Friendly Introduction to Numerical Analysis, A

Brian Bradie  
Total pages
April 2005
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Friendly Introduction to Numerical Analysis, A
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For one or two-semester undergraduate/graduate-level courses in Numerical Analysis/Methods in mathematics departments, CS departments, and all engineering departments.

This student-friendly text develops concepts and techniques in a clear, concise, easy-to-read manner, followed by fully-worked examples. Application problems drawn from the literature of many different fields prepares students to use the techniques covered to solve a wide variety of practical problems.


  • A theme of comparing/contrasting numerical methods for accuracy, error, boundaries, and speed of convergence


  • Unique topical coverage—Provides extensive coverage of material (especially PDEs and boundary value problems) not typically covered, or only briefly discussed, in other texts. E.g.,
    • Improper integrals.

    • Non-Dirichlet boundary conditions.

    • The handling of artificial singularities for one-dimensional boundary value problems.

    • The multigrid method and irregular domains for elliptic partial differential equations.

    • Source and decay terms, polar coordinates and problems in two space dimensions for parabolic partial differential equations.

    • One-dimensional hyperbolic partial differential equations.

    • Numerical dispersion and diffusion and the convection-diffusion equation.

    • Prepares students for the practical application of numerical methods; offers instructors flexibility in coverage—they can touch as lightly or as in depth as desired and design courses around students' interests.

  • More than 200 fully-worked examples—The examples are each tied carefully to some new concepts.
    • Helps students grasp the sequence of calculations associated with a particular method and gain better insight into algorithm operation.

  • An extensive set of application problems—Used both as worked examples and exercises. Problems are drawn from the literature of many different fields (physics, biology, chemistry, chemical engineering, thermodynamics, heat transfer, electrostatics, ecology, manufacturing, sociology, etc.).
    • Shows students how numerical methods can be applied within the context of real-world problems, and motivates their study of the various numerical techniques. Gives instructors the opportunity to discuss practical implementation issues.

  • Chapters organized thematically around mathematical problems—Each chapter is devoted to a single type of problem. Within each chapter, the presentation begins with the simplest, most basic methods and progresses gradually to more advanced topics.
    • Helps students find parallels and better comprehend the topics.

  • Chapter Overviews—Presents several real-world problems relating to the specific mathematical problem that will be treated in the chapter.
    • Places the material into perspective for students and motivates the reader with the broad applicability of numerical methods to real-world problems.

  • Exercise Sets—Features roughly 1000 numbered exercises (many with multiple parts).  An appropriate balance of theoretical, applications, and coding questions.
    • Provides students with the opportunity to practice (with paper, pencil and calculator) the sequence of calculations associated with a particular method.

    • Gives students extensive practice in using numerical methods.

  • Code available for instructors—Code for all methods discussed in the text will be available for faculty for both Maple and MATLAB (through a web site).
    • Gives instructors a reference guide.

  • Absence of pseudocode—The author believes that with pseudocode provided, students feel like they don't need to really understand the techniques; they just have to be able to convert the pseudocode into whatever the language of choice happens to be.
    • Requires students to program the techniques themselves.

Table of Contents

(NOTE: Each chapter begins with An Overview.)

1. Getting Started.

Algorithms. Convergence. Floating Point Numbers. Floating Point Arithmetic.

2. Rootfinding.

Bisection Method. Method of False Position. Fixed Point Iteration. Newton's Method. The Secant Method and Muller's Method. Accelerating Convergence. Roots of Polynomials.

3. Systems of Equations.

Gaussian Elimination. Pivoting Strategies. Norms. Error Estimates. LU Decomposition. Direct Factorization. Special Matrices. Iterative Techniques for Linear Systems: Basic Concepts and Methods. Iterative Techniques for Linear Systems: Conjugate-Gradient Method. Nonlinear Systems.

4. Eigenvalues and Eigenvectors.

The Power Method. The Inverse Power Method. Deflation. Reduction to Tridiagonal Form. Eigenvalues of Tridiagonal and Hessenberg Matrices.

