Author | Stroud, A. H. author |
---|---|
Title | Numerical Quadrature and Solution of Ordinary Differential Equations [electronic resource] : A Textbook for a Beginning Course in Numerical Analysis / by A. H. Stroud |
Imprint | New York, NY : Springer New York, 1974 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-6390-6 |
Descript | XI, 338 p. online resource |
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 Rolleโs 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 Taylorโs formula with integral form of remainder -- 1.21 Taylorโs formula with usual form of remainder -- 1.22 Taylorโs 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 Simpsonโs 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 Taylorโs series methods -- 4.3 Convergence of Taylorโs 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 yโ = ?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