Table of Contents

1 Background Information.- 1.1 Significant figures and round-off error.- 1.2 Computers and floating-point arithmetic.- 1.3 Complex numbers.- 1.4 Inequalities for numbers.- 1.5 Convergence of a sequence of numbers.- 1.6 Polynomials and their roots.- 1.7 Systems of linear equations.- 1.8 The Vandermonde matrix.- 1.9 Continuous functions; piecewise continuous functions.- 1.10 Mean value theorem and Rolle’s theorem.- 1.11 Convergence of a sequence of functions.- 1.12 The chain rule for derivatives.- 1.13 Definite integrals and Riemann sums.- 1.14 Linear transformation of one interval onto another.- 1.15 Change of variables in an integral.- 1.16 Mean value theorem for integrals.- 1.17 Inequalities for integrals.- 1.18 The class of functions Wm[Mm; a,b].- 1.19 The function (x - t)+k.- 1.20 Taylor’s formula with integral form of remainder.- 1.21 Taylor’s formula with usual form of remainder.- 1.22 Taylor’s formula for functions of two variables.- 1.23 Difference equations.- 1.24 Linear difference equations with constant coefficients.- 1.25 Linear functional.- 1.26 Tri-diagonal linear systems.- References for Chapter 1.- 2 Interpolation.- 2.1 Existence of interpolating polynomials.- 2.2 Construction of the interpolating polynomial by solution of a linear system.- 2.3 One form for the error in the interpolation.- 2.4 Convergence of a sequence of interpolations.- 2.5 The Weierstrass approximation theorem.- 2.6 Iterated interpolation.- 2.7 Peano estimates for the error in interpolation.- 2.8 Interpolation by rational functions.- 2.9 Interpolation by cubic spline functions.- 2.10 Additional reading.- References for Chapter 2.- 3 Quadrature.- 3.1 Introductory remarks and definitions.- 3.2 Existence of formulas exact for polynomials.- 3.3 Newton-Cotes formulas and theirproperties.- 3.4 Linear transformations of formulas.- 3.5 Repeated trapezoidal formula; repeated midpoint formula; repeated Simpson’s formula.- 3.6 Introduction to Gauss formulas.- 3.7 Orthogonal polynomials and their zeros.- 3.8 Existence of Gauss formulas.- 3.9 Convergence of a sequence of Gauss formulas for a continuous integrand.- 3.10 Introduction to Romberg formulas.- 3.11 Romberg formulas and their properties.- 3.12 Peano error estimates for quadrature formulas.- 3.13 Gauss-Legendre formulas are Riemann sums.- 3.14 The merits of Gauss-Legendre formulas.- 3.15 Formulas exact for trigonometric polynomials.- 3.16 Numerical integration by rational extrapolation.- 3.17 Numerical integration by cubic splines.- 3.18 Additional reading.- References for Chapter 3.- 4 Initial Value Problems for Ordinary Differential Equations.- 4.1 Introduction.- 4.2 Taylor’s series methods.- 4.3 Convergence of Taylor’s series methods.- 4.4 Runge-Kutta methods.- 4.5 Derivation of Runge-Kutta methods.- 4.6 The need for automatic choice of stepsize; the earth-moon-spaceship problem.- 4.7 Runge-Kutta methods with automatic choice of stepsize; methods of Zonneveld.- 4.8 Explicit multistep methods or predictor methods.- 4.9 Implicit multistep methods or corrector methods.- 4.10 Practical use of corrector methods; predictor-corrector methods.- 4.11 Stability of multistep methods for y’ =—y.- 4.12 Stability of multistep methods for general equations.- 4.13 A method based on the midpoint formula and rational extrapolation.- 4.14 Additional reading.- References for Chapter 4.- Appendix A Tables of Orthogonal Polynomials.- Appendix B Tables of Peano Error Constants for Various Quadrature Formulas.- Appendix C Tables of Quadrature Formulas.- Index of Symbols.