Overview

This book is the first part of a modern, two-volume introduction to numerical computation, which strongly emphasizes software aspects. It can serve as a textbook for courses on numerical analysis, particularly for engineers. The book can also be used as a reference book and it includes an extensive bibliography. The author is a well-known specialist in numerical analysis who was involved in the creation of the software package QUADPACK.

Full Product Details

Author:   Christoph W. Ueberhuber
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   1997 ed.
Dimensions:   Width: 15.50cm , Height: 2.50cm , Length: 23.50cm
Weight:   1.520kg
ISBN:  

9783540620587

ISBN 10:   3540620583
Pages:   474
Publication Date:   27 February 1997
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Replaced By:   9783540425441
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 Scientific Modeling.- 1.1 Reality Versus Model.- 1.2 The Model Subject and the Model.- 1.2.1 The Purpose of a Model.- 1.3 The Model Subject and Reality.- 1.4 Model Building.- 1.4.1 Problem Specification.- 1.4.2 Creating a Structural Concept.- 1.4.3 Choosing and Designing a Model.- 1.4.4 Establishing Parameter Values.- 1.4.5 Tests and Validation.- 2 Fundamental Principles of Numerical Methods.- 2.1 From Application Problems to their Numerical Solution.- 2.1.1 Case Study: Pendulum.- 2.1.2 Qualitative and Quantitative Problems.- 2.2 Numerical Problems.- 2.2.1 Numerical Problems.- 2.2.2 Categories of Numerical Problems.- 2.2.3 The Accuracy of Numerical Results.- 2.3 Types of Errors in Numerics.- 2.3.1 Model Errors.- 2.3.2 Data Errors.- 2.3.3 Algorithm Errors.- 2.3.4 Rounding Errors.- 2.3.5 Error Hierarchy.- 2.4 The Condition of Mathematical Problems.- 2.4.1 Perturbed and Unperturbed Problems.- 2.4.2 The Absolute Condition Number.- 2.4.3 The Relative Condition Number.- 2.4.4 Narrowing the Problem Class.- 2.4.5 Calculating Condition Numbers Using Differentiation.- 2.4.6 Case Study: Quadratic Equation.- 2.4.7 Ill-Conditioned Problems.- 2.4.8 Ill-Posed Problems.- 2.5 The Condition of Application Problems.- 2.6 The Mathematical Elements of Condition Estimation.- 2.6.1 The Condition of Direct Mathematical Problems.- 2.6.2 The Condition of Inverse Mathematical Problems.- 2.7 Validation of Numerical Computations.- 2.7.1 The Uncertainty of Numerical Computations.- 2.7.2 Validation of Mathematical Models.- 2.7.3 Sensitivity Analysis and Error Estimation.- 2.7.4 Validation of Numerical Software.- 3 Computers for Numerical Data Processing.- 3.1 Processors.- 3.1.1 Pipelining.- 3.1.2 Superpipeline Architecture.- 3.1.3 Superscalar Architectures.- 3.1.4 Vector Processors.- 3.2 Memory.- 3.2.1 Memory Hierarchy.- 3.2.2 Addressing Schemes.- 3.2.3 Registers.- 3.2.4 Cache Memory.- 3.2.5 Virtual Memory.- 3.2.6 Interleaved Memory.- 3.3 Performance Quantification.- 3.3.1 The Notion of Performance (Power).- 3.3.2 Time as a Factor in Performance.- 3.4 Analytical Performance Assessment.- 3.4.1 Peak Performance.- 3.4.2 Instruction Performance.- 3.4.3 Performance of Vector Processors.- 3.4.4 The Influence of Memory on Performance.- 3.5 Empirical Performance Assessment.- 3.5.1 Temporal Performance.- 3.5.2 Empirical Instruction Performance.- 3.5.3 Empirical Floating-Point Performance.- 3.5.4 Empirical Performance of Vector Processors.- 4 Numerical Data and Numerical Operations.- 4.1 Mathematical Data.- 4.1.1 Fundamental Mathematical Data.- 4.1.2 Algebraic Data.- 4.1.3 Analytic Data.- 4.2 Numerical Data on Computers.- 4.2.1 Fundamental Numerical Data.- 4.2.2 Algebraic Data.- 4.2.3 Analytic Data.- 4.2.4 Numerical Data Types.- 4.3 Operations on Numerical Data.- 4.3.1 Arithmetic Operations.- 4.3.2 Algebraic Operations.- 4.3.3 Array Processing in Fortran 90.- 4.3.4 Analytic Operations.- 4.4 Number Systems on Computers.- 4.4.1 INTEGER Number Systems.- 4.4.2 Fixed-Point Number Systems.- 4.4.3 Floating-Point Number Systems.- 4.5 Structure of Floating-Point Systems.