An Introduction to Numerical Analysis

Citation

Süli, E., & Mayers, D. F. (2003). An Introduction to Numerical Analysis. Cambridge University Press.

Chapter Summary

The book has fourteen chapters and two short appendices in 444 pages. In the preface the authors explain that it grew out of printed notes for lectures given to undergraduate mathematicians at Oxford over many years, that it has a wider scope than is currently taught, and that it is aimed at students in the second year of a university mathematics course. Chapter 2 deals with linear systems, which can be solved in a finite number of arithmetical operations; in the rest of the book, the preface says, the solution of a problem is sought as the limit of an infinite sequence, so the output of the algorithm is an approximate solution. Cambridge's description lists the problems covered as the approximation of functions and integrals and the approximate solution of algebraic, transcendental, differential and integral equations, with attention throughout to stability, accuracy, reliability and efficiency.

Chapter 1: Solution of Equations by Iteration

At 38 pages, one of the longer chapters. It opens the book with the iterative solution of equations, in line with the preface's description of solutions as limits of sequences.

Chapters 2–3: Solution of Systems of Linear Equations; Special Matrices

Chapter 2 (48 pages) treats linear systems, the one problem the preface identifies as solvable exactly in finitely many operations. Chapter 3 is a short chapter on special matrices.

Chapters 4–5: Simultaneous Nonlinear Equations; Eigenvalues and Eigenvectors of a Symmetric Matrix

Chapter 4 extends equation solving to simultaneous nonlinear equations. Chapter 5, one of the longest chapters at 46 pages, turns to the eigenvalue problem for symmetric matrices.

Chapters 6–8: Polynomial Interpolation; Numerical Integration – I; Polynomial Approximation in the ∞-Norm

Chapter 6 introduces polynomial interpolation. Chapter 7 is the first of two chapters on numerical integration. Chapter 8 studies best polynomial approximation of continuous functions in the ∞-norm.

Chapters 9–11: Approximation in the 2-Norm; Numerical Integration – II; Piecewise Polynomial Approximation

Chapter 9 begins from the observation that the choice of norm changes the best approximation, so best approximation in the 2-norm must be developed separately; it is tied to orthogonality and so begins by recalling inner product spaces. Chapter 10 is the second chapter on numerical integration. Chapter 11 contrasts the earlier global constructions (its introduction names Lagrange and Hermite interpolation and best approximation in both norms) with piecewise approximation by low-degree polynomials: splines of degree n, with knots at the ends of subintervals and a prescribed smoothness.

Chapters 12–14: Initial Value Problems for ODEs; Boundary Value Problems for ODEs; The Finite Element Method

The last three chapters turn to differential equations. Chapter 12 (the longest, at 51 pages) treats initial value problems, Chapter 13 boundary value problems, and Chapter 14 closes with the finite element method.

Appendices

Appendix A is a four-page overview of results from real analysis; Appendix B is a one-page list of WWW resources. A bibliography and index follow.

Key Concepts

Chapters 1–5: Equations and matrices

• Solution by iteration: Chapter 1's approach to equations, in which a solution is the limit of a sequence.
• Stability, accuracy, reliability and efficiency: the four qualities the preface and publisher description single out in a numerical algorithm.
• Symmetric-matrix eigenproblem: the subject of Chapter 5.

Chapters 6–11: Interpolation, approximation and integration

• Interpolation and best approximation: Lagrange and Hermite interpolation, and best approximation in the ∞-norm and the 2-norm.
• Inner product space and orthogonality: the framework recalled at the start of Chapter 9.
• Splines and knots: the piecewise polynomial approximations of Chapter 11.

Chapters 12–14: Differential equations

• Initial value and boundary value problems: Chapters 12 and 13.
• Finite element method: the closing chapter.

Critical Analysis

Strengths

• Praised for rigour combined with readability. Eugene Allgower writes that the book "offers a good level of rigor for the intended readership", and a review in Zentralblatt MATH calls it a very good textbook, "very carefully written" and recommendable for second- or third-year mathematics students, adding that the authors have chosen introductory topics well. Both reviews are quoted on the publisher's page.
• Born of lectures. A review in Internationale Mathematische Nachrichten explains the book's ability to answer questions students often ask by the fact that it comes from many years of Oxford lectures; Mathematics Today calls it easy to read and flowing.
• Motivated treatment of approximation. The opening of Chapter 9 explains why the 2-norm needs its own theory by showing that the best approximation depends on the norm, and Chapter 11 sets piecewise polynomials against the global methods of Chapters 6–9, so each new tool is introduced by a reason.
• A route to modern methods. The sequence from interpolation and approximation to differential equations ends with the finite element method. The Zentralblatt MATH review adds that the book has many historical references to mathematicians.

Limitations

• Analysis background is expected. Allgower says the book suits students "who come armed with some background in analysis", and Appendix A, which summarises results from real analysis, is only four pages long.
• Little on implementation. The contents contain no chapter on programming or software, and Appendix B is a single page of web resources in a book published in 2003. A student who wants to code the algorithms will need a companion text or course.
• Narrower scope than some course texts. No chapter is named for partial differential equations; the finite element method is the closing chapter.

Where It Fits in the Course

This is the mathematically rigorous undergraduate option in the numerical analysis list. Burden, Faires and Burden's Numerical Analysis is an undergraduate introduction that requires only a calculus background and ranges more widely through twelve chapters including numerical differentiation and partial differential equations; it is a gentler first book, with Süli and Mayers as the natural second pass for students who want proofs of convergence and error bounds. 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 the theory here. Trefethen and Bau's Numerical Linear Algebra and Stoer and Bulirsch's Introduction to Numerical Analysis are graduate-level next steps, the first for linear algebra in depth and the second as a comprehensive treatise.

How to Use This Book

The book is aimed at second-year mathematics students with a grounding in real analysis and linear algebra, and it suits readers who want to understand why numerical algorithms work, not only how to run them.

A practical plan:

• Core (Chapters 1–2, 6–7, 12): iteration for equations, linear systems, interpolation, the first chapter on numerical integration, and initial value problems for ODEs make a compact first course.
• Linear algebra and nonlinear systems (Chapters 3–5): take as time allows; Chapter 5 on symmetric eigenvalue problems is the longest of the three.
• Approximation theory (Chapters 8–11): a second block on best approximation, integration and splines, for readers who want the theory behind Chapters 6–7.
• Advanced (Chapters 13–14): boundary value problems and the finite element method suit a second semester or independent study.

Review Appendix A before starting, and pair the reading with a programming exercise for each method to see the stability and accuracy issues the book discusses.