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* This book is an introduction to numerical analysis and intends to strike a balance between analytical rigor and the treatment of particular methods for engineering problems * Emphasizes the earlier stages of numerical analysis for engineers with real-life problem-solving solutions applied to computing and engineering * Includes MATLAB oriented examples An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.
* This book is an introduction to numerical analysis and intends to strike a balance between analytical rigor and the treatment of particular methods for engineering problems * Emphasizes the earlier stages of numerical analysis for engineers with real-life problem-solving solutions applied to computing and engineering * Includes MATLAB oriented examples An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.
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Autorenporträt
Christopher?J. Zarowski, PhD,?is an associate professor?in the? Department of Electrical and Computer Engineering at the ?University of Alberta, Canada. His research areas include digital communications applications (wireless, wireline, optical fiber), biomedical applications (e.g., circadian rhythm parameter estimation), structured matrix algebra, wavelet methods, signal detection and parameter estimation, computationally efficient and numerically reliable algorithms, and parallel processing algorithms. He has authored over 100 journal articles and conference papers, and?is a senior member of the IEEE.
Inhaltsangabe
Preface xiii 1 Functional Analysis Ideas 1 1.1 Introduction 1 1.2 Some Sets 2 1.3 Some Special Mappings: Metrics, Norms, and Inner Products 4 1.3.1 Metrics and Metric Spaces 6 1.3.2 Norms and Normed Spaces 8 1.3.3 Inner Products and Inner Product Spaces 14 1.4 The Discrete Fourier Series (DFS) 25 Appendix 1.A Complex Arithmetic 28 Appendix 1.B Elementary Logic 31 References 32 Problems 33 2 Number Representations 38 2.1 Introduction 38 2.2 Fixed-Point Representations 38 2.3 Floating-Point Representations 42 2.4 Rounding Effects in Dot Product Computation 48 2.5 Machine Epsilon 53 Appendix 2.A Review of Binary Number Codes 54 References 59 Problems 59 3 Sequences and Series 63 3.1 Introduction 63 3.2 Cauchy Sequences and Complete Spaces 63 3.3 Pointwise Convergence and Uniform Convergence 70 3.4 Fourier Series 73 3.5 Taylor Series 78 3.6 Asymptotic Series 97 3.7 More on the Dirichlet Kernel 103 3.8 Final Remarks 107 Appendix 3.A COordinate Rotation DIgital Computing (CORDIC) 107 3.A.1 Introduction 107 3.A.2 The Concept of a Discrete Basis 108 3.A.3 Rotating Vectors in the Plane 112 3.A.4 Computing Arctangents 114 3.A.5 Final Remarks 115 Appendix 3.B Mathematical Induction 116 Appendix 3.C Catastrophic Cancellation 117 References 119 Problems 120 4 Linear Systems of Equations 127 4.1 Introduction 127 4.2 Least-Squares Approximation and Linear Systems 127 4.3 Least-Squares Approximation and Ill-Conditioned Linear Systems 132 4.4 Condition Numbers 135 4.5 LU Decomposition 148 4.6 Least-Squares Problems and QR Decomposition 161 4.7 Iterative Methods for Linear Systems 176 4.8 Final Remarks 186 Appendix 4.A Hilbert Matrix Inverses 186 Appendix 4.B SVD and Least Squares 191 References 193 Problems 194 5 Orthogonal Polynomials 207 5.1 Introduction 207 5.2 General Properties of Orthogonal Polynomials 207 5.3 Chebyshev Polynomials 218 5.4 Hermite Polynomials 225 5.5 Legendre Polynomials 229 5.6 An Example of Orthogonal Polynomial Least-Squares Approximation 235 5.7 Uniform Approximation 238 References 241 Problems 241 6 Interpolation 251 6.1 