- 4.5.1 The Number of Floating-Point Numbers.- 4.5.2 Largest and Smallest Floating-Point Numbers.- 4.5.3 Distances Between Floating-Point Numbers.- 4.5.4 Relative Distances Between Floating-Point Numbers.- 4.5.5 Case Study: IF(10, 6, -9, 9, true).- 4.6 Standardization of Floating-Point Number Systems.- 4.6.1 The IEEE Standard for Floating-Point Numbers.- 4.6.2 Hidden Bit.- 4.6.3 Infinity, NaNs, and Signed Zero.- 4.6.4 Impact on Programming Languages.- 4.7 Arithmetics for Floating-Point Systems.- 4.7.1 Rounding Functions.- 4.7.2 Rounding Error.- 4.7.3 Rounding and Arithmetic Operations.- 4.7.4 Implementation of a Floating-Point Arithmetic.- 4.7.5 Multiple-Precision Software.- 4.8 Inquiry Functions and Manipulation of Numbers in Fortran 90.- 4.8.1 Parameters for Floating-Point Numbers.- 4.8.2 Characteristics of Floating-Point Numbers.- 4.8.3 The Distance Between Floating-Point Numbers, Rounding.- 4.8.4 Manipulation of Floating-Point Numbers.- 4.8.5 Parameters in INTEGER Numbers.- 4.8.6 Case Study: Multiple Products.- 4.9 Operations with Algebraic Data.- 4.10 Operations with Arrays.- 4.10.1 BLAS.- 4.11 Operations with Analytic Data.- 4.11.1 Representation of Functions.- 4.11.2 Implementation of Functions.- 4.11.3 Operations with Functions.- 4.11.4 Functions as Results.- 5 Numerical Algorithms.- 5.1 The Intuitive Notion of an Algorithm.- 5.2 Properties of Algorithms.- 5.2.1 Abstraction.- 5.2.2 Generality.- 5.2.3 Finiteness.- 5.2.4 Termination.- 5.2.5 Deterministic Algorithms.- 5.2.6 Determinate Algorithms.- 5.3 Existence of Algorithms.- 5.3.1 Precise Definitions of Algorithm .- 5.3.2 Computable Functions.- 5.4 Practical Solvability of Problems.- 5.5 Complexity of Algorithms.- 5.5.1 Abstract Computer Models.- 5.5.2 Theoretical Execution Cost.- 5.5.3 Asymptotic Complexity of Algorithms.- 5.5.4 Problem Complexity.- 5.5.5 Case Study: Multiplication of Matrices.- 5.5.6 Practical Effort Determination.- 5.6 Representation of Algorithms.- 5.6.1 Fortran 90.- 5.6.2 Pseudocode.- 5.7 Influence of Rounding Errors on Numerical Algorithms.- 5.7.1 Arithmetic Algorithms.- 5.7.2 Implementation of Arithmetic Algorithms.- 5.7.3 Error Propagation.- 5.7.4 Error Propagation Analysis.- 5.8 Case Study: Floating-Point Summation.- 5.8.1 Pairwise Summation.- 5.8.2 Compensated Summation.- 5.8.3 Comparison of Summation Methods.- 5.8.4 Ascending Summation.- 6 Numerical Programs.- 6.1 The Quality of Numerical Programs.- 6.1.1 Reliability.- 6.1.2 Portability.- 6.1.3 Efficiency.- 6.2 Reasons for Poor Efficiency.- 6.2.1 Lack of Parallelism.- 6.2.2 Lack of Locality of Reference.- 6.2.3 Inadequate Memory Access Patterns.- 6.2.4 Overhead.- 6.3 The Measurement of Performance Indices.- 6.3.1 Measurement of the Workload.- 6.3.2 Measurement of the Time.- 6.3.3 Profiling.- 6.3.4 Spotting Performance Decreasing Factors.- 6.4 Performance Optimization.- 6.5 Architecture Independent Optimizations.- 6.6 Loop Optimizations.- 6.6.1 Loop Unrolling.- 6.6.2 Unrolling Nested Loops.- 6.6.3 Loop Fusion.- 6.6.4 The Elimination of Loop Branches.- 6.6.5 Associative Transformations.- 6.6.6 Loop Interchanges.- 6.7 Blocked Memory Access.- 6.7.1 Hierarchical Blocking.- 6.8 Case Study: Multiplication of Matrices.- 6.8.1 Loop Interchanges.- 6.8.2 Loop Unrolling.- 6.8.3 Blocking.- 6.8.4 Block Copying.- 6.8.5 Blocking, Copying and Loop Unrolling.- 6.8.6 Optimizing System Software.- 7 Available Numerical Software.- 7.1 The Cost of Software.- 7.2 Sources of Numerical Software.- 7.2.1 Numerical Application Software.- 7.2.2 Printed Programs.- 7.2.3 Numerical Software Libraries.- 7.2.4 Numerical Software Packages.- 7.2.5 References to Software in this Book.- 7.3 Software and the Internet.- 7.3.1 The Internet.- 7.3.2 Communication in the Internet, E-Mail.- 7.3.3 Forums for Discussion on the Internet.- 7.3.4 Resource Sharing on the Internet.- 7.3.5 Finding Software in the Internet.- 7.3.6 Internet Front Ends.- 7.3.7 NETLIB.- 7.3.8 eLib.