Preface to the original edition

It might seem that computing should simply be a matter of translating formulas from the page to the machine. But even when such formulas are known, applying them in a numerical fashion requires care. For instance, rounding off numbers at the sixteenth significant digit can lay low such stalwarts as the quadratic formula! Fortunately, the consequences of applying a numerical method to a mathematical problem are quite understandable from the right perspective. In fact, it is our mastery of what can go wrong in some approaches that gives us confidence in the rest of them.

If mathematical modeling is the process of turning real phenomena into mathematical abstractions, then numerical computation is largely about the transformation from abstract mathematics to concrete reality. Many science and engineering disciplines have long benefited from the tremendous value of the correspondence between quantitative information and mathematical manipulation. Other fields, from biology to history to political science, are rapidly catching up. In our opinion, a young mathematician who is ignorant of numerical computation in the 21st century has much in common with one who was ignorant of calculus in the 18th century.

1To the student¶

Welcome! We expect that you have had lessons in manipulating power series, solving linear systems of equations, calculating eigenvalues of matrices, and obtaining solutions of differential equations. We also expect that you have written computer programs that take a nontrivial number of steps to perform a well-defined task, such as sorting a list. Even if you have rarely seen how these isolated mathematical and computational tasks interact with one another, or what they have to do with practical realities, you are in the audience for this book.

Based on our experiences teaching this subject, our guess is that some rough seas may lie ahead of you. Probably you do not remember learning all parts of calculus, linear algebra, differential equations, and computer science with equal skill and fondness. This book draws from all of these areas at times, so your weaknesses are going to surface once in a while. Furthermore, this may be the first course you have taken that does not fit neatly within a major discipline. Von Neumann’s use of “impurer” in the quote above is a telling one: numerical computation is about solving problems, and the search for solution methods that work well can take us across intellectual disciplinary boundaries. This mindset may be unfamiliar and disorienting at times.

Don’t panic! There is a great deal to be gained by working your way through this book. It goes almost without saying that you can acquire computing skills that are in much demand for present and future use in science, engineering, and mathematics—and increasingly, in business, social science, and humanities, too. There are less tangible benefits as well. Having a new context to wrestle with concepts like Taylor series and matrices may shed new light on why they are important enough to learn. It can be exhilarating to apply skills to solve relatable problems. Finally, the computational way of thought is one that complements other methods of analysis and can serve you well.

2To the instructor¶

The plausibly important introductory material on numerical computation for the majority of undergraduate students easily exceeds the capacity of two semesters—and of one textbook. As instructors and as authors, we face difficult choices as a result. We set aside the goal of creating an agreeable canon. Instead we hope for students to experience an echo of that “obscene interest” that von Neumann so gleefully described and pursued. For while there are excellent practical reasons to learn about numerical computing, it also stands as a subject of intellectual and even emotional relevance. We have seen students excited and motivated by applications of their newly found abilities to problems in mechanics, biology, networks, finance, and more—problems that are of unmistakable importance in the jungle beyond university textbooks, yet largely impenetrable using only the techniques learned within our well-tended gardens.

In writing this book, we have not attempted to be encyclopedic. We’re sorry if some of your favorite topics don’t appear or are minimized in the book. (It happened to us too; many painful cuts were made from prior drafts.) But in an information-saturated world, the usefulness of a textbook lies with teaching process, not content. We have aimed not for a cookbook but for an introduction to the principles of cooking.

Still, there are lots of recipes in the book—it’s hard to imagine how one could become a great chef without spending time in the kitchen! Our language for these recipes is MATLAB for a number of reasons: it is precise, it is executable, it is as readable as could be hoped for our purposes, it rewards thinking at the vector and matrix level, and (at this writing) it is widespread and worth knowing. There are 46 standalone functions and over 150 example scripts, all of them downloadable exactly as seen in the text. Some of our codes are quite close to production quality, some are serviceable but lacking, and others still are meant for demonstration only. Ultimately our codes are mainly meant to be clear, not ideal. We try to at least be explicit about the shortcomings of our implementations.

Just as good coding and performance optimization are secondary objectives of the book, we cut some corners in the mathematics as well. We state and in some cases prove the most essential and accessible theorems, but this is not a theorem-oriented book, and in some cases we are content with less precise arguments. We have tried to make clear where solid proof ends and where approximation, estimation, heuristics, and other indispensable tools of numerical analysis begin.

The examples and exercises are meant to show and encourage a numerical mode of thought. As such they often focus on issues of convergence, experimentation leading to abstraction, and attempts to build on or refine presented ideas. Some exercises follow the premise that an algorithm’s most interesting mode of operation is failure. We expect that any learning resulting from the book is likely to take place mostly from careful study of the examples and working through the problems.

Contents¶

Chapter 1 explains how computers represent numbers and arithmetic, and what doing so means for mathematical expressions. Chapter 2 discusses the solution of square systems of linear equations and, more broadly, basic numerical linear algebra. Chapter 3 extends the linear algebra to linear least squares. These topics are the bedrock of scientific computing, because “everything” has multiple dimensions, and while “everything” is also nonlinear, our preferred starting point is to linearize.

Chapters 4 through 6 introduce what we take to be the rest of the most common problem types in scientific computing: roots and minimization of algebraic functions, piecewise approximation of functions, numerical analogs of differentiation and integration, and initial-value problems for ordinary differential equations. We also explain some of the most familiar and reliable ways to solve these problems, effective up to a certain point of size and/or difficulty. Chapters 1 through 6 can serve for a single-semester survey course. If desired, Chapter 6 could be left out in favor of one of Chapters 7, 8, or 9.

The remaining chapters are intended for a second course in the material. They go into more sophisticated types of problems (eigenvalues and singular values, boundary value problems, and partial differential equations), as well as more advanced techniques for problems from the first half (Krylov subspace methods, spectral approximation, stiff problems, boundary conditions, and tensor-product discretizations).