Introduction 251 6.2 Lagrange Interpolation 252 6.3 Newton Interpolation 257 6.4 Hermite Interpolation 266 6.5 Spline Interpolation 269 References 284 Problems 285 7 Nonlinear Systems of Equations 290 7.1 Introduction 290 7.2 Bisection Method 292 7.3 Fixed-Point Method 296 7.4 Newton-Raphson Method 305 7.4.1 The Method 305 7.4.2 Rate of Convergence Analysis 309 7.4.3 Breakdown Phenomena 311 7.5 Systems of Nonlinear Equations 312 7.5.1 Fixed-Point Method 312 7.5.2 Newton-Raphson Method 318 7.6 Chaotic Phenomena and a Cryptography Application 323 References 332 Problems 333 8 Unconstrained Optimization 341 8.1 Introduction 341 8.2 Problem Statement and Preliminaries 341 8.3 Line Searches 345 8.4 Newton's Method 353 8.5 Equality Constraints and Lagrange Multipliers 357 Appendix 8.A MATLAB Code for Golden Section Search 362 References 364 Problems 364 9 Numerical Integration and Differentiation 369 9.1 Introduction 369 9.2 Trapezoidal Rule 371 9.3 Simpson's Rule 378 9.4 Gaussian Quadrature 385 9.5 Romberg Integration 393 9.6 Numerical Differentiation 401 References 406 Problems 406 10 Numerical Solution of Ordinary Differential Equations 415 10.1 Introduction 415 10.2 First-Order ODEs 421 10.3 Systems of First-Order ODEs 442 10.4 Multistep Methods for ODEs 455 10.4.1 Adams-Bashforth Methods 459 10.4.2 Adams-Moulton Methods 461 10.4.3 Comments on the Adams Families 462 10.5 Variable-Step-Size (Adaptive) Methods for ODEs 464 10.6 Stiff Systems 467 10.7 Final Remarks 469 Appendix 10.A MATLAB Code for Example 10.8 469 Appendix 10.B MATLAB Code for Example 10.13 470 References 472 Problems 473 11 Numerical Methods for Eigenproblems 480 11.1 Introduction 480 11.2 Review of Eigenvalues and Eigenvectors 480 11.3 The Matrix Exponential 488 11.4 The Power Methods 498 11.5 QR Iterations 508 References 518 Problems 519 12 Numerical Solution of Partial Differential Equations 525 12.1 Introduction 525 12.2 A Brief Overview of Partial Differential Equations 525 12.3 Applications of Hyperbolic PDEs 528 12.3.1 The Vibrating String 528 12.3.2 Plane Electromagnetic Waves 534 12.4 The Finite-Difference (FD) Method 545 12.5 The Finite-Difference Time-Domain (FDTD) Method 550 Appendix 12.A MATLAB Code for Example 12.5 557 References 560 Problems 561 13 An Introduction to MATLAB 565 13.1 Introduction 565 13.2 Startup 565 13.3 Some Basic Operators, Operations, and Functions 566 13.4 Working with Polynomials 571 13.5 Loops 572 13.6 Plotting and M-Files 573 References 577 Index 579
Preface xiii 1 Functional Analysis Ideas 1 1.1 Introduction 1 1.2 Some Sets 2 1.3 Some Special Mappings: Metrics, Norms, and Inner Products 4 1.3.1 Metrics and Metric Spaces 6 1.3.2 Norms and Normed Spaces 8 1.3.3 Inner Products and Inner Product Spaces 14 1.4 The Discrete Fourier Series (DFS) 25 Appendix 1.A Complex Arithmetic 28 Appendix 1.B Elementary Logic 31 References 32 Problems 33 2 Number Representations 38 2.1 Introduction 38 2.2 Fixed-Point Representations 38 2.3 Floating-Point Representations 42 2.4 Rounding Effects in Dot Product Computation 48 2.5 Machine Epsilon 53 Appendix 2.A Review of Binary Number Codes 54 References 59 Problems 59 3 Sequences and Series 63 3.1 Introduction 63 3.2 Cauchy Sequences and Complete Spaces 63 3.3 Pointwise Convergence and Uniform Convergence 70 3.4 Fourier Series 73 3.5 Taylor Series 78 3.6 Asymptotic Series 97 3.7 More on the Dirichlet Kernel 103 3.8 Final Remarks 107 Appendix 3.A COordinate Rotation DIgital Computing (CORDIC) 107 3.A.1 Introduction 107 3.A.2 The Concept of a Discrete Basis 108 3.A.3 Rotating Vectors in the Plane 112 3.A.4 Computing Arctangents 114 3.A.5 Final Remarks 115 Appendix 3.B Mathematical Induction 116 Appendix 3.C Catastrophic Cancellation 117 References 119 Problems 120 4 Linear Systems of Equations 127 4.1 Introduction 127 4.2 Least-Squares Approximation and Linear Systems 127 4.3 Least-Squares Approximation and Ill-Conditioned Linear Systems 132 4.4 Condition Numbers 135 4.5 LU Decomposition 148 4.6 Least-Squares Problems and QR Decomposition 161 4.7 Iterative Methods for Linear Systems 176 4.8 Final Remarks 186 Appendix 4.A Hilbert Matrix Inverses 186 Appendix 4.B SVD and Least Squares 191 References 193 Problems 194 5 Orthogonal Polynomials 207 5.1 Introduction 207 5.2 General Properties of Orthogonal Polynomials 207 5.3 Chebyshev Polynomials 218 5.4 Hermite Polynomials 225 5.5 Legendre Polynomials 229 5.6 An Example of Orthogonal Polynomial Least-Squares Approximation 235 5.7 Uniform Approximation 238 References 241 Problems 241 6 Interpolation 251 6.1 Introduction 251 6.2 Lagrange Interpolation 252 6.3 Newton Interpolation 257 6.4 Hermite Interpolation 266 6.5 Spline Interpolation 269 References 284 Problems 285 7 Nonlinear Systems of Equations 290 7.1 Introduction 290 7.2 Bisection Method 292 7.3 Fixed-Point Method 296 7.4 Newton-Raphson Method 305 7.4.1 The Method 305 7.4.2 Rate of Convergence Analysis 309 7.4.3 Breakdown Phenomena 311 7.5 Systems of Nonlinear Equations 312 7.5.1 Fixed-Point Method 312 7.5.2 Newton-Raphson Method 318 7.6 Chaotic Phenomena and a Cryptography Application 323 References 332 Problems 333 8 Unconstrained Optimization 341 8.1 Introduction 341 8.2 Problem Statement and Preliminaries 341 8.3 Line Searches 345 8.4 Newton's Method 353 8.5 Equality Constraints and Lagrange Multipliers 357 Appendix 8.A MATLAB Code for Golden Section Search 362 References 364 Problems 364 9 Numerical Integration and Differentiation 369 9.1 Introduction 369 9.2 Trapezoidal Rule 371 9.3 Simpson's Rule 378 9.4 Gaussian Quadrature 385 9.5 Romberg Integration 393 9.6 Numerical Differentiation 401 References 406 Problems 406 10 Numerical Solution of Ordinary Differential Equations 415 10.1 Introduction 415 10.2 First-Order ODEs 421 10.3 Systems of First-Order ODEs 442 10.4 Multistep Methods for ODEs 455 10.4.1 Adams-Bashforth Methods 459 10.4.2 Adams-Moulton Methods 461 10.4.3 Comments on the Adams Families 462 10.5 Variable-Step-Size (Adaptive) Methods for ODEs 464 10.6 Stiff Systems 467 10.7 Final Remarks 469 Appendix 10.A MATLAB Code for Example 10.8 469 Appendix 10.B MATLAB Code for Example 10.13 470 References 472 Problems 473 11 Numerical Methods for Eigenproblems 480 11.1 Introduction 480 11.2 Review of Eigenvalues and Eigenvectors 480 11.3 The Matrix Exponential 488 11.4 The Power Methods 498 11.5 QR Iterations 508 References 518 Problems 519 12 Numerical Solution of Partial Differential Equations 525 12.1 Introduction 525 12.2 A Brief Overview of Partial Differential Equations 525 12.3 Applications of Hyperbolic PDEs 528 12.3.1 The Vibrating String 528 12.3.2 Plane Electromagnetic Waves 534 12.4 The Finite-Difference (FD) Method 545 12.5 The Finite-Difference Time-Domain (FDTD) Method 550 Appendix 12.A MATLAB Code for Example 12.5 557 References 560 Problems 561 13 An Introduction to MATLAB 565 13.1 Introduction 565 13.2 Startup 565 13.3 Some Basic Operators, Operations, and Functions 566 13.4 Working with Polynomials 571 13.5 Loops 572 13.6 Plotting and M-Files 573 References 577 Index 579
Rezensionen
"Zarkowski (Univ. of Alberta) offers this book as a general, advanced undergraduate work in numerical analysis, containing all of the usual topics." ( CHOICE , October 2004)
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