- 7.3.9 GAMS.- 7.4 Interactive Multifunctional Systems.- 7.4.1 Exploratory Systems.- 7.4.2 Numerical Systems.- 7.4.3 Symbolic Systems.- 7.4.4 Simulation Systems.- 7.5 Problem Solving Environments.- 7.5.1 Available Problem Solving Environments.- 7.6 Case Study: Software for Elliptic PDEs.- 7.6.1 Problem Classification.- 7.6.2 Software Packages for Elliptic Problems.- 7.6.3 Numerical Program Library Sections.- 7.6.4 TOMS Programs.- 8 Using Approximation in Mathematical Model Building.- 8.1 Analytic Models.- 8.1.1 Elementary Functions as Models.- 8.1.2 Algorithms Used as Mathematical Models.- 8.2 Information and Data.- 8.2.1 Obtaining Algebraic Data from Discrete Information.- 8.2.2 Obtaining Analytic Data from Continuous Information.- 8.2.3 The Discretization of Continuous Information.- 8.2.4 The Homogenization of Discrete Data.- 8.3 Discrete Approximation.- 8.3.1 Discrete Approximation Data.- 8.3.2 Interpolation.- 8.3.3 Case Study: Water Resistance of Vessels.- 8.4 Function Approximation.- 8.4.1 Nonadaptive Discretization.- 8.4.2 Adaptive Discretization.- 8.4.3 Homogenization Algorithms.- 8.4.4 Additional Information.- 8.5 Choosing a Model Function.- 8.5.1 Uniform or Piecewise Approximation.- 8.5.2 Linear Approximation.- 8.5.3 Nonlinear Approximation.- 8.5.4 Global or Local Approximation.- 8.5.5 Continuity and Differentiability.- 8.5.6 Condition.- 8.5.7 Invariance under Scaling.- 8.5.8 Constraints.- 8.5.9 Appearance.- 8.6 Choice of the Distance Function.- 8.6.1 Mathematical Foundations.- 8.6.2 Norms for Finite-Dimensional Spaces.- 8.6.3 Norms for Infinite-Dimensional Spaces.- 8.6.4 Weighted Norms.- 8.6.5 Hamming Distance.- 8.6.6 Robust Distance Functions.- 8.6.7 Orthogonal Approximation.- 8.7 Transformation of the Problem.- 8.7.1 Ordinate Transformation.- 8.7.2 Curves.- 9 Interpolation.- 9.1 Interpolation Problems.- 9.1.1 Choosing a Class of Functions.- 9.1.2 Determination of the Parameters of an Interpolation Function.- 9.1.3 Manipulation of the Interpolation Function.- 9.2 Mathematical Foundations.- 9.2.1 The General Interpolation Problem.- 9.2.2 Interpolation of Function Values.- 9.3 Univariate Polynomial Interpolation.- 9.3.1 Univariate Polynomials.- 9.3.2 Representation Forms of Univariate Polynomials.- 9.3.3 The Calculation of Coefficients.- 9.3.4 The Evaluation of Polynomials.- 9.3.5 Error and Convergence Behavior.- 9.3.6 Algorithm Error in Polynomial Interpolation.- 9.3.7 The Convergence of Interpolation Polynomials.- 9.3.8 The Conditioning of Polynomial Interpolation.- 9.3.9 Choosing Interpolation Nodes.- 9.3.10 Hermite Interpolation.- 9.4 Univariate, Piecewise, Polynomial Interpolation.- 9.4.1 The Accuracy of the Approximation.- 9.4.2 Evaluation.- 9.5 Polynomial Splines.- 9.5.1 Oscillations and Sensitivity to Perturbations.- 9.5.2 The Representation of Polynomial Splines.- 9.6 B-Splines.- 9.6.1 Choice of the B-Spline Break Points.- 9.6.2 B-Splines in Graphical Data Processing.- 9.6.3 Software for B-Splines.- 9.7 Cubic Spline Interpolation.- 9.7.1 Boundary Conditions.- 9.7.2 Extremal Property.- 9.7.3 Error and Convergence Behavior.- 9.7.4 The Calculation of Coefficients.- 9.7.5 The Evaluation of Spline Functions.- 9.7.6 Condition.- 9.8 Splines Without Undesirable Oscillations.- 9.8.1 Exponential Splines.- 9.8.2 v-Splines.- 9.8.3 The Akima Subspline Interpolation.- 9.9 Multivariate Interpolation.- 9.9.1 Tensor Product Interpolation.- 9.9.2 Triangulation.- 9.10 Multivariate Polynomial Interpolation.- 9.11 Multivariate (Sub-) Spline Interpolation.- 9.11.1 Tensor Product Spline Functions.- 9.11.2 Polynomial Interpolation on Triangles.- 9.12 Related Problems and Methods.- Glossary of Notation.- Author Index.

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

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

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<p>... 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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