Introduction to Numerical Analysis

Citation

Stoer, J., & Bulirsch, R. (2002). Introduction to Numerical Analysis (3rd ed.; R. Bartels, W. Gautschi & C. Witzgall, Trans.). Springer. Texts in Applied Mathematics, vol. 12.

Chapter Summary

The third edition runs to about 740 pages in eight chapters, followed by a bibliography, a list of general literature on numerical methods and an index. It is a translation of the authors' German text, first published as an English book in 1980. The chapters follow the usual arc of a numerical analysis course, from the arithmetic of errors, through approximation and integration and the two main branches of numerical linear algebra, to zeros and minimization and finally differential equations. Chapter lengths show where the weight lies: ordinary differential equations (154 pages), iterative methods for large systems (111 pages), interpolation (108 pages) and eigenvalue problems (101 pages) are the largest.

Chapter 1: Error Analysis

The first chapter (36 pages) deals with the errors that arise when numerical computations are done in finite-precision arithmetic and with how they propagate through an algorithm. It sets the vocabulary of error and stability used in all later chapters.

Chapter 2: Interpolation

At 108 pages, the longest of the first four chapters. It treats the construction of functions that agree with given data.

Chapter 3: Topics in Integration

A shorter chapter (45 pages) on numerical integration, the computation of definite integrals from function values.

Chapter 4: Systems of Linear Equations

The numerical solution of systems of linear equations, in about a hundred pages. It is the first of three chapters on matrix problems, with Chapters 6 and 8.

Chapter 5: Finding Zeros and Minimum Points by Iterative Methods

Iterative methods for finding zeros of nonlinear equations and minimum points of functions of several variables. A reviewer for Acta Scientiarum Mathematicarum singles out the thorough discussion of the most important minimization methods.

Chapter 6: Eigenvalue Problems

Methods for computing eigenvalues and eigenvectors of matrices, a topic that occupies about a hundred pages.

Chapter 7: Ordinary Differential Equations

The numerical solution of ordinary differential equations. The same reviewer highlights the solution of stiff or implicit differential equations and of differential-algebraic systems among the book's strengths.

Chapter 8: Iterative Methods for the Solution of Large Systems of Linear Equations. Additional Methods

Iterative solvers for large linear systems, where direct methods become impractical. The same reviewer lists the basics of multigrid methods among the book's features, which fits this chapter's subject.

Key Concepts

Chapters 1–3: Errors, Interpolation, Integration

• Error propagation: how rounding and data errors grow or shrink as a computation proceeds, the basis for judging any algorithm.
• Interpolation: constructing a function from data so that approximation, differentiation and integration can be carried out on it.
• Quadrature: approximating a definite integral from sampled values.

Chapters 4–6: Linear Algebra, Zeros and Minima, Eigenvalues

• Linear systems: the numerical solution of systems of linear equations, the subject of Chapter 4 and, for large systems, Chapter 8.
• Iterative minimization: producing a sequence of approximations that converges to a minimum point.
• Eigenvalue computation: algorithms for the eigenvalues and eigenvectors of matrices.

Chapters 7–8: Differential Equations and Large Systems

• Stiff and implicit differential equations: problems that need special methods for stable numerical solution.
• Iterative solution of large linear systems: methods that exploit size and structure, including multigrid ideas.

Critical Analysis

Strengths

• Breadth in one volume. The eight chapters cover the central areas of numerical analysis from errors to large linear systems, and the Gazette review (Leversha, 2004) calls it a comprehensive introduction and an invaluable reference.
• Both classical and modern methods. The reviewer in the EMS Newsletter (2003) describes the book as well written, containing classical methods and modern approaches, with many interesting examples and carefully selected exercises.
• Notable topics. The Acta review highlights fast Fourier transform methods, minimization methods, stiff and implicit ODEs, differential-algebraic systems and the basics of multigrid methods as especially well presented.
• A reference as well as a text. The Gazette review says the table of contents makes it easy to find individual sections, and the book carries a bibliography of general literature on numerical methods.

Limitations

• Demanding as a first course. The reviews place it for the final-year undergraduate or beginning graduate student, and the publisher's series description gives advanced undergraduate and beginning graduate courses as the target; students without prior mathematical maturity will need a gentler book first.
• Long. At about 740 pages the book contains more than a single course can cover, so a course must select chapters.
• Computation is not the focus. Nothing in the chapter list points to programming practice or software; a reader who wants code and implementation will need a computing-oriented text alongside.

Where It Fits in the Course

Stoer and Bulirsch is the broad, theory-rich text in this course list. Burden, Faires and Burden's Numerical Analysis is an undergraduate introduction to numerical methods and the mathematics behind them; Süli and Mayers's An Introduction to Numerical Analysis is a mathematically rigorous introduction. Either is a sensible first book. Moler's Numerical Computing with MATLAB and Kong, Siauw and Bayen's Python-based text teach the subject through programming and are good companions for the hands-on side.

Trefethen and Bau's Numerical Linear Algebra is a concise graduate textbook devoted to one area, covering the subject of Stoer and Bulirsch's Chapters 4, 6 and 8 as a book in its own right. A reader would take Stoer and Bulirsch for the full sweep and Trefethen and Bau when numerical linear algebra becomes the main interest.

How to Use This Book

The book suits final-year undergraduates and beginning graduate students in mathematics, and researchers who want a reference. Familiarity with analysis and linear algebra is needed, and a prior numerical methods course makes the pace comfortable.

• Core: Chapters 1 to 4 form a natural first course: errors, interpolation, integration and linear systems.
• Next: Chapters 5 and 6 for nonlinear equations, minimization and eigenvalues, then Chapter 7 for differential equations.
• Later: Chapter 8 for large sparse systems, and as a reference chapter for specialized methods.

Treat the exercises as the main practice, and implement the algorithms yourself in a language of your choice to see the error behaviour the text analyses.