Numerical Quadrature and Solution of Ordinary Differential Equations: A Textbook for a Beginning Course in Numerical Analysis

ISBN-13: 9780387901008 / Angielski / Miękka / 1974 / 338 str.

A. H. Stroud

Opis

This is a textbook for a one semester course on numerical analysis for senior undergraduate or beginning graduate students with no previous knowledge of the subject. The prerequisites are calculus, some knowledge of ordinary differential equations, and knowledge of computer programming using Fortran. Normally this should be half of a two semester course, the other semester covering numerical solution of linear systems, inversion of matrices and roots of polynomials. Neither semester should be a prerequisite for the other. This would prepare the student for advanced topics on numerical analysis such as partial differential equations. We are philosophically opposed to a one semester surveyor "numerical methods" course which covers all of the above mentioned topics, plus perhaps others, in one semester. We believe the student in such a course does not learn enough about anyone topic to develop an appreciation for it. For reference Chapter I contains statements of results from other branches of mathematics needed for the numerical analysis. The instructor may have to review some of these results. Chapter 2 contains basic results about interpolation. We spend only about one week of a semester on interpolation and divide the remainder of the semester between quadrature and differential equations. Most of the sections not marked with an * can be covered in one semester. The sections marked with an * are included as a guide for further study.

Spis treści

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 their properties.- 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.