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A First Course in Numerical Analysis

Anthony Ralston and Philip Rabinowitz
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

First published in 1965, this has been a standard text and then a well-known reference for many topics in numerical analysis. A second edition with revisions and additional material was published in 1977.

Although the fundamentals of numerical analysis haven’t changed since the book was first published fifty years ago, the environment in which we do it is radically different. With packages like Mathematica or Matlab and a variety of high quality libraries of mathematical software available, two things have happened. A good deal more numerical analysis is being done in one form or other, and far fewer people are writing their own computer code to do it.

Where does the current book fit in this new milieu? It is useful to compare it to two other books, one now almost twenty years and the other less than five years old. Numerical Recipes, aptly named and with probably the widest reach of any numerical analysis book, emphasizes computer implementation, offers guidance for selection and use of algorithms, and provides some analysis. The newer Numerical Methods: Design, Analysis and Computer Implementation of Algorithms by Greenbaum and Chartier gives largely equal coverage to the areas that the title names and, coincidentally or not, echoes precisely the three parts of numerical analysis as a mathematical science that Ralston and Rabinowitz (hereafter simply R&R) identify early in their book. R&R’s strengths are in the analysis component, and particularly in aspects of numerical analysis that have clear connections to real analysis.

R&R is not the place to go for the newest or the slickest algorithms. But the treatment of the basic elements of numerical analysis is solid all the way through. The strongest parts are probably the chapters on interpolation, functional approximation and the solution of nonlinear equations. The basic elements of numerical integration, differentiation and numerical solution of ordinary differential equations all get thorough treatments. The final two chapters on numerical linear algebra (solution of systems of linear equations and calculation of eigenvalues) are adequate but weaker than the rest of the book. A treatment of the fast Fourier transform was added in the second edition, but it is the barest of introductions.

It is difficult to envision using R&R as a textbook now, especially given the number of good modern alternatives, so it is probably best regarded as a reference. It does still have its place; there is a lot of very good analysis here. It is also a great source of exercises. 

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.


