Classical and Modern Numerical Analysis: Theory, Methods and Practice provides a sound foundation in numerical analysis for more specialized topics, such as finite element theory, advanced numerical linear algebra, and optimization. It prepares graduate students for taking doctoral examinations in numerical analysis.

The text covers the main areas of introductory numerical analysis, including the solution of nonlinear equations, numerical linear algebra, ordinary differential equations, approximation theory, numerical integration, and boundary value problems. Focusing on interval computing in numerical analysis, it explains interval arithmetic, interval computation, and interval algorithms. The authors illustrate the concepts with many examples as well as analytical and computational exercises at the end of each chapter.

This advanced, graduate-level introduction to the theory and methods of numerical analysis supplies the necessary background in numerical methods so that students can apply the techniques and understand the mathematical literature in this area. Although the book is independent of a specific computer program, MATLAB(r) code is available on the authors' website to illustrate various concepts.

Primary Audience for the Book * Specialists in numerical computations who are interested in algorithms with automatic result verification. * Engineers, scientists, and practitioners who desire results with automatic verification and who would therefore benefit from the experience of suc cessful applications. * Students in applied mathematics and computer science who want to learn these methods. Goal Of the Book This book contains surveys of applications of interval computations, i. e. , appli cations of numerical methods with automatic result verification, that were pre sented at an international workshop on the subject in EI Paso, Texas, February 23-25, 1995. The purpose of this book is to disseminate detailed and surveyed information about existing and potential applications of this new growing field. Brief Description of the Papers At the most fundamental level, interval arithmetic operations work with sets: The result of a single arithmetic operation is the set of all possible results as the operands range over the domain. For example, [0. 9,1. 1] + [2. 9,3. 1] = [3. 8,4. 2], where [3. 8,4. 2] = {x + ylx E [0. 9,1. 1] and y E [3. 8,4. 2]}. The power of interval arithmetic comes from the fact that (i) the elementary operations and standard functions can be computed for intervals with formulas and subroutines; and (ii) directed roundings can be used, so that the images of these operations (e. g.

This work grew out of several years of research, graduate seminars and talks on the subject. It was motivated by a desire to make the technology accessible to those who most needed it or could most use it. It is meant to be a self-contained introduction, a reference for the techniques, and a guide to the literature for the underlying theory. It contains pointers to fertile areas for future research. It also serves as introductory documentation for a Fortran 90 software package for nonlinear systems and global optimization. The subject of the monograph is deterministic, automatically verified or r- orous methods. In such methods, directed rounding and computational fix- point theory are combined with exhaustive search (branch and bound) te- niques. Completion of such an algorithm with a list of solutions constitutes a rigorous mathematical proof that all of the solutions within the original search region are within the output list. The monograph is appropriate as an introduction to research and technology in the area, as a desk reference, or as a graduate-level course reference. Kno- edge of calculus, linear algebra, and elementary numerical analysis is assumed.

Interval computing combined with fuzzy logic has become an emerging tool in studying artificial intelligence and knowledge processing (AIKP) applications since it models uncertainties frequently raised in the field. This book provides introductions for both interval and fuzzy computing in a very accessible style.

Application algorithms covered in this book include quantitative and qualitative data mining with interval valued datasets, decision making systems with interval valued parameters, interval valued Nash games and interval weighted graphs. Successful applications in studying finance and economics, etc are also included.

This book can serve as a handbook or a text for readers interested in applying interval and soft computing for AIKP.

Interval computing combined with fuzzy logic has become an emerging tool in studying artificial intelligence and knowledge processing (AIKP) applications since it models uncertainties frequently raised in the field. This book provides introductions for both interval and fuzzy computing in a very accessible style.

Application algorithms covered in this book include quantitative and qualitative data mining with interval valued datasets, decision making systems with interval valued parameters, interval valued Nash games and interval weighted graphs. Successful applications in studying finance and economics, etc are also included.

This book can serve as a handbook or a text for readers interested in applying interval and soft computing for AIKP.

Reliable computing techniques are essential if the validity of the output of a - merical algorithm is to be guaranteed to be correct. Our society relies more and more on computer systems. Usually, our systems appear to work successfully, but there are sometimes serious, and often minor, errors. Validated computing is one essential technology to achieve increased software reliability. Formal - gor in the de?nition of data types, the computer arithmetic, in algorithm design, and in program execution allows us to guarantee that the stated problem has (or does not have) a solution in an enclosing interval we compute. If the enclosure is narrow, we are certain that the result can be used. Otherwise, we have a clear warning that the uncertainty of input values might be large and the algorithm and the model have to be improved. The use of interval data types and al- rithms with controlled rounding and result veri?cation capture uncertainty in modeling and problem formulation, in model parameter estimation, in algorithm truncation, in operation round-o?, and in model interpretation. The techniques of validated computing have proven their merits in many scienti?c and engineering applications. They are based on solid and interesting theoretical studies in mathematics and computer science. Contributions from ?elds including real, complex and functional analysis, semigroups, probability, statistics, fuzzyintervalanalysis, fuzzylogic, automaticdi?erentiation, computer hardware, operating systems, compiler construction, programming languages, object-oriented modeling, parallel processing, and software engineering are all essent

The book provides a self-contained introduction to underlying techniques, as well as a compendium of theory and a guide to the author's Fortran 90 software for nonlinear algebraic systems and global, constrained optimization with automatic result verification. Besides introductory and survey material, the book contains unique research results. The book also contains non-traditional ideas concerning non-smooth optimization. Thus, the book should be a valuable reference to applied mathematicians and computational scientists and engineers. With numerous examples and exercises, as well as leads for future research, the book can be used as a graduate text or reference on interval arithmetic, automatic differentiation and interval fixed point theory. The book can also serve as a user's guide for the nonlinear equations and optimization software, available free of charge from the author, for the author's general Fortran 90 interval arithmetic package, or for the associated automatic differentiation package. Audience: Researchers in operations research, numerical analysis, computational chemistry, computer-aided geometric design and computational geometry, robot kinematics, remote sensing. Also suitable for graduate and topics courses in numerical analysis, optimization, and operations research.

Primary Audience for the Book * Specialists in numerical computations who are interested in algorithms with automatic result verification. * Engineers, scientists, and practitioners who desire results with automatic verification and who would therefore benefit from the experience of suc cessful applications. * Students in applied mathematics and computer science who want to learn these methods. Goal Of the Book This book contains surveys of applications of interval computations, i. e. , appli cations of numerical methods with automatic result verification, that were pre sented at an international workshop on the subject in EI Paso, Texas, February 23-25, 1995. The purpose of this book is to disseminate detailed and surveyed information about existing and potential applications of this new growing field. Brief Description of the Papers At the most fundamental level, interval arithmetic operations work with sets: The result of a single arithmetic operation is the set of all possible results as the operands range over the domain. For example, [0. 9,1. 1] + [2. 9,3. 1] = [3. 8,4. 2], where [3. 8,4. 2] = {x + ylx E [0. 9,1. 1] and y E [3. 8,4. 2]}. The power of interval arithmetic comes from the fact that (i) the elementary operations and standard functions can be computed for intervals with formulas and subroutines; and (ii) directed roundings can be used, so that the images of these operations (e. g.