5. Interpolation and Curve Fitting.

Lagrange Form of the Interpolating Polynomial. Neville's Algorithm. The Newton Form of the Interpolating Polynomial and Divided Differences. Optimal Interpolating Points. Piecewise Linear Interpolation. Hermite and Hermite Cubic Interpolation. Regression.

6. Numerical Differentiation and Integration.

Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.

7. Numerical Methods for Initial Value Problems of Ordinary Differential Equations.

Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.

8. Second-Order One-Dimensional Two-Point Boundary Value Problems.

Finite Difference Method, Part I: The Linear Problem with Dirichlet Boundary Conditions. Finite Difference Method, Part II: The Linear Problem with Non-Dirichlet Boundary Conditions. Finite Difference Method, Part III: Nonlinear Problems. The Shooting Method, Part I: Linear Boundary Value Problems. The Shooting Method, Part II: Nonlinear Boundary Value Problems.

9. Finite Difference Method for Elliptic Partial Differential Equations.

The Poisson Equation on a Rectangular Domain, I: Dirichlet Boundary Conditions. The Poisson Equation on a Rectangular Domain, II: Non-Dirichlet Boundary Conditions. Solving the Discrete Equations: Relaxation Schemes. Local Mode Analysis of Relaxation and the Multigrid Method. Irregular Domains.

10. Finite Difference Method for Parabolic Partial Differential Equations.

The Heat Equation with Dirichlet Boundary Conditions. Stability. More General Parabolic Equations. Non-Dirichlet Boundary Conditions. Polar Coordinates. Problems in Two Space Dimensions.

11. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation.

Advection Equation, I: Upwind Differencing. Advection Equation, II: MacCormack Method. Convection-Diffusion Equation. The Wave Equation.


Appendix A. Important Theorems from Calculus. Appendix B. Algorithm for Solving a Tridiagonal System of Linear Equations.



Answers to Selected Problems.

Reader Review(s)

"I am extremely impressed with Bradie's book. His passion for explaining things as clearly and understandably as possible, his thorough research of the literature for bringing relevant and pedagogically sound examples from outside mathematics, and his crisp and clear style will certainly make this text an instant success. This is one of the better texts in Numerical Analysis that I have ever seen, and I congratulate the author for producing such a gem." - Alejandro Engel, Rochester Institute of Technology

"The chapters in this book are of uniformly high standards. Chapter 1 in particular is a gem. The treatments of floating point number systems and of floating point arithmetic are especially good. These are topics that are often glossed over in other books, and which are often difficult for students to grasp. The book is extremely well written: the style is clear, the prose flows smoothly, the pace is unhurried, the tone is friendly and conversational, the examples and exercises are interesting and-relevant, and the amount of detail is far greater than in any textbook of its kind that I have ever seen. For these reasons, it will certainly appeal to my students." - Richard Zalik, Auburn University

"I think the tone will appeal to my students: It is relaxed and friendly without being wordy and effusive. The style is a very readable compromise between proof and technical detail on the one hand, and concepts with applications on the other. I think he addresses this fundamental challenge in a way that my students would like. Bradie has decided to include lots of worked examples accompanied by plots. The plots facilitate the inclusion of such a large number of examples, by succinctly communicating the point of each. This reduces the effort needed to understand the ideas behind the example, (I think students simply will not read the book if it takes too much effort. Bradie can include more exercises than is typical because the illustrations ease the communication.)" - Mark Arnold, University of Arkansas

"I like the way Bradie presents the materials in each chapter. He gives a mathematics review on what is needed at the beginning of each chapter. After refreshing students' memories, he begins with the simplest, most basic methods and then progresses gradually to more advanced topics. The book is well written and student-friendly. It provides a lot of examples and exercise problems. The book is written in the way that is easy for students to read. For instance, for each method, there is at least one fully worked example that helps students to understand the concept and the method." - Kuiyuan Li, University of West Florida