Contents vii 5.3. Summary of Method Behavior on Model Problems 169 5.4. Paired Methods: Error, Step-Size, Order Control 177 5.5. Behavior of Example Methods on a Model 2 × 2 System 180 5.6. Stiff Systems and the Method of Lines 187 5.7. Convergence Analysis: Euler’s Method 213 Appendix A. Linear Algebra and Analysis 225 A.1. Metric and Normed Spaces 225 A.2. Inner-Product Spaces 227 Appendix B. The Magic of Iteration 233 B.1. The Banach Contraction Principle 233 B.2. Newton’s Method 238 B.3. The Inverse Function Theorem 240 B.4. The Existence and Uniqueness Theorem for ODE 241 Appendix C. Vector Fields as Differential Operators 243 Appendix D. Coordinate Systems and Canonical Forms 247 D.1. Local Coordinates 247 D.2. Some Canonical Forms 250 Appendix E. Parametrized Curves and Arclength 255 Appendix F. Smoothness with Respect to Initial Conditions 257 Appendix G. Canonical Form for Linear Operators 259 G.1. The Spectral Theorem 259 Appendix H. Runge-Kutta Methods 263

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