What we plan not to include

Some of them we may include, provided there is sufficient demand. Mostly, we would limit ourselves to literature references.

• There is a variety of ``non-linear'' eigenproblems which can be linearized in various ways and converted to the above problems
  1. Rational problems
  2. General nonlinear problems
  3. Polynomial system zero finding

• The SVD can be used to solve a variety of least squares problems, and these may best be solved by representing the SVD in ``factored'' form. These factored SVD algorithms are somewhat complicated and hard to motivate without the least squares application, but there are a great many least squares problems, but including too many of them (beyond references to the literature) would greatly expand the scope of this project. The same comments apply to the generalized SVD, of which there are even more variations.

• Eigenproblems from systems and control. There is a large source of eigenproblems, many of which can be reduced to finding certain kinds of spectral information about possibly singular pencils . These problems arise with many kinds of special structures, many of which have been exploited in special packages, some of them can be solved by general algorithms. For example, solutions of Riccati equations are often reduced to eigenproblems. A comprehensive treatment of these problems would greatly expand the scope of the book, but including some treatment seems important.

• Structured singular value problems, also called -analysis by its practitioners These problems must generally be solved by optimization techniques, and indeed NP-completeness results exist for some of them. We believe that the algorithms in our proposed book would be called within inner loops for these optimization algorithms, and we should simply refer to the literature for this application. The same comment applies for other optimization problems based on eigenvalues (e.g. finding the nearest unstable matrix).

• Generalized singular value decompositions of more than two matrices

• Computing matrix functions like or , or for multivariate eigenvalue problems , whose solutions are typically continua.