Preface

This volume contains lecture notes prepared by the speakers for the

American Mathematical Society Short Course on Numerical Analysis given in

Atlanta, Georgia, 3-4 January 1978.

We were very pleased that the Short Course Advisory Subcommittee

decided to hold a short course on Numerical Analysis, and even more

pleased by the large attendance. We are indebted to our colleagues for

their enthusiastic cooperation and efforts which made the Short Course

and these published proceedings possible.

The choice of topics was influenced rather strongly by the subcom-

mittee's objectives for these short courses. These objectives are that

the short courses provide an entree to an area and lead up to current re-

search problems in that area. We have tried to emphasize those areas

where research activity is greatest and the present state of understanding

is not satisfactory. Consequently, many classical problem areas are

hardly mentioned, or ignored.

The term numerical analysis is too narrow as a description of the

area as it is generally viewed today, since the construction of algorithms

is an important aspect of the subject as well. In understanding why a

particular algorithm "works" or "does not work" one often is led to better

algorithms. Thus, the constructive and analytical aspects are not inde-

pendent.

It is often said that the "computations are ahead of the analysis."

This means that known algorithms perform in an inexplicable manner when

they are tested on problems with known solutions. They often work better

than we can guarantee them to work--our error estimates are not sharp

enough or don't exist. This illustrates the strong influence that compu-

tations performed in the physical sciences and engineering have on the

subject.

The papers given here are mainly of a mathematical nature. The

results presented describe properties of computational METHODS that are

only relevant in the context of that computation. It is the need to per-

form the computation which presents the problems to the subject and

justifies it. For example, in the emerging field of Computational Physics

METHODS are developed as they are needed for various problems. These

METHODS are usually constructed via physical reasoning, experience, and

intuition. They are often tested on problems with known solutions, but

their validity is often judged on their behavior in physical terms. It is

then the numerical analyst who attempts to give error estimates and des-

cribe the numerical behavior of these METHODS. The convergence results

needed here differ from those of classical constructive analysis. Error

estimates which hold for finite values of the discretization parameters

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