In this monograph, the authors develop a methodology that allows one to construct and substantiate optimal and suboptimal algorithms to solve problems in computational and applied mathematics. Throughout the book, the authors explore well-known and proposed algorithms with a view toward analyzing their quality and the range of their efficiency. The concept of the approach taken is based on several theories (of computations, of optimal algorithms, of interpolation, interlination, and interflatation of functions, to name several). Theoretical principles and practical aspects of testing the quality of algorithms and applied software, are a major component of the exposition. The computer technology in construction of T-efficient algorithms for computing ε-solutions to problems of computational and applied mathematics, is also explored. The readership for this monograph is aimed at scientists, postgraduate students, advanced students, and specialists dealing with issues of developing algorithmic and software support for the solution of problems of computational and applied mathematics.

In this monograph, the authors develop a methodology that allows one to construct and substantiate optimal and suboptimal algorithms to solve problems in computational and applied mathematics. Throughout the book, the authors explore well-known and proposed algorithms with a view toward analyzing their quality and the range of their efficiency. The concept of the approach taken is based on several theories (of computations, of optimal algorithms, of interpolation, interlination, and interflatation of functions, to name several). Theoretical principles and practical aspects of testing the quality of algorithms and applied software, are a major component of the exposition. The computer technology in construction of T-efficient algorithms for computing -solutions to problems of computational and applied mathematics, is also explored. The readership for this monograph is aimed at scientists, postgraduate students, advanced students, and specialists dealing with issues of developing algorithmic and software support for the solution of problems of computational and applied mathematics.

The purpose of this monograph is to create a general framework for the study of optimal algorithms for problems that are solved approximately. For generality the setting is abstract, but we present many applications to practical problems and provide examples to illustrate concepts and major theorems. The work presented here is motivated by research in many fields. Influential have been questions, concepts, and results from complexity theory, algorithmic analysis, applied mathematics and numerical analysis, the mathematical theory of approximation (particularly the work on n-widths in the sense of Gelfand and Kolmogorov), applied approximation theory (particularly the theory of splines), as well as earlier work on optimal algorithms. But many of the questions we ask (see Overview) are new. We present a different view of algorithms and complexity and must request the reader's

In the Russian edition published in 1989, this book was called "Minimax Algorithms in Problems of Numerical Analysis". The new title is better related to the subject of the book and its style. The basis for every decision or inference concerning the ways to solve a given problem is the computa tion model. Thus, the computation model is the epicenter of any structure studied in the book. Algorithms are not constructed here, they are rather derived from computation models. Quality of an algorithm depends entirely on consistency of the model with the real-life problem. So, constructing a model is an art, deriving an algorithm is a science. We study only minimax or, in other words, worst-case computation models. However, one of the characteristic features of the book is a new approach to the notion of the worst-case conditions in dynamic processes. This approach leads to the concept of sequentially optimal algorithms, which play the central role in the book. In conclusion, I would like to express my gratitude to Prof. Dr. Heinz J. Skala and Dr. Sergei A. Orlovsky for encouraging translation of this book. I also greatly appreciate the highly professional job of Dr. Olga R. Chuyan who translated the book.

This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.