A small group of specialists gathered together in the fascinating city of Split on the Croatian Coast. The goal was to share their knowledge of methods to solve the problem $Ax = \lambda Bx$ (or its singular value pendant) to as much accuracy as the data warrants. In particular, there is a need to describe the structure of matrices $A,B$, which determine their small eigenvalues (the ones of interest in most, but not all applications) to high relative accuracy. For example, consider the stiffness and mass matrices, usually called $K$, $M$, obtained by finite differences approximation. For one-dimensional problems the unassembled form $NDN^t$ does determine the eigensolution $(NDN^t - \lambda M)v = 0$ to high relative accuracy, but more general finite element models, especially those in high dimensions, are still undecided.

While it was already known that some factorizations like QR may essentially improve the computed accuracy of the singular values, we have now learned that the same is true of completely pivoted LU - a deeply non-unitary process. Also the highly accurate singular values of a bidiagonal were shown to tune the subsequent inverse iterations to keep both their high speed and the necessary orthogonality.

The organizers, Professors Barlow, Slapnicar and Veselic, thought hard about the format of the meeting. The participants fell into 2 groups, about 10 senior researchers active on these problems (Demmel, Drmac, Hari, Ipsen, Parlett, Ruhe, and the organizers) and about 15 graduate students who are writing dissertations in this area. All participants stayed in one of the student dormitories of the University of Split, the formal host of the Workshop. After breakfast a small caravan of cars made its way to the Department of ME, EE and Naval Arch. There were just 3 one hour talks in the morning and at most one in the afternoon. This left time for group discussions on Open Problems, Software and a possible monograph.

The local organizer, Ivan Slapnicar, had arranged an impressive program of entertainment, including among others, a guided tour of Diocletian's palace (from which the city grew in the 5th century), a full day on a wonderful Zlatni Rat Beach on the island of Brac and attending an open air performance of Verdi's Nabucco. The weather was as usual, superb. The participants were looked after very well by local organizers.

How can you tell a workshop from a regular meeting? Regular meetings are dominated by overhead projectors, workshops make use of the blackboard and all talks are peppered with questions from senior researchers while graduate students can learn the habit of interrupting.

Although there were only 15 talks in 4 working days there was a consensus that the remaining time must allow more space for informal technical work. The list of talks and the abstracts are available at the web site http://www.fesb.hr/~slap/workshop.

At the end of the Workshop, Jesse Barlow announced the follow-up meeting at the Pennsylvania State University, State College, PA, in two years.

Beresford Parlett