1st Edition

A Theoretical Introduction to Numerical Analysis

    552 Pages 50 B/W Illustrations
    by Chapman & Hall

    552 Pages 50 B/W Illustrations
    by Chapman & Hall

    A Theoretical Introduction to Numerical Analysis presents the general methodology and principles of numerical analysis, illustrating these concepts using numerical methods from real analysis, linear algebra, and differential equations. The book focuses on how to efficiently represent mathematical models for computer-based study.

    An accessible yet rigorous mathematical introduction, this book provides a pedagogical account of the fundamentals of numerical analysis. The authors thoroughly explain basic concepts, such as discretization, error, efficiency, complexity, numerical stability, consistency, and convergence. The text also addresses more complex topics like intrinsic error limits and the effect of smoothness on the accuracy of approximation in the context of Chebyshev interpolation, Gaussian quadratures, and spectral methods for differential equations. Another advanced subject discussed, the method of difference potentials, employs discrete analogues of Calderon’s potentials and boundary projection operators. The authors often delineate various techniques through exercises that require further theoretical study or computer implementation.

    By lucidly presenting the central mathematical concepts of numerical methods, A Theoretical Introduction to Numerical Analysis