Numerical Mathematics and Computing

1. MATHEMATICAL PRELIMINARIES AND FLOATING-POINT REPRESENTATION. Introduction, Mathematical Preliminaries. Floating-Point Representation. Loss of Significance. 2. LINEAR SYSTEMS. Naive Gaussian Elimination. Gaussian Elimination with Scaled Partial Pivoting. Tridiagonal and Banded Systems. 3. NONLINEAR EQUATIONS. Bisection Method. Newton's Method, Secant Method. 4. INTERPOLATION AND NUMBERICAL DIFFERENTIATION. Polynomial Interpolation. Errors in Polynomial Interpolation. Estimating Derivatives and Richardson Extrapolation. 5. NUMERICAL INTEGRATION. Trapezoid Method. Romberg Algorithm. Simpson's Rules and Newton-Cotes Rules. Gaussian Quadrature Formulas. 6. SPLINE FUNCTIONS. First-Degree and Second-Degree Splines. Natural Cubic Splines. B Splines: Interpolation and Approximation. 7. INITIAL VALUES PROBLEMS. Taylor Series Methods. Runge-Kutta Methods. Adaptive Runge-Kutta and Multistep Methods. Methods for First and Higher-Order Systems. Adams-Bashforth-Moulton Methods. 8. MORE ON LINEAR SYSTEMS. Matrix Factorizations. Eigenvalues and Eigenvectors. Power Method. Iterative Solutions of Linear Systems. 9. LEAST SQUARES METHODS AND FOURIER SERIES. Method of Least Squares. Orthogonal Systems and Chebyshev Polynomials. Examples of the Least-Squares Principle. Fourier Series. 10. MONTE CARLO METHODS AND SIMULATION. Random Numbers. Estimation of Areas and Volumes by Monte Carlo Techniques. Simulation. 11. BOUNDARY-VALUE PROBLEMS. Shooting Method. A Discretization Method. 12. PARTIAL DIFFERENTIAL EQUATIONS. Parabolic Problems. Hyperbolic Problems. Elliptic Problems. 13. MINIMIZATION OF FUNTIONS. One-Variable Case. Multivariable Case. 14. LINEAR PROGRAMMING PROBLEMS. Standard Forms and Duality. Simplex Method, Inconsistent Linear Systems. APPENDIX A. ADVICE ON GOOD PROGRAMMING PRACTICES. Programming Suggestions. APPENDIX B. REPRESENTATION OF NUMBERS IN DIFFERENT BASES. Representation of Numbers in Different Bases. APPENDIX C. ADDITIONAL DETAILS ON IEEE FLOATING-POINT ARITHMETIC. More on IEEE Standard Floating-Point Arithmetic. APPENDIX D. LINEAR ALGEBRA CONCEPTS AND NOTATION. Elementary Concepts. ANSWERS FOR SELECTED EXERCISES. BIBLIOGRAPHY. INDEX.