this way!

If you have any information to add, please put it in the “Leave a Reply” box below.

Acknowledgement: thanks to Christopher Baker and David Silvester for comments on a draft of this post.

Mathematics and Digital Photography

A few months ago I wrote a post Mathematics in Color, in which I discussed some mathematical aspects of color and showed how a simple change of basis from RGB color space to LAB color space can enable dramatic color changes to be done very easily.

Over the last decade most recent developments in digital imaging software such as Adobe Lightroom and Adobe Photoshop have been based on advanced mathematics, yet many of the most powerful and useful transformations that one can make to an image are based on elementary mathematics and have been possible since the early versions of the products. For example, with every click of the healing brush—which might, for example, be used to remove a stray piece of litter in a landscape or a skin imperfection in a portrait—Photoshop solves a partial differential equation. Yet global operations such as changes to colour and contrast can be done with commands (and in particular, masks) that amount to simple addition and multiplication operations.

I’ve just launched a new blog on photography and digital imaging in which one theme will be exploiting elementary mathematics in digital imaging. The first post shows how the much-used Clarity tool in Lightroom and Photoshop can be applied in a more effective way by taking what amounts to a componentwise linear combination of the images before and after Clarity has been applied.

Head over to the post Refined Use of the Clarity Tool in Photoshop to find out more.

Original (cropped) image.

With Clarity applied in blended fashion. Compare the water, clouds, and trees with the original.

ORCID: Open Researcher and Contributor ID


An Open Researcher and Contributor ID, or ORCID, is a unique identifier for a researcher that allows research outputs to be associated with that researcher. If you have a common name, or have moved institutions during your career, then it can be very difficult for people to determine which papers returned in a Google Scholar search (say) are by you rather than by someone else with the same name. If at some point in your career you change your name, or how you list it on papers, the difficulty of attribution may be even greater. Having your ORCID associated with your publications solves this problem.

Such an identifier scheme has existed for some time in the form of ResearcherID from Thomson Reuters, but this is commercial and linked to the Web of Science. The ORCID organization is open and not-for-profit, and its software is open source. The ORCID web site says that as well as the registry of identifiers, ORCID provides “APIs that support system-to-system communication and authentication”. This is what makes ORCID particularly interesting, as it makes it possible to have one’s list of publications generated in an automatic way, either by entering each on the ORCID site and then the list propagating elsewhere, or by ORCID automatically pulling in publication metadata and associating it with you.

The starting point for exploiting ORCIDs is for publishers to collect them at the time of submission. The Royal Society has been collecting ORCIDs with submissions since 2014, and since the turn of the year it has required authors submitting to its journals to provide an ORCID. As the Royal Society points out, “Once you have created an ORCID identifier and connected it with your publications, grants, and affiliations, your details will automatically be entered when using any compatible system”. Other publishers are following suit, as this open letter indicates.

I recently received an email saying “You have 1 new notification in your ORCID inbox”. When I went to the inbox I found a message saying that “Crossref [another not-for-profit organization] would like your permission to interact with your ORCID Record as a trusted party”. I gave permission and now when publishers send information about my new publications to Crossref they will be added to my ORCID record. For more on this important Crossref-ORCID link, see this article.

For academics used to having to repeatedly enter the same information into different systems this automatic updating of an ORCID record is great news.

The University of Manchester recently made it compulsory for an academic to have an ORCID in order to be part of its internal research assessment exercise (the university has a good page about ORCID that answers some FAQs). Research Councils UK recently joined ORCID and will soon start capturing ORCIDs on its grant systems. With these organizations actively supporting the scheme, and many other organizations members of the scheme (including the American Mathematical Society, but not yet SIAM), it seems that ORCID has a bright future. However, my impression is that many academics are unaware of ORCID. There is plenty of information about it on the web, but probably not in the places academics tend to look.

I expect interest will increase rapidly as ORCID becomes better known. I notice that Springer has started to provide links from an author name on a paper to the ORCID page for that author. This is just the sort of added value feature that will make academics want to register for an ORCID.

Manchester Numerical Analysis Reports

This post is an edited and updated version of an article that I published in 2006 in the IMANA Newsletter (“Newsletter of the Numerical Analysis Group of the Institute of Mathematics and its Applications”). Very few issues of the Newsletter appear to be electronically available, so I thought it worthwhile to reproduce the article here.

