# AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

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1 AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication

2 CONTENTS Preface xiii 1 Introductory Concepts and Calculus Review Basic Tools of Calculus Taylor's Theorem Mean Value and Extreme Value Theorems Error, Approximate Equality, and Asymptotic Order Notation Error Notation: Approximate Equality Notation: Asymptotic Order A Primer on Computer Arithmetic A Word on Computer Languages and Software Simple Approximations Application: Approximating the Natural Logarithm 34 References 37 2 A Survey of Simple Methods and Tools Horner's Rule and Nested Multiplication Difference Approximations to the Derivative Application: Euler's Method for Initial Value Problems 52

3 Vi CONTENTS 2.4 Linear Interpolation Application The Trapezoid Rule Solution of Tridiagonal Linear Systems Application: Simple Two-Point Boundary Value Problems 81 3 Root-Finding The Bisection Method Newton's Method: Derivation and Examples How to Stop Newton's Method Application: Division Using Newton's Method The Newton Error Formula Newton's Method: Theory and Convergence Application: Computation of the Square Root The Secant Method: Derivation and Examples Fixed Point Iteration Special Topics in Root-finding Methods Extrapolation and Acceleration Variants of Newton's method The Secant Method: Theory and Convergence Multiple Roots In Search of Fast Global Convergence: Hybrid Algorithms Literature and Software Discussion 156 References Interpolation and Approximation Lagrange Interpolation Newton Interpolation and Divided Differences Interpolation Error Application: Muller's Method and Inverse Quadratic Interpolation Application: More Approximations to the Derivative Hermite Interpolation Piecewise Polynomial Interpolation An Introduction to Splines Definition of the Problem Cubic B-Splines Application: Solution of Boundary Value Problems Least Squares Concepts in Approximation An Introduction to Data Fitting Least Squares Approximation and Orthogonal Polynomials Advanced Topics in Interpolation Error 237

4 CONTENTS Vii Stability of Polynomial Interpolation The Runge Example The Chebyshev Nodes Literature and Software Discussion 250 References 5 Numerical Integration References 5.1 A Review of the Definite Integral 5.2 Improving the Trapezoid Rule 5.3 Simpson's Rule and Degree of Precision 5.4 The Midpoint Rule 5.5 Application: Stirling's Formula 5.6 Gaussian Quadrature 5.7 Extrapolation Methods 5.8 Special Topics in Numerical Integration Romberg Integration Quadrature with Non-smooth Integrands Adaptive Integration Peano Estimates for the Trapezoid Rule 5.9 Literature and Software Discussion 6 Numerical Methods for Ordinary Differential Equations 6.1 The Initial Value Problem Background 6.2 Euler's Method 6.3 Analysis of Euler's Method 6.4 Variants of Euler's Method The Residual and Truncation Error Implicit Methods and Predictor Corrector Schemes Starting Values and Multistep Methods The Midpoint Method and Weak Stability 6.5 Single Step Methods Runge Kutta 6.6 Multi-step Methods The Adams Families The BDF Family 6.7 Stability Issues Stability Theory for Multistep Methods Stability Regions 6.8 Application to Systems of Equations

5 Vi i I CONTENTS Implementation Issues and Examples Stiff Equations A-Stability Adaptive Solvers Boundary Value Problems Simple Difference Methods Shooting Methods Literature and Software Discussion 403 References Numerical Methods for the Solution of Systems of Equations Linear Algebra Review Linear Systems and Gaussian Elimination Operation Counts The LU Factorization Perturbation, Conditioning, and Stability Vector and Matrix Norms The Condition Number and Perturbations Estimating the Condition Number Iterative Refinement SPD Matrices and the Cholesky Decomposition Iterative Methods for Linear Systems A Brief Survey Nonlinear Systems: Newton's Method and Related Ideas Newton's Method Fixed Point Methods Application: Numerical Solution of Nonlinear Boundary Value Problems Literature and Software Discussion 465 References Approximate Solution of the Algebraic Eigenvalue Problem Eigenvalue Review Reduction to Hessenberg Form Power Methods An Overview of the QR Iteration Literature and Software Discussion 510 References 511

6 CONTENTS IX 9 A Survey of Finite Difference Methods for Partial Differential Equations Difference Methods for the Diffusion Equation The Basic Problem The Explicit Method and Stability Implicit Methods and the Crank Nicolson Method Difference Methods for Poisson Equations Discretization Banded Cholesky Solvers Iteration and the Method of Conjugate Gradients Literature and Software Discussion 545 References 547 Appendix A: Proofs of Selected Theorems, and Other Additional Material 549 A.1 Proofs of the Interpolation Error Theorems 549 A.2 Proof of the Stability Result for Smooth and Uniformly Monotone Decreasing Initial Value Problems 551 A.3 Stiff Systems of Differential Equations and Eigenvalues 552 A.4 The Matrix Perturbation Theorem 553 A.5 Answers to Selected Exercises 555 Index 569

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