  Preface to the Dover Edition; Preface to the Second Edition; Notation
Chapter 1. Introduction and Preliminaries
  1.1 What Is Numerical Analysis?
  1.2 Sources of Error
  1.3 Error Definitions and Related Matters
    1.3-1 Significant digits; 1.3-2 Error in functional Evaluation; 1.3-3 Norms
  1.4 Roundoff Error
    1.4-1 The Probabilistic Approach to Roundoff: A Particular Example
  1.5 Computer Arithmetic
    1.5-1 Fixed-Point Arithmetic; 1.5-2 Floating-Point Numbers; 1.5-3 Floating-Point Arithmetic; 1.5-4 Overflow and Underflow; 1.5-5 Single- and Double-Precision Arithmetic
  1.6 Error Analysis
    1.6-1 Backward Error Analysis
  1.7 Condition and Stability
  Bibliographic Notes; Bibliography; Problems
Chapter 2. Approximation and Algorithms
  2.1 Approximation
    2.1-1 Classes of Approximating Functions; 2.1-2 Types of Approximations; 2.1-3 The Case for Polynomial Approximation
  2.2 Numerical Algorithms
  2.3 Functionals and Error Analysis
  2.4 The Method of Undetermined Coefficients
  Bibliographic Notes; Bibliography; Problems
Chapter 3. Interpolation
  3.1 Introduction
  3.2 Lagrangian Interpolation
  3.3 Interpolation at Equal Intervals
    3.3-1 Lagrangian Interpolation at Equal Intervals; 3.3-2 Finite Differences
  3.4 The use of Interpolation Formulas
  3.5 Iterated Interpolation
  3.6 Inverse Interpolation
  3.7 Hermite Interpolation
  3.8 Spline Interpolation
  3.9 Other Methods of Interpolation; Extrapolation
  Bibliographic Notes; Bibliography; Problems
Chapter 4. Numerical Differentiation, Numerical Quadrature, and Summation
  4.1 Numerical Differentiation of Data
  4.2 Numerical Differentiation of Functions
  4.3 Numerical Quadrature: The General Problem
    4.3-1 Numerical Integration of Data
  4.4 Guassian Quadrature
  4.5 Weight Functions
  4.6 Orthogonal Polynomials and Gaussian Quadrature
  4.7 Gaussian Quadrature over Infinite Inte
  4.8 Particular Gaussian Quadrature Formulas
    4.8-1 Gauss-Jacobi Quadrature; 4.8-2 Gauss-Chebyshev Quadrature; 4.8-3 Singular Integrals
  4.9 Composite Quadrature Formulas
  4.10 Newton-Cotes Quadrature Formulas
    4.10-1 Composite Newton-Cotes Formulas; 4.10-2 Romberg Integration
  4.11 Adaptive Integration
  4.12 Choosing a Quadrature Formula
  4.13 Summation
    4.13-1 The Euler-Maclaurin Sum Formula; 4.13-2 Summation of Rational Functions; Factorial Functions; 4.13-3 The Euler Transformation
  Bibliographic Notes; Bibliography; Problems
Chapter 5. The Numerical Solution of Ordinary Differential Equations
  5.1 Statement of the Problem
  5.2 Numerical Integration Methods
    5.2-1 The Method of Undetermined Coefficients
  5.3 Truncation Error in Numerical Integration Methods
  5.4 Stability of Numerical Integration Methods
    5.4-1 Convergence and Stability; 5.4-2 Propagated-Error Bounds and Estimates
  5.5 Predictor-Corrector Methods
    5.5-1 Convergence of the Iterations; 5.5-2 Predictors and Correctors; 5.5-3 Error Estimation; 5.5-4 Stability
  5.6 Starting the Solution and Changing the Interval
    5.6-1 Analytic Methods; 5.6-2 A Numerical Method; 5.6-3 Changing the Interval
  5.7 Using Predictor-Corrector Methods
    5.7-1 Variable-Order--Variable-Step Methods; 5.7-2 Some Illustrative Examples
  5.8 Runge-Kutta Methods
    5.8-1 Errors in Runge-Kutta Methods; 5.8-2 Second-Order Methods; 5.8-3 Third-Order Methods; 5.8-4 Fourth-Order Methods; 5.8-5 Higher-Order Methods; 5.8-6 Practical Error Estimation;
      5.8-7 Step-size Strategy; 5.8-8 Stability; 5.8-9 Comparison of Runge-Kutta and Predictor-Corrector Methods
  5.9 Other Numerical Integration Methods
    5.9-1 Methods Based on Higher Derivatives; 5.9-2 Extrapolation Methods
  5.10 Stiff Equations
  Bibliographic Notes; Bibliography; Problems
Chapter 6. Functional Approximation: Least-Squares Techniques
  6.1 Introduction
  6.2 The Principle of Least Squares
  6.3 Polynomial Least-Squares Approximations
    6.3-1 Solution of the Normal Equations; 6.3-2 Choosing the Degree of the Polyn
  6.4 Orthogonal-Polynomial Approximations
  6.5 An Example of the Generation of Least-Squares Approximations
  6.6 The Fourier Approximation
    6.6-1 The Fast Fourier Transform; 6.6-2 Least-Squares Approximations and Trigonometric Interpolation
  Bibliographic Notes; Bibliography; Problems
Chapter 7. Functional Approximation: Minimum Maximum Error Techniques
  7.1 General Remarks
  7.2 Rational Functions, Polynomials, and Continued Fractions
  7.3 Padé Approximations
  7.4 An Example
  7.5 Chebyshev Polynomials
  7.6 Chebyshev Expansions
  7.7 Economization of Rational Functions
    7.7-1 Economization of Power Series; 7.7-2 Generalization to Rational Functions
  7.8 Chebyshev's Theorem of Minimax Approximations
  7.9 Constructing Minimax Approximations
    7.9-1 The Second Algorithm of Remes; 7.9-2 The Differential Correction Algorithm
  Bibliographic Notes; Bibliography; Problems
Chapter 8. The Solution of Nonlinear Equations
  8.1 Introduction
  8.2 Functional Iteration
    8.2-1 Computational Efficiency
  8.3 The Secant Method
  8.4 One-Point Iteration Formulas
  8.5 Multipoint Iteration Formulas
    8.5-1 Iteration Formulas Using General Inverse Interpolation; 8.5-2 Derivative Estimated Iteration Formulas
  8.6 Functional Iteration at a Multiple Root
  8.7 Some Computational Aspects of Functional Iteration
    8.7-1 The delta superscript 2 Process
  8.8 Systems of Nonlinear Equations
  8.9 The Zeros of Polynomials: The Problem
    8.9-1 Sturm Sequences
  8.10 Classical Methods
    8.10-1 Bairstow's Method; 8.10-2 Graeffe's Root-squaring Method; 8.10-3 Bernoulli's Method; 8.10-4 Laguerre's Method
  8.11 The Jenkins-Traub Method
  8.12 A Newton-based Method
  8.13 The Effect of Coefficient Errors on the Roots; Ill-conditioned Polynomials
  Bibliographic Notes; Bibliography; Problems
Chapter 9. The Solution of Simultaneous Linear Equations
  9.1 The Basic theorem and the Pr
  9.2 General Remarks
  9.3 Direct Methods
    9.3-1 Gaussian Elimination; 9.3-2 Compact forms of Gaussian Elimination; 9.3-3 The Doolittle, Crout, and Cholesky Algorithms; 9.3-4 Pivoting and Equilibration
  9.4 Error Analysis
    9.4-1 Roundoff-Error Analysis
  9.5 Iterative Refinement
  9.6 Matrix Iterative Methods
  9.7 Stationary Iterative Processes and Related Matters
    9.7-1 The Jacobi Iteration; 9.7-2 The Gauss-Seidel Method; 9.7-3 Roundoff Error in Iterative Methods; 9.7-4 Acceleration of Stationary Iterative Processes
  9.8 Matrix Inversion
  9.9 Overdetermined Systems of Linear Equations
  9.10 The Simplex Method for Solving Linear Programming Problems
  9.11 Miscellaneous topics
  Bibliographic Notes; Bibliography; Problems
Chapter 10. The Calculation of Eigenvalues and Eigenvectors of Matrices
  10.1 Basic Relationships
    10.1-1 Basic Theorems; 10.1-2 The characteristic Equation; 10.1-3 The Location of, and Bo
    10.2-1 Acceleration of convergence; 10.2-2 The Inverse Power Method
  10.3 The Eigenvalues and Eigenvectors of Symmetric Matrices
    10.3-1 The Jacobi Method; 10.3-2 Givens' Method; 10.3-3 Householder's Method
  10.4 Methods for Nonsymmetric Matrices
    10.4-1 Lanczos' Method; 10.4-2 Supertriangularization; 10.4-3 Jacobi-Type Methods
  10.5 The LR and QR Algorithms
    10.5-1 The Simple QR Algorithm; 10.5-2 The Double QR Algorithm
  10.6 Errors in Computed eigenvalues and Eigenvectors
  Bibliographic Notes; Bibliography; Problems
  Index; Hints and Answers to Problems