Scientific Computing

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Interpolation and Approximation

   John A. Trangenstein

   This chapter begins with a discussion of interpolation. Polynomial interpolation for a function of a single variable is analyzed, and implemented through Newton, Lagrange and Hermite forms. Intelligent selection of interpolation points is discussed, and extensions to multi-dimensional polynomial interpolation are presented. Rational polynomial interpolation is studied next, and connected to quadric surfaces. Then the discussion turns to piecewise polynomial interpolation and splines. The study of interpolation concludes with a presentation of parametric curves. Afterwards, the chapter moves on to least squares approximation, orthogonal polynomials and trigonometric polynomials. Trigonometric polynomial interpolation or approximation is implemented by the fast Fourier transform. The chapter concludes with wavelets, as well as their application to discrete data and continuous functions.

3. Chapter 2. Differentiation and Integration

   John A. Trangenstein

   Abstract

   This chapter develops numerical methods for computing derivatives and integrals. Numerical differentiation of polynomials can be performed by synthetic division, or through special properties of trigonometric polynomials or orthogonal polynomials. For derivatives of more general functions, finite differences lead to difficulties with rounding errors that can be largely overcome by clever post-processing, such as Richardson extrapolation. Integration is a more complicated topic. The Lebesgue integral is related to Monte Carlo methods, and Riemann sums are improved by trapezoidal and midpoint rules. Analysis of the errors leads to the Euler-MacLaurin formula. Various polynomial interpolation techniques lead to specialized numerical integration methods. The chapter ends with discussions of tricks for difficult integrals, adaptive quadrature, and integration in multiple dimensions.

4. Chapter 3. Initial Value Problems

   John A. Trangenstein

   This chapter is devoted to initial values problems for ordinary differential equations. It discusses theory for existence, uniqueness and continuous dependence on the data of the problem. Special techniques for linear ordinary differential equations with constant coefficients are discussed in terms of matrix exponentials and their approximations. Next, linear multistep methods are introduced and analyzed, leading to a presentation of important families of linear multistep methods and their stability. These methods are implemented through predictor-corrector methods, and techniques for automatically selecting stepsize and order are discussed. Afterwards, deferred correction and Runge-Kutta methods are examined. The chapter ends with the selection of numerical methods for stiff problems, and a discussion of nonlinear stability.

5. Chapter 4. Boundary Value Problems

   John A. Trangenstein

   Abstract

   This chapter is devoted to boundary value problems for ordinary differential equations. It begins with analysis of the existence and uniqueness of solutions to these problems, and the effect of perturbations to the problem. The first numerical approach is the shooting method. This is followed by finite differences and collocation. Finite elements allow for the development of very high order methods for many boundary value problems, but their analysis typically requires sophisticated ideas from real analysis. The chapter ends with the application of deferred correction to both collocation and finite elements.

6. Backmatter