An introduction to some current research in numerical computational complexity

This is an expository invited paper presented at a Conference on the Influence of Computing on Mathematical Research and Education, at the summer meeting of the American Mathematical Society at Missoula, Montana, August, 1973. AN INTRODUCTION TO SOME CURRENT RESEARCH IN NUMERICAL COMPUTATIONAL COMPLEXITY J. F. Traub This is a Conference on the Influence of Computing on Mathematical Research and Education. I want to talk about a particular area of numerical mathematics, numerical computational complexity, which has a very large intersection with mathematics. On the one hand, it requires the development of new mathematical techniques while on the other hand, certain of the questions it tries to answer are primarily mathematical. In this talk I will focus on research rather than educational issues. However, I will mention that some non-trivial results and questions can be formulated so as to be accessible to both undergraduate and graduate students. This is at least partially the case because the field is so new. I ll not attempt a survey of current work in computational complexity, rather, I'll restrict myself to problems and algorithms from numerical mathematics. Later in this conference, Professor Karp will discuss non-numerical complexity. Within numerical computational complexity, I will try to give you the flavor of some of the recent work and state a few open problems. I will draw on work done at Carnegie-Mellon University for many of my examples. A number of the papers I'll cite were presented at a Symposium on Complexity of Sequential and Parallel Numerical Algorithms at Carnegie-Mellon University in May, 1973. Incidentally, there have been at least four Symposia in the United States from April to July, 1973 devoted entirely or in major part to computational complexity.