Three different cover designs, from 1988, 1996, and 2005.

The University of Manchester Numerical Analysis (NA) Report series began in 1974. The two key movers in setting up the series were Ian Gladwell, a member of the Department of Mathematics at the University of Manchester (now retired from the Department of Mathematics at Southern Methodist University, Dallas), and Charlie Van Loan, an SERC-funded postdoctoral visitor to the department in 1974–1975 (and subsequently a professor in the Department of Computer Science at Cornell University). The first report was

Charles F. Van Loan, Least Squares Problems with Emphasis Upon Singular Value Techniques, Numerical Analysis Report No. 1, September 1974.

and Charlie wrote four of the first 10 reports. Particularly notable is

Charles F. Van Loan. A study of the matrix exponential. Numerical Analysis Report No. 10, August 1975.

This was an early version of the classic, highly-cited article “Nineteen Dubious Ways to Compute the Exponential of a Matrix” written with Cleve Moler and subsequently published in SIAM Review in 1978, with an updated reprint in SIAM Review in 2003. The report has been reissued as MIMS EPrint 2206.397. Ian’s early contributions include the often-cited

J. L. Siemieniuch and I. Gladwell On time discretization for linear time-dependent partial differential equations. Numerical Analysis Report No. 5, September 1974.

One of the main aims of the series was to provide a vehicle for pre-publication of a preliminary version of a piece of work, prior it to being submitted to a journal. Right from the start this aim was achieved, with at least 15 of the first 20 reports known to have appeared in refereed journals. Nevertheless, a number of important early reports, such as Number 5 mentioned above, were not submitted but surely would have been in today’s academic climate.

The contents of the series naturally reflect the interests of the numerical analysts at the University of Manchester and UMIST over the years. The first 125 reports (taking us up to October 1986) include contributions on stiff differential equations (George Hall, Jack Williams), complex approximation (Jack Williams), Volterra integral equations (Christopher Baker), polynomial zero-finding (Len Freeman), methods for second order ordinary differential equations (Ian Gladwell, Ruth Thomas), multigrid (Joan Walsh), numerical linear algebra (Nick Higham), and numerical analysis of partial differential equations (Ian Gladwell, David Silvester, Ron Thatcher, Joan Walsh).

As well as containing preprints of research papers, the series includes all thirteen Annual Reports of the Manchester Centre for Computational Mathematics and the proceedings of two 1982 meetings:

Ian Gladwell (ed.), Proceedings of a One-Day Colloquium On Numerical Linear Algebra and Its Applications. Numerical Analysis Report No. 78, July 1982.

George Hall and Jack Williams (eds), Proceedings of a One-Day Colloquium on the Numerical Solution of Ordinary Differential Equations. Numerical Analysis Report No. 84, December 1982.

The reports illustrate the changes in typesetting mathematics since the 1970s. Early reports were typewritten, sometimes with equations written in by hand. In the 1980s many of the reports were wordprocessed using Vuwriter—a technical wordprocessor produced by Vuman Ltd., a spin-off company of the University of Manchester, targeted at the Apricot microcomputer. I wordprocessed several reports on a Commodore 64 microcomputer using an Epson printer, with Greek letters and mathematical characters produced in the printer’s graphics mode (see this earlier post for more details).

The first TeXed reports were around 1986/1987 and by the early 1990s most reports were produced in \LaTeX, as they are today.

The printed reports retained their distinctive green card cover to the end, but a major change came in May 1993 when they were first made available over the internet—originally by anonymous ftp from and then from the Manchester Centre for Computational Mathematics (MCCM) web site set up in 1994. The web page from which the reports are available, now located here, was automatically created from a BibTeX bib file, the latter being maintained by hand, as was the repository of PDF and PS files.

In 2005, the NA Report series was folded into the new MIMS EPrints archive ( which hosts research outputs of members of the School of Mathematics and associated researchers. EPrints entries are assigned an AMS subject classification and can be searched by those numbers. Reports that would have appeared in the old NA Report series can now generally be found under the classification 65 Numerical Analysis.

On a recent visit to the University of Manchester library I was pleased to find that many of the NA reports up to 2001 are still available in hard copy on the shelves. (A search of the catalogue for “numerical analysis report” reveals the details.)