SOUL SEARCHING FOR LINEAR ALGEBRA

by Osni Marques


An edited version of this article appeared in SIAM News, vol. 34,
no. 1, p. 20 (2001).


The 7th SIAM Conference on Applied Linear Algebra was held at the McKimmon
Conference Center, North Carolina State University (NCSU), Raleigh, on
October 23-25, 2000. The previous two conferences in the series (1994 and
1997) were held in the Snowbird Ski and Summer Resort, in Snowbird, Utah.
While places like Snowbird offer a plethora of outdoor options for more
adventurous travelers, places like Raleigh seem to be more convenient for
travelers to reach. In addition, it was also in Raleigh that the first SIAM
Conference on Applied Linear Algebra took place two decades ago. The mild
temperatures, the varied colors of the leaves at that time of the year and
the (now customary) box lunches provided by SIAM contributed to an
attractive ambiance for discussion and interaction.

This year's conference was shortened by one day, thanks to the inclusion of
sessions on Tuesday evening. Although this arrangement decreed a busy day
for the attendees, the outcome was good; the work I co-authored was
presented in one of the evening sessions and so I was there to witness the
attendance. The three-day format helped to reduce expenses as well! In
total, there were 8 invited plenary talks, 9 invited concurrent talks, 20
concurrent sessions and 9 minisymposia. There were also 8 posters on
display. The conference organizers, Ilse Ipsen and Carl Meyer (NCSU), pulled
all that together in fine style.

The conference featured ``classic'' topics such as eigenvalue problems
(including recent developments in Lanczos-type algorithms), direct and
iterative methods for the solution of systems of equations and
preconditioning for iterative methods. Also represented were the computation
of invariant subspaces, totally nonnegative matrices, matrix completions and
pseudoinverses, and matrix calculations in image processing, just to name a
few subjects. The attendees were further entertained by excellent invited
plenary talks on random perturbations of special matrices, the ubiquitous
Kronecker product, interior-point methods for optimization, and displacement
structure, among others.

A challenging topic for discussion during a conference is the status of the
field in the scientific community. This self-examination was the goal of the
panel ``Linear Algebra: What's it Worth?'' which was moderated by Daniel
Pierce (The Boeing Company). The truth is that Linear Algebra is at the
heart of most computational sciences problems but many of us feel that it
does not receive adequate (or the expected) credit. The difficulty, one
panelist pointed out, is that Linear Algebra is in the base of the pyramid
(``food chain'') and that it almost always arises in another context, such as
optimization. Does this explain why scientists are sometimes reluctant to
tell their funding agencies about the importance of research on Linear
Algebra? Another panelist recalled that in the past, industries hired a
numerical linear algebraist to tackle a common problem whereas nowadays
industries hire an expert in the application area hoping that she/he knows
enough Linear Algebra. Cleve Moler (The MathWorks, Inc) pointed out that
MathWorks' Matlab is used in many different real applications, from finances
to car and cell phone design, yet the trend is for Matlab to hide matrix
computations from the users. By the way, The Mathworks' market value was
said to be as high as Ben and Jerry's ice cream business. In any case, the
time required for Linear Algebra research, implementation and then
simulation or business calculations is usually very long, which makes it
tougher to assess the value of the initial research. The panel did not come
out with definitive answers. However, it was suggested that people working
in Applied Linear Algebra are perhaps not good in public relations. Also,
there may exist a communication problem: people working in Linear Algebra
could maybe try to master other groups's language and then talk to them in
their own terms. One panelist suggested these synonyms should be widely
disseminated in the Linear Algebra community.

The 2000 Linear Algebra prize was shared by Olga Holtz (University of
Wisconsin, Madison), and Alan Edelman (MIT), Eric Elmroth and Bo Kagstrom
(both from Umea University, Sweden). Olga was awarded the prize for her work
on GKK (after Gantmacher, Krein and Kotelyansky) Tau-matrices described in
the paper ``Not all GKK tau-matrices are stable? (LAA, 291:235-244, 1999). In
this paper she disproved four conjectures (three of them more than 20 years
old) about the properties (positivity of the principal minors, weak sign
symmetry, eigenvalue motonicity and positive stability) attributed to such
matrices. Next, Alan, Eric and Bo were awarded the prize for the work
described in the paper ``A Geometric Approach to Perturbation Theory of
Matrices and Matrix Pencils. Part I: Versal Deformations'' (SIAM Journal on
Matrix Analysis and Applications, 18:653-692, 1997) where they derived
versal deformations of the Kronecker canonical form (the three authors
managed to use numerical Linear Algebra language in their derivations).
Roughly speaking, a deformation of a matrix is versal if it captures all
possible Jordan form behavior, near the matrix.

As a warm-up for the conference, a workshop on computation issues of
Information Retrieval (IR) took place on Sunday, October 22. The workshop
was organized by Michael Berry (University of Tennessee at Knoxville). At
the previous conference in Snowbird, the numerical aspects of IR were
limited to a few presentations (in the form of posters, with one of them
winning a special recognition). Since then the field has evolved, gained
momentum and researchers? attention. As a result, the workshop claimed an
almost full occupancy. The key issue is that the data associated to the
nowadays huge collection of documents, such as those coming from web related
applications, mandate an adequate extraction of features or reduction of the
models by projecting them into subspaces. Low-rank approximations by means
of singular value decompositions (SVD) provide the framework for techniques
like Latent Semantic Indexing (LSI), for example. Following those lines, the
workshop covered issues arising in dimension reduction, concept
decompositions, data preprocessing schemes, as well as a probabilistic model
for IR which lead to the conclusion that LSI/SVD are optimal. A comparative
analysis of strategies for LSI was also presented. In total, there were 15
invited and contributed talks in the workshop.

The panel ``Linear Algebra: What's it Worth?'' may not have come out with
definite answers (that was not the goal) but some real applications
certainly speak for themselves.  What about ``Cruising at (approximately)
41000 feet - Iterative Methods at Boeing'' or ``Linear Algebra or Nuclear
Testing: The Lesser of Two Devils'' (talks given by John Lewis, The Boeing
Company, and Bruce Hendrickson, Sandia National Laboratories, respectively)?
Isn't that convincing enough? Three cheers for Applied Linear Algebra!

Osni Marques works at the National Energy Research Scientific Computing
Center Division (NERSC) of the Lawrence Berkeley National Laboratory (LBNL),
Berkeley, California.