Nicolas Fillion

Numerical Solutions of Boundary Value Problems

Boundary value problems (BVP) for ordinary differential equations are more than a simple extension of initial-value problems; indeed, it may be more fairly said that initial-value problems are a degenerate case of BVP. The Matlab codes for solving BVPs are remarkably similar to call and assess although their detailed behavior and algorithms are quite different. In this chapter, we look at linear problems, quasilinearization to solve nonlinear problems, and the principle of equidistribution on an optimal mesh. ⊲

Numerical Solution of ODEs

We use a continuously differentiable representation of the numerical solution in order to define a residual, also known as a defect or deviation. This allows the use of backward error analysis on the numerical solution of initial-value problems (IVP) for ordinary differential equations. Of course, this means that we should also examine the conditioning or sensitivity of an IVP. We use the Matlab ODE Suite as an example of high-quality software. We pay some attention to chaotic problems, stiff problems, and singular problems. ⊲

Numerical Solution of PDEs

We look at the method of lines using standard initial-value problem (IVP) software for stiff problems. Both spectral methods and compact finite differences are used for the spatial derivatives. We look briefly at the transverse method of lines, which instead uses standard boundary value problem (BVP) software that has automatic mesh selection. We also briefly consider Fourier transform methods for Poisson’s equation. ⊲

Rootfinding and Function Evaluation

We introduce general methods to evaluate functions and to find roots (or zeros) of functions of all kinds. We examine various approximation methods and study their respective numerical accuracy, by examining their backward error and the condition numbers for evaluation and rootfinding. ⊲

Structured Linear Systems

We define structured linear systems to include sparse systems or systems with correlated entries or both. We define the structured backward error and a structured condition number. We give examples of various classes of structured linear systems and examples of algorithms that take advantage of the special structure. ⊲

The Discrete Fourier Transform

This short chapter introduces the discrete (finite) and fast Fourier transform. The numerical stability and conditioning of the Fourier matrix is mentioned. Applications using convolution and circulant matrices are considered, as is the periodogram or power spectral density. ◃

Polynomials and Series

This chapter introduces the reader to the numerical aspects of polynomials. In particular, we examine different polynomial bases such as the monomial, the Chebyshev, and the Lagrange basis; we provide algorithms to evaluate polynomials in many of those bases and examine the different condition numbers in different bases. We give a first look at the important problem of numerically finding zeros and pseudozeros of polynomials. We give an algorithmic overview of the numerical computation of truncated power series including Taylor series. Finally, we give a brief discussion of asymptotics. ⊲

Computer Arithmetic and Fundamental Concepts of Computation

This chapter introduces the main concepts of error analysis used in this book. The chapter defines reference problems and modified problems and notation to distinguish them. Two kinds of modified problems are shown to be particularly important in numerical analysis, namely, engineered and reverse-engineered problems. The reader is introduced to three concepts of error: (forward error, backward error, and residual), to the concept of conditioning, and to residual-based backward error analysis—which is the method favored in this book. We define numerical properties of algorithms, including stability and cost. Finally, we apply those concepts to floating-point arithmetic. ⊲

Numerical Methods for ODEs

This chapter gives a very brief survey of Runge–Kutta methods, including continuous explicit Runge–Kutta methods. We also discuss multistep methods and the Taylor series method (i.e., analytic continuation). We talk about implicit methods for stiff problems and backward error by the method of modified equations. We mention some of the several flavors of stability of a numerical method for solving IVP that are sometimes useful. We talk about interpolants and the residual and compare the residual with local error per unit step. We sketch methods for adaptive step-size control. ⊲

Polynomial and Rational Interpolation

This chapter gives a detailed discussion of barycentric Lagrange and Hermite interpolation and extends this to rational interpolation with a specified denominator. We discuss the conditioning of these interpolants. A numerically stable method to find roots of polynomials expressed in barycentric form via a generalized eigenvalue problem is given. We conclude with a section on piecewise polynomial interpolants. ⊲