Numerical Analysis

Citation

Burden, R. L., Faires, J. D., & Burden, A. M. (2016). Numerical Analysis (10th ed.). Cengage Learning.

Chapter Summary

The tenth edition runs to roughly 900 pages (library records give 896) and succeeds an edition of 2011. Cengage describes it as a text that introduces the theory and application of modern numerical approximation techniques for students taking a one- or two-semester course, with a calculus prerequisite; the authors explain how, why and when approximation techniques can be expected to work, and why in some situations they fail. The chapter titles below follow the library catalogue record of the previous (ninth) edition, which lists twelve chapters; the tenth edition's description covers the same ground. They are listed at chapter level only.

Chapters 1–3: Mathematical Preliminaries and Error Analysis; Solutions of Equations in One Variable; Interpolation and Polynomial Approximation

The book begins with the background for everything that follows, including error analysis, then turns to solving equations in one variable and to interpolation and polynomial approximation. These three chapters are the foundation for the later numerical methods.

Chapters 4–5: Numerical Differentiation and Integration; Initial-Value Problems for Ordinary Differential Equations

Chapter 4 covers numerical differentiation and integration, and Chapter 5 methods for initial-value problems for ordinary differential equations.

Chapters 6–7: Direct Methods for Solving Linear Systems; Iterative Techniques in Matrix Algebra

The numerical linear algebra block deals with direct methods for linear systems and then with iterative techniques.

Chapters 8–10: Approximation Theory; Approximating Eigenvalues; Numerical Solutions of Nonlinear Systems of Equations

Chapter 8 is on approximation theory, Chapter 9 on approximating eigenvalues, and Chapter 10 on nonlinear systems of equations, extending the single-variable equation solving of Chapter 2 to several unknowns.

Chapters 11–12: Boundary-Value Problems for Ordinary Differential Equations; Numerical Solutions to Partial Differential Equations

The last two chapters treat boundary-value problems for ordinary differential equations and numerical solutions to partial differential equations.

Key Concepts

Chapters 1–5: Preliminaries, equations, approximation and integration

• Error analysis: named in the title of Chapter 1 as part of the preliminaries.
• Interpolation and polynomial approximation: the subject of Chapter 3.
• Numerical differentiation and integration: the subject of Chapter 4.
• Initial-value problems: the numerical solution of ODEs in Chapter 5.

Chapters 6–12: Linear algebra, further approximation and differential equations

• Direct and iterative methods for linear systems: Chapters 6 and 7.
• Approximation theory and eigenvalue approximation: Chapters 8 and 9.
• Nonlinear systems: Chapter 10.
• Boundary-value problems and PDEs: Chapters 11 and 12.

Critical Analysis

Strengths

• Low entry barrier. The publisher states that the treatment is accessible and requires only a calculus prerequisite, which puts the book within reach of students who have not yet taken real analysis.
• Attention to failure as well as success. Cengage says the authors explain when approximation techniques can be expected to work and why in some situations they fail, which is the point at which numerical analysis separates from recipe-following.
• Examples and exercises. The description stresses a wealth of examples and exercises that build intuition and show practical applications in mathematics, computing, engineering and the physical sciences.
• Broad coverage and long history. Twelve chapters run from error analysis through linear algebra and ordinary differential equations to partial differential equations, and the book has been revised repeatedly since it was first written more than thirty years ago; the tenth edition succeeds one published in 2011.

Limitations

• Linear algebra comes late. Direct methods for linear systems start only in Chapter 6, after initial-value problems in Chapter 5, so a course that wants linear systems early has to reorder the chapters.
• More material than a single course. At close to 900 pages and twelve chapters, the book contains more than a one- or two-semester course can use; instructors and self-learners must choose.
• Limited public detail. The sources available show chapter titles and a short publisher description only, so anyone relying on the book for a particular method should check the section-level contents of the edition they will use.

Where It Fits in the Course

Burden, Faires and Burden is the gentler and broader of the two undergraduate texts in this course list. Süli and Mayers's An Introduction to Numerical Analysis is a more rigorous treatment based on Oxford lectures, aimed at second-year mathematics students who have a grounding in analysis, and emphasises proofs of convergence and error bounds; it is a natural second book for students who want to see why the algorithms work. Moler's Numerical Computing with MATLAB and Kong, Siauw and Bayen's Python Programming and Numerical Methods are programming-first introductions that pair well with Burden's coverage of the same methods. Trefethen and Bau's Numerical Linear Algebra and Stoer and Bulirsch's Introduction to Numerical Analysis are graduate-level next steps: the first for numerical linear algebra in depth, the second as a comprehensive treatise.

How to Use This Book

The book is meant for students in mathematics, computer science, engineering and the physical sciences who have completed a calculus sequence and want a first course in numerical methods and their error behaviour.

A practical plan:

• Core (Chapters 1–5): read in order. These chapters take a student from error analysis through equation solving, interpolation, integration and differentiation to initial-value problems, which makes a plausible spine for a one-semester course.
• Linear algebra (Chapters 6–7, 9): direct and iterative methods for linear systems, and eigenvalue approximation, for a second semester or for students whose goals are numerical linear algebra. Chapter 10 on nonlinear systems is a natural follow-on to Chapters 2 and 6.
• Advanced (Chapters 8, 11–12): approximation theory, boundary-value problems and partial differential equations, for the second half of a two-semester course.

Work the exercises with a computer, since numerical analysis is learned by running and comparing methods. Because linear systems appear after differential equations, a course that wants them earlier can take Chapters 6–7 right after Chapter 3.