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Numerical Analysis for Engineers and Scientists

Book Description

Striking a balance between theory and practice, this graduate-level text is perfect for students in the applied sciences. The author provides a clear introduction to the classical methods, how they work and why they sometimes fail. Crucially, he also demonstrates how these simple and classical techniques can be combined to address difficult problems. Many worked examples and sample programs are provided to help the reader make practical use of the subject material. Further mathematical background, if required, is summarized in an appendix. Topics covered include classical methods for linear systems, eigenvalues, interpolation and integration, ODEs and data fitting, and also more modern ideas like adaptivity and stochastic differential equations.

Table of Contents

  1. Cover
  2. Half title
  3. Title
  4. Copyright
  5. Table of Contents
  6. Preface
  7. 1 Numerical error
    1. 1.1 Types of error
    2. 1.2 Floating point numbers
    3. 1.3 Algorithms and error
    4. 1.4 Approximation error vs. algorithm error
    5. 1.5 An important example
    6. 1.6 Backward error
    7. Problems
  8. 2 Direct solution of linear systems
    1. 2.1 Gaussian elimination
    2. 2.2 Pivot selection
    3. 2.3 Errors in Gaussian elimination
    4. 2.4 Householder reduction
    5. 2.5 Cholesky decomposition
    6. 2.6 The residual correction method
    7. Problems
  9. 3 Eigenvalues and eigenvectors
    1. 3.1 Gerschgorin’s estimate
    2. 3.2 The power method
    3. 3.3 The QR algorithm
    4. 3.4 Singular value decomposition
    5. 3.5 Hyman’s method
    6. Problems
  10. 4 Iterative approaches for linear systems
    1. 4.1 Conjugate gradient
    2. 4.2 Relaxation methods
    3. 4.3 Jacobi
    4. 4.4 Irreducibility
    5. 4.5 Gauss–Seidel
    6. 4.6 Multigrid
    7. Problems
  11. 5 Interpolation
    1. 5.1 Modified Lagrange interpolation and the barycentric form
    2. 5.2 Neville’s algorithm
    3. 5.3 Newton
    4. 5.4 Hermite
    5. 5.5 Discrete Fourier transform
    6. Problems
  12. 6 Iterative methods and the roots of polynomials
    1. 6.1 Convergence and rates
    2. 6.2 Bisection
    3. 6.3 Regula falsi
    4. 6.4 The secant method
    5. 6.5 Newton–Raphson
    6. 6.6 Roots of a polynomial
    7. 6.7 Newton–Raphson on the complex plane
    8. 6.8 Bairstow’s method
    9. 6.9 Improving convergence
    10. Problems
  13. 7 Optimization
    1. 7.1 1D: Bracketing
    2. 7.2 1D: Refinement by interpolation
    3. 7.3 1D: Refinement by golden section search
    4. 7.4 nD: Variable metric methods
    5. 7.5 Linear programming
    6. 7.6 Quadratic programming
    7. Problems
  14. 8 Data fitting
    1. 8.1 Least squares
    2. 8.2 An application to the Taylor series
    3. 8.3 Data with experimental error
    4. 8.4 Error in x and y
    5. 8.5 Nonlinear least squares
    6. 8.6 Fits in other norms
    7. 8.7 Splines
    8. Problems
  15. 9 Integration
    1. 9.1 Newton–Cotes
    2. 9.2 Extrapolation
    3. 9.3 Adaptivity
    4. 9.4 Gaussian quadrature
    5. 9.5 Special cases
    6. Problems
  16. 10 Ordinary differential equations
    1. 10.1 Initial value problems I: one-step methods
    2. 10.2 Initial value problems II: multistep methods
    3. 10.3 Adaptivity
    4. 10.4 Boundary value problems
    5. 10.5 Stiff systems
    6. Problems
  17. 11 Introduction to stochastic ODEs
    1. 11.1 White noise and the Wiener processs
    2. 11.2 Itô and Stratonovich calculus
    3. 11.3 Itô’s formula
    4. 11.4 The Itô–Taylor series
    5. 11.5 Orders of accuracy
    6. 11.6 Strong convergence
    7. 11.7 Weak convergence
    8. 11.8 Modeling
    9. Problems
  18. 12 A big integrative example
    1. 12.1 The Schrödinger equation
    2. 12.2 Gaussian basis functions
    3. 12.3 Results I: H2
    4. 12.4 Angular momentum
    5. 12.5 Rys polynomials
    6. 12.6 Results II: H2O
  19. Appendix A Mathematical background
    1. A.1 Continuity
    2. A.2 Triangle inequality
    3. A.3 Rolle’s theorem
    4. A.4 Mean value theorem
    5. A.5 Geometric series
    6. A.6 Taylor series
    7. A.7 Linear algebra
    8. A.8 Complex numbers
  20. Appendix B Sample codes
    1. B.1 Utility routines
    2. B.2 Gaussian elimination
    3. B.3 Householder reduction
    4. B.4 Cholesky reduction
    5. B.5 The QR method with shifts for symmetric real matrices
    6. B.6 Singular value decomposition
    7. B.7 Conjugate gradient
    8. B.8 Jacobi, Gauss–Seidel, and multigrid
    9. B.9 Cooley–Tukey FFT
    10. B.10 Variable metric methods
    11. B.11 The simplex method for linear programming
    12. B.12 Quadratic programming for convex systems
    13. B.13 Adaptive Simpson’s rule integration
    14. B.14 Adaptive Runge–Kutta ODE example
    15. B.15 Adaptive multistep ODE example
    16. B.16 Stochastic integration and testing
    17. B.17 Big example: Hartree–Fock–Roothaan
  21. Solutions
  22. References
  23. Index