Scientific

A general matrix A can be reduced to tridiagonal form by orthogonal
transformations on the left and right: UTAV = T. We can arrange that the
rst columns of U and V are proportional to given vectors b and c. An iterative
form of this process was given by Saunders, Simon, and Yip (SINUM 1988) and
used to solve square systems Ax = b and ATy = c simultaneously. (One of the
resulting solvers becomes MINRES when A is symmetric and b = c.)

The approach was rediscovered by Reichel and Ye (NLAA 2008) with emphasis
on rectangular A and least-squares problems Ax ~ b. The resulting solver was
regarded as a generalization of LSQR (although it doesn't become LSQR in
any special case). Careful choice of c was shown to improve convergence.

In his last year of life, Gene Golub became interested in \GLSQR" for
estimating cTx = bTy without computing x or y. Golub, Stoll, and Wathen
(ETNA 2008) revealed that the orthogonal tridiagonalization is equivalent to a
certain block Lanczos process. This reminds us of Golub, Luk, and Overton
(TOMS 1981): a block Lanczos approach to computing singular vectors.

On solving indefinite least squares problems via anti-triangular factorizations:
Nicola Mastronardi, IAC-CNR, Bari, Italy and Paul Van Dooren, UCL, Louvain-la-Neuve, Belgium

Communication-­ Avoiding Algorithms for Linear Algebra and Beyond

Convex Optimization for Finding Influential Nodes in Social Networks. Joint work with Lisa Elkin and Ting Kei Pong of Waterloo.

Differential equations for the approximation of the distance to the closest defective matrix. Joint work with P. Butta' and S. Noschese (Università di Roma La Sapienza) and M. Manetta (Università dell'Aquila).