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An Introduction to Numerical Methods and Analysis, 2nd Edition

Book Description

Praise for the First Edition

". . . outstandingly appealing with regard to its style, contents, considerations of requirements of practice, choice of examples, and exercises."—Zentralblatt MATH

". . . carefully structured with many detailed worked examples."—The Mathematical Gazette

The Second Edition of the highly regarded An Introduction to Numerical Methods and Analysis provides a fully revised guide to numerical approximation. The book continues to be accessible and expertly guides readers through the many available techniques of numerical methods and analysis.

An Introduction to Numerical Methods and Analysis, Second Edition reflects the latest trends in the field, includes new material and revised exercises, and offers a unique emphasis on applications. The author clearly explains how to both construct and evaluate approximations for accuracy and performance, which are key skills in a variety of fields. A wide range of higher-level methods and solutions, including new topics such as the roots of polynomials, spectral collocation, finite element ideas, and Clenshaw-Curtis quadrature, are presented from an introductory perspective, and the Second Edition also features:

  • Chapters and sections that begin with basic, elementary material followed by gradual coverage of more advanced material

  • Exercises ranging from simple hand computations to challenging derivations and minor proofs to programming exercises

  • Widespread exposure and utilization of MATLAB

  • An appendix that contains proofs of various theorems and other material

  • The book is an ideal textbook for students in advanced undergraduate mathematics and engineering courses who are interested in gaining an understanding of numerical methods and numerical analysis.

    Table of Contents

    1. Cover
    2. Half Title page
    3. Title page
    4. Copyright page
    5. Dedication
    6. Preface
    7. Chapter 1: Introductory Concepts and Calculus Review
      1. 1.1 Basic Tools of Calculus
      2. 1.2 Error, Approximate Equality, and Asymptotic Order Notation
      3. 1.3 A Primer on Computer Arithmetic
      4. 1.4 A Word on Computer Languages and Software
      5. 1.5 Simple Approximations
      6. 1.6 Application: Approximating the Natural Logarithm
      7. 1.7 A Brief History of Computing
      8. 1.8 Literature Review
      9. References
    8. Chapter 2: A Survey of Simple Methods and Tools
      1. 2.1 Horner’s Rule and Nested Multiplication
      2. 2.2 Difference Approximations to the Derivative
      3. 2.3 Application: Euler’s Method for Initial Value Problems
      4. 2.4 Linear Interpolation
      5. 2.5 Application—The Trapezoid Rule
      6. 2.6 Solution of Tridiagonal Linear Systems
      7. 2.7 Application: Simple Two-Point Boundary Value Problems
    9. Chapter 3: Root-Finding
      1. 3.1 The Bisection Method
      2. 3.2 Newton’s Method: Derivation and Examples
      3. 3.3 How to Stop Newton’s Method
      4. 3.4 Application: Division Using Newton’s Method
      5. 3.5 The Newton Error Formula
      6. 3.6 Newton’s Method: Theory and Convergence
      7. 3.7 Application: Computation of the Square Root
      8. 3.8 The Secant Method: Derivation and Examples
      9. 3.9 Fixed-Point Iteration
      10. 3.10 Roots of Polynomials, Part 1
      11. 3.11 Special Topics in Root-Finding Methods
      12. 3.12 Very High-Order Methods and the Efficiency Index
      13. 3.13 Literature and Software Discussion
      14. References
    10. Chapter 4: Interpolation and Approximation
      1. 4.1 Lagrange Interpolation
      2. 4.2 Newton Interpolation and Divided Differences
      3. 4.3 Interpolation Error
      4. 4.4 Application: Muller’s Method and Inverse Quadratic Interpolation
      5. 4.5 Application: More Approximations to the Derivative
      6. 4.6 Hermite Interpolation
      7. 4.7 Piecewise Polynomial Interpolation
      8. 4.8 An Introduction to Splines
      9. 4.9 Application: Solution of Boundary Value Problems
      10. 4.10 Tension Splines
      11. 4.11 Least Squares Concepts in Approximation
      12. 4.12 Advanced Topics in Interpolation Error
      13. 4.13 Literature and Software Discussion
      14. References
    11. Chapter 5: Numerical Integration
      1. 5.1 A Review of the Definite Integral
      2. 5.2 Improving the Trapezoid Rule
      3. 5.3 Simpson’s Rule and Degree of Precision
      4. 5.4 The Midpoint Rule
      5. 5.5 Application: Stirling’s Formula
      6. 5.6 Gaussian Quadrature
      7. 5.7 Extrapolation Methods
      8. 5.8 Special Topics in Numerical Integration
      9. 5.9 Literature and Software Discussion
      10. References
    12. Chapter 6: Numerical Methods for Ordinary Differential Equations
      1. 6.1 The Initial Value Problem: Background
      2. 6.2 Euler’s Method
      3. 6.3 Analysis of Euler’s Method
      4. 6.4 Variants of Euler’s Method
      5. 6.5 Single-Step Methods: Runge–Kutta
      6. 6.6 Multistep Methods
      7. 6.7 Stability Issues
      8. 6.8 Application to Systems of Equations
      9. 6.9 Adaptive Solvers
      10. 6.10 Boundary Value Problems
      11. 6.11 Literature and Software Discussion
      12. References
    13. Chapter 7: Numerical Methods for the Solution of Systems of Equations
      1. 7.1 Linear Algebra Review
      2. 7.2 Linear Systems and Gaussian Elimination
      3. 7.3 Operation Counts
      4. 7.4 The LU Factorization
      5. 7.5 Perturbation, Conditioning, and Stability
      6. 7.6 SPD Matrices and the Cholesky Decomposition
      7. 7.7 Iterative Methods for Linear Systems: A Brief Survey
      8. 7.8 Nonlinear Systems: Newton’s Method and Related Ideas
      9. 7.9 Application: Numerical Solution of Nonlinear Boundary Value Problems
      10. 7.10 Literature and Software Discussion
      11. References
    14. Chapter 8: Approximate Solution of the Algebraic Eigenvalue Problem
      1. 8.1 Eigenvalue Review
      2. 8.2 Reduction to Hessenberg Form
      3. 8.3 Power Methods
      4. 8.4 An Overview of the QR Iteration
      5. 8.5 Application: Roots of Polynomials, Part II
      6. 8.6 Literature and Software Discussion
      7. References
    15. Chapter 9: A Survey of Numerical Methods for Partial Differential Equations
      1. 9.1 Difference Methods for the Diffusion Equation
      2. 9.2 Finite Element Methods for the Diffusion Equation
      3. 9.3 Difference Methods for Poisson Equations
      4. 9.4 Literature and Software Discussion
      5. References
    16. Chapter 10: An Introduction to Spectral Methods
      1. 10.1 Spectral Methods for Two-Point Boundary Value Problems
      2. 10.2 Spectral Methods for Time-Dependent Problems
      3. 10.3 Clenshaw–Curtis Quadrature
      4. 10.4 Literature and Software Discussion
      5. References
    17. Appendix A: Proofs of Selected Theorems, and Additional Material
      1. A.1 Proofs of the Interpolation Error Theorems
      2. A.2 Proof of the Stability Result for ODEs
      3. A.3 Stiff Systems of Differential Equations and Eigenvalues
      4. A.4 The Matrix Perturbation Theorem
    18. Index