Computational science is fundamentally changing how technological questions are addressed. The design of aircraft, automobiles, and even racing sailboats is now done by computational simulation. The mathematical foundation of this new approach is numerical analysis, which studies algorithms for computing expressions defined with real numbers. Emphasizing the theory behind the computation, this book provides a rigorous and self-contained introduction to numerical analysis and presents the advanced mathematics that underpin industrial software, including complete details that are missing from most textbooks. Using an inquiry-based learning approach, Numerical Analysis is written in a narrative style, provides historical background, and includes many of the proofs and technical details in exercises. Students will be able to go beyond an elementary understanding of numerical simulation and develop deep insights into the foundations of the subject. They will no longer have to accept the mathematical gaps that exist in current textbooks. For example, both necessary and sufficient conditions for convergence of basic iterative methods are covered, and proofs are given in full generality, not just based on special cases. The book is accessible to undergraduate mathematics majors as well as computational scientists wanting to learn the foundations of the subject. * Presents the mathematical foundations of numerical analysis * Explains the mathematical details behind simulation software * Introduces many advanced concepts in modern analysis * Self-contained and mathematically rigorous * Contains problems and solutions in each chapter * Excellent follow-up course to Principles of Mathematical Analysis by Rudin

Table of Contents:
Preface xi Chapter 1. Numerical Algorithms 1 1.1 Finding roots 2 1.2 Analyzing Heron's algorithm 5 1.3 Where to start 6 1.4 An unstable algorithm 8 1.5 General roots: effects of floating-point 9 1.6 Exercises 11 1.7 Solutions 13 Chapter 2. Nonlinear Equations 15 2.1 Fixed-point iteration 16 2.2 Particular methods 20 2.3 Complex roots 25 2.4 Error propagation 26 2.5 More reading 27 2.6 Exercises 27 2.7 Solutions 30 Chapter 3. Linear Systems 35 3.1 Gaussian elimination 36 3.2 Factorization 38 3.3 Triangular matrices 42 3.4 Pivoting 44 3.5 More reading 47 3.6 Exercises 47 3.7 Solutions 50 Chapter 4. Direct Solvers 51 4.1 Direct factorization 51 4.2 Caution about factorization 56 4.3 Banded matrices 58 4.4 More reading 60 4.5 Exercises 60 4.6 Solutions 63 Chapter 5. Vector Spaces 65 5.1 Normed vector spaces 66 5.2 Proving the triangle inequality 69 5.3 Relations between norms 71 5.4 Inner-product spaces 72 5.5 More reading 76 5.6 Exercises 77 5.7 Solutions 79 Chapter 6. Operators 81 6.1 Operators 82 6.2 Schur decomposition 84 6.3 Convergent matrices 89 6.4 Powers of matrices 89 6.5 Exercises 92 6.6 Solutions 95 Chapter 7. Nonlinear Systems 97 7.1 Functional iteration for systems 98 7.2 Newton's method 103 7.3 Limiting behavior of Newton's method 108 7.4 Mixing solvers 110 7.5 More reading 111 7.6 Exercises 111 7.7 Solutions 114 Chapter 8. Iterative Methods 115 8.1 Stationary iterative methods 116 8.2 General splittings 117 8.3 Necessary conditions for convergence 123 8.4 More reading 128 8.5 Exercises 128 8.6 Solutions 131 Chapter 9. Conjugate Gradients 133 9.1 Minimization methods 133 9.2 Conjugate Gradient iteration 137 9.3 Optimal approximation of CG 141 9.4 Comparing iterative solvers 147 9.5 More reading 147 9.6 Exercises 148 9.7 Solutions 149 Chapter 10. Polynomial Interpolation 151 10.1 Local approximation: Taylor's theorem 151 10.2 Distributed approximation: interpolation 152 10.3 Norms in infinite-dimensional spaces 157 10.4 More reading 160 10.5 Exercises 160 10.6 Solutions 163 Chapter 11. Chebyshev and Hermite Interpolation 167 11.1 Error term ! 167 11.2 Chebyshev basis functions 170 11.3 Lebesgue function 171 11.4 Generalized interpolation 173 11.5 More reading 177 11.6 Exercises 178 11.7 Solutions 180 Chapter 12. Approximation Theory 183 12.1 Best approximation by polynomials 183 12.2 Weierstrass and Bernstein 187 12.3 Least squares 191 12.4 Piecewise polynomial approximation 193 12.5 Adaptive approximation 195 12.6 More reading 196 12.7 Exercises 196 12.8 Solutions 199 Chapter 13. Numerical Quadrature 203 13.1 Interpolatory quadrature 203 13.2 Peano kernel theorem 209 13.3 Gregorie-Euler-Maclaurin formulas 212 13.4 Other quadrature rules 219 13.5 More reading 221 13.6 Exercises 221 13.7 Solutions 224 Chapter 14. Eigenvalue Problems 225 14.1 Eigenvalue examples 225 14.2 Gershgorin's theorem 227 14.3 Solving separately 232 14.4 How not to eigen 233 14.5 Reduction to Hessenberg form 234 14.6 More reading 237 14.7 Exercises 238 14.8 Solutions 240 Chapter 15. Eigenvalue Algorithms 241 15.1 Power method 241 15.2 Inverse iteration 250 15.3 Singular value decomposition 252 15.4 Comparing factorizations 253 15.5 More reading 254 15.6 Exercises 254 15.7 Solutions 256 Chapter 16. Ordinary Differential Equations 257 16.1 Basic theory of ODEs 257 16.2 Existence and uniqueness of solutions 258 16.3 Basic discretization methods 262 16.4 Convergence of discretization methods 266 16.5 More reading 269 16.6 Exercises 269 16.7 Solutions 271 Chapter 17. Higher-order ODE Discretization Methods 275 17.1 Higher-order discretization 276 17.2 Convergence conditions 281 17.3 Backward differentiation formulas 287 17.4 More reading 288 17.5 Exercises 289 17.6 Solutions 291 Chapter 18. Floating Point 293 18.1 Floating-point arithmetic 293 18.2 Errors in solving systems 301 18.3 More reading 305 18.4 Exercises 305 18.5 Solutions 308 Chapter 19. Notation 309 Bibliography 311 Index 323