Survey and Review

Abstract

It is a great pleasure to assume the role of Section Editor for the Survey and Review section of this journal. I'd like to thank Nick Trefethen for editing this section so ably over the past four years, and guiding its development since the format of SIAM Review was substantially overhauled in 1999. I will have the help of an excellent editorial board, and we intend to continue the tradition of publishing informative and well-written papers on topics of broad interest to the applied mathematics community. I encourage prospective authors to read the revised guidelines at the front of this journal. SIAM Review has played a preeminent role in the Society for Industrial and Applied Mathematics since Volume 1 was published in 1959. Over the years many important papers have appeared in these pages, and as part of SIAM's 50th anniversary celebration we thought it would be appropriate to reprint a classic SIAM Review paper from the past. The one we have chosen, "Nineteen Dubious Ways to Compute the Exponential of a Matrix," by Cleve Moler and Charles Van Loan, is one of the most frequently cited papers ever to appear in this journal. More importantly, it's still worth reading, and worth introducing to a new generation of students and young researchers. It's a great paper for a journal club; working through the nineteen dubious ways gives a tour of a broad spectrum of techniques and sets the stage for discussing and exploring many related issues. Pade approximation, ODE solvers, the Cayley--Hamilton theorem, polynomial interpolation, Laplace transforms, the Jordan canonical form, Schur decompositions, and the Trotter product formula---these are just some of the approaches that make an appearance. Understanding why they are dubious leads to the study of conditioning and stability, inverse error analysis, logarithmic norms, and an appreciation for the effects of nonnormal matrices and coincident eigenvalues. All this in 37 very readable pages. Of course, much progress has been made in the years since this paper first appeared. Many of the issues discussed are better understood today than they were at the time, and new algorithms have appeared that may be less dubious. The behavior of nonnormal matrices and the "hump problem" are clarified via the theory of pseudospectra and related notions. Particularly noteworthy on the algorithmic front is the development of Krylov space methods that give good accuracy and efficient computation of the exponential of large matrices. These methods are now frequently used in various contexts, such as quantum dynamics, Markov chains, and differential equations. The original "Nineteen Ways" paper is an excellent starting point for exploring a host of other topics. To aid in this exploration, Moler and Van Loan have kindly provided a summary of some recent developments, with a few entry points into the vast literature that has appeared since the original paper. Their new contribution follows the original paper, starting on page 40.