1st Edition

Classical and Modern Numerical Analysis Theory, Methods and Practice

    628 Pages 40 B/W Illustrations
    by Chapman & Hall

    Classical and Modern Numerical Analysis: Theory, Methods and Practice provides a sound foundation in numerical analysis for more specialized topics, such as finite element theory, advanced numerical linear algebra, and optimization. It prepares graduate students for taking doctoral examinations in numerical analysis.

    The text covers the main areas of introductory numerical analysis, including the solution of nonlinear equations, numerical linear algebra, ordinary differential equations, approximation theory, numerical integration, and boundary value problems. Focusing on interval computing in numerical analysis, it explains interval arithmetic, interval computation, and interval algorithms. The authors illustrate the concepts with many examples as well as analytical and computational exercises at the end of each chapter.

    This advanced, graduate-level introduction to the theory and methods of numerical analysis supplies the necessary background in numerical methods so that students can apply the techniques and understand the mathematical literature in this area. Although the book is independent of a specific computer program, MATLAB® code is available on the authors' website to illustrate various concepts.

    Mathematical Review and Computer Arithmetic

    Mathematical Review

    Computer Arithmetic

    Interval Computations

    Numerical Solution of Nonlinear Equations of One Variable


    Bisection Method

    The Fixed Point Method

    Newton’s Method (Newton–Raphson Method)

    The Univariate Interval Newton Method

    Secant Method and Müller’s Method

    Aitken Acceleration and Steffensen’s Method

    Roots of Polynomials

    Additional Notes and Summary

    Numerical Linear Algebra

    Basic Results from Linear Algebra

    Normed Linear Spaces

    Direct Methods for Solving Linear Systems

    Iterative Methods for Solving Linear Systems

    The Singular Value Decomposition

    Approximation Theory


    Norms, Projections, Inner Product Spaces, and Orthogonalization in Function Spaces

    Polynomial Approximation

    Piecewise Polynomial Approximation

    Trigonometric Approximation

    Rational Approximation

    Wavelet Bases

    Least Squares Approximation on a Finite Point Set

    Eigenvalue-Eigenvector Computation

    Basic Results from Linear Algebra

    The Power Method

    The Inverse Power Method


    The QR Method

    Jacobi Diagonalization (Jacobi Method)

    Simultaneous Iteration (Subspace Iteration)

    Numerical Differentiation and Integration

    Numerical Differentiation

    Automatic (Computational) Differentiation

    Numerical Integration

    Initial Value Problems for Ordinary Differential Equations


    Euler’s Method

    Single-Step Methods: Taylor Series and Runge–Kutta

    Error Control and the Runge–Kutta–Fehlberg Method

    Multistep Methods

    Predictor-Corrector Methods

    Stiff Systems

    Extrapolation Methods

    Application to Parameter Estimation in Differential Equations

    Numerical Solution of Systems of Nonlinear Equations

    Introduction and Fréchet Derivatives

    Successive Approximation (Fixed Point Iteration) and the Contraction Mapping Theorem

    Newton’s Method and Variations

    Multivariate Interval Newton Methods

    Quasi-Newton Methods (Broyden’s Method)

    Methods for Finding All Solutions


    Local Optimization

    Constrained Local Optimization

    Constrained Optimization and Nonlinear Systems

    Linear Programming

    Dynamic Programming

    Global (Nonconvex) Optimization

    Boundary-Value Problems and Integral Equations

    Boundary-Value Problems

    Approximation of Integral Equations

    Appendix: Solutions to Selected Exercises



    Exercises appear at the end of each chapter.


    Azmy S. Ackleh is Dr. Ray P. Authement/BORSF Eminent Scholar Endowed Chair in Computational Mathematics at the University of Louisiana. Dr. Ackleh has more than fifteen years experience in mathematical biology with an emphasis on the long-time behavior of discrete and continuous population models and numerical methods for structured-population models.

    Edward James Allen is a professor of mathematics at Texas Tech University. Dr. Allen works primarily on the derivation and computation of stochastic differential equation models in biology and physics and on the development and analysis of numerical methods for problems in neutron transport.

    R. Baker Kearfott is a professor of mathematics at the University of Louisiana, with over thirty years experience teaching numerical analysis. Dr. Kearfott’s research focuses on nonlinear equations, nonlinear optimization, and mathematically rigorous numerical analysis.

    Padmanabhan Seshaiyer is an associate professor of mathematical sciences at George Mason University. Dr. Seshaiyer has done extensive work on the theoretical and computational aspects of finite element methods and applications of numerical methods to biological and bio-inspired problems.

    …this book provides useful background knowledge for graduate study in any area of applied mathematics … this is a thorough, well-written treatment of an important subject.

    Computing Reviews, May 2010