Numerical Mathematics and Computing

Preface

1. Introduction   1.1 Preliminary Remarks
   1.2 Review of Taylor Series
2. Floating-Point Representation and Errors 2.1 Floating-Point Representation
   2.2 Loss of Significance
3. Locating Roots of Equations 3.1 Bisection Method
   3.2 Newton's Method
   3.3 Secant Method
4. Interpolation and Numerical Differentiation 4.1 Polynomial Interpolation
   4.2 Errors in Polynomial Interpolation
   4.3 Estimating Derivatives and Richardson Extrapolation
5. Numerical Integration 5.1 Lower and Upper Sums
   5.2 Trapezoid Rule
   5.3 Romberg Algorithm
6. Additional Topics on Numerical Integration 6.1 Simpson's Rule and Adaptive Simpson's Rule
   6.2 Gaussian Quadrature Formulas
7. Systems of Linear Equations 7.1 Naive Gaussian Elimination
   7.2 Gaussian Elimination with Scaled Partial Pivoting
   7.3 Tridiagonal and Banded Systems
8. Additional Topics on Systems of Linear Equations 8.1 Matrix Factorizations
   8.2 Iterative Solution of Linear Systems
   8.3 Eigenvalues and Eigenvectors
   8.4 Power Methods
9. Approximation by Spline Functions 9.1 First-Degree and Second-Degree Splines
   9.2 Natural Cubic Splines
   9.3 B Splines: Interpolation and Approximation
10. Ordinary Differential Equations 10.1 Taylor Series Methods
    10.2 Runge-Kutta Methods
    10.3 Stability, Adaptive Runge-Kutta Methods, and Multistep Methods
11. Systems of Ordinary Differential Equations 11.1 Methods for First-Order Systems
    11.2 Higher-Order Equations and Systems
    11.3 Adams-Bashforth-Moulton Methods
12. Smoothing of Data and the Method of Least Squares 12.1 Method of Least Squares
    12.2 Orthogonal Systems and Chebyshev Polynomials
    12.3 Other Examples of the Least Squares Principle
13. Monte Carlo Methods and Simulation 13.1 Random Numbers
    13.2 Estimation of Areas and Volumes by Monte Carlo Techniques
    13.3 Simulation
14. Boundary Value Problems for Ordinary Differential Equations 14.1 Shooting Method
    14.2 A Discretization Method
15. Partial Differential Equations 15.0 Some Partial Differential Equations from Applied Problems
    15.1 Parabolic Problems
    15.2 Hyperbolic Problems
    15.3 Elliptic Problems
16. Minimization of Multivariate Functions 16.1 One-Variable Case
    16.2 Multivariate Case
17. Linear Programming 17.1 Standard Forms and Duality
    17.2 Simplex Method
    17.3 Approximate Solution of Inconsistent Linear Systems

Appendix A: Advice on Good Programming Practices

Appendix B: Representation of Numbers in Different Bases

Appendix C: Additional Details on IEEE Floating-Point Arithmetic

Appendix D: Linear Algebra Concepts and Notation

Answers for Selected Problems

Bibliography

Index  

C3   C6   C7