Numerical Computation 1: Methods, Software, and Analysis / Edition 1

by Christoph W. Ueberhuber

Overview

This book deals with various aspects of scientific numerical computing. No at­ tempt was made to be complete or encyclopedic. The successful solution of a numerical problem has many facets and consequently involves different fields of computer science. Computer numerics- as opposed to computer algebra- is thus based on applied mathematics, numerical analysis and numerical computation as well as on certain areas of computer science such as computer architecture and operating systems. Applied Mathemalies I I I Numerical Analysis Analysis, Algebra I I Numerical Computation Symbolic Computation I Operating Systems Computer Hardware Each chapter begins with sample situations taken from specific fields of appli­ cation. Abstract and general formulations of mathematical problems are then presented. Following this abstract level, a general discussion about principles and methods for the numerical solution of mathematical problems is presented. Relevant algorithms are developed and their efficiency and the accuracy of their results is assessed. It is then explained as to how they can be obtained in the form of numerical software. The reader is presented with various ways of applying the general methods and principles to particular classes of problems and approaches to extracting practically useful solutions with appropriately chosen numerical software are developed. Potential difficulties and obstacles are examined, and ways of avoiding them are discussed. The volume and diversity of all the available numerical software is tremendous.

Product Details

ISBN-13:	9783540620587
Publisher:	Springer Berlin Heidelberg
Publication date:	04/11/1997
Series:	Numerical Computation Series
Edition description:	1997
Pages:	474
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

1 Scientific Modeling.- 1.1 Reality Versus Model.- 1.2 The Model Subject and the Model.- 1.3 The Model Subject and Reality.- 1.4 Model Building.- 2 Fundamental Principles of Numerical Methods.- 2.1 From Application Problems to their Numerical Solution.- 2.2 Numerical Problems.- 2.3 Types of Errors in Numerics.- 2.4 The Condition of Mathematical Problems.- 2.5 The Condition of Application Problems.- 2.6 The Mathematical Elements of Condition Estimation.- 2.7 Validation of Numerical Computations.- 3 Computers for Numerical Data Processing.- 3.1 Processors.- 3.2 Memory.- 3.3 Performance Quantification.- 3.4 Analytical Performance Assessment.- 3.5 Empirical Performance Assessment.- 4 Numerical Data and Numerical Operations.- 4.1 Mathematical Data.- 4.2 Numerical Data on Computers.- 4.3 Operations on Numerical Data.- 4.4 Number Systems on Computers.- 4.5 Structure of Floating-Point Systems.- 4.6 Standardization of Floating-Point Number Systems.- 4.7 Arithmetics for Floating-Point Systems.- 4.8 Inquiry Functions and Manipulation of Numbers in Fortran 90.- 4.9 Operations with Algebraic Data.- 4.10 Operations with Arrays.- 4.11 Operations with Analytic Data.- 5 Numerical Algorithms.- 5.1 The Intuitive Notion of an Algorithm.- 5.2 Properties of Algorithms.- 5.3 Existence of Algorithms.- 5.4 Practical Solvability of Problems.- 5.5 Complexity of Algorithms.- 5.6 Representation of Algorithms.- 5.7 Influence of Rounding Errors on Numerical Algorithms.- 5.8 Case Study: Floating-Point Summation.- 6 Numerical Programs.- 6.1 The Quality of Numerical Programs.- 6.2 Reasons for Poor Efficiency.- 6.3 The Measurement of Performance Indices.- 6.4 Performance Optimization.- 6.5 Architecture Independent Optimizations.- 6.6 Loop Optimizations.- 6.7 Blocked Memory Access.- 6.8 Case Study: Multiplication of Matrices.- 7 Available Numerical Software.- 7.1 The Cost of Software.- 7.2 Sources of Numerical Software.- 7.3 Software and the Internet.- 7.4 Interactive Multifunctional Systems.- 7.5 Problem Solving Environments.- 7.6 Case Study: Software for Elliptic PDEs.- 8 Using Approximation in Mathematical Model Building.- 8.1 Analytic Models.- 8.2 Information and Data.- 8.3 Discrete Approximation.- 8.4 Function Approximation.- 8.5 Choosing a Model Function.- 8.6 Choice of the Distance Function.- 8.7 Transformation of the Problem.- 9 Interpolation.- 9.1 Interpolation Problems.- 9.2 Mathematical Foundations.- 9.3 Univariate Polynomial Interpolation.- 9.4 Univariate, Piecewise, Polynomial Interpolation.- 9.5 Polynomial Splines.- 9.6 B-Splines.- 9.7 Cubic Spline Interpolation.- 9.8 Splines Without Undesirable Oscillations.- 9.9 Multivariate Interpolation.- 9.10 Multivariate Polynomial Interpolation.- 9.11 Multivariate (Sub-) Spline Interpolation.- 9.12 Related Problems and Methods.- Glossary of Notation.- Author Index.

Editorial Reviews

"The two volumes can be highly recommended for newcomers in the area as well as for people working for a long time in or with computer numerics." - EUROSIM - Simulation News Europe

"...This book is highly recommended to students, scientists, and engineers interested in the numerical solution of mathematical problems. It is very useful as a handbook for both newcomers and experts. Every science/engineering\-/mathematics/computer science library should have a copy of this book."—MATHEMATICAL REVIEWS

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