Steve Smale - Böcker

Computational complexity theory provides a framework for understanding the cost of solving computational problems, as measured by the requirement for resources such as time and space. The objects of study are algorithms defined within a formal model of computation. Upper bounds on the computational complexity of a problem are usually derived by constructing and analyzing specific algorithms. Meaningful lower bounds on computational complexity are harder to come by, and are not available for most problems of interest. The dominant approach in complexity theory is to consider algorithms as oper­ ating on finite strings of symbols from a finite alphabet. Such strings may represent various discrete objects such as integers or algebraic expressions, but cannot rep­ resent real or complex numbers, unless the numbers are rounded to approximate values from a discrete set. A major concern of the theory is the number of com­ putation steps required to solve a problem, as a function of the length of the input string.

The lectures in this volume are the proceedings from the 1995 AMS-SIAM Summer Seminar in Applied Mathematics held in Park City, UT. The mathematical theory of real number algorithms was the subject of the conference, with emphasis on geometrical, algebraic, analytic, and foundational perspectives. Investigations on efficiency played a special role. The goal of the conference was to give the topic of numerical analysis greater coherence by focusing on the mathematical side. Particular attention was aimed at strengthening the unity of mathematics and numerical analysis and narrowing the gap between pure and applied mathematics.The conference was international in character, with strong representation from the most mathematically developed parts of numerical analysis. Seminars in the following areas were held: linear algebra, nonlinear systems-path following, differential equations, linear programming, interval arithmetic, algebraic questions, foundations, information based complexity, lower bounds, and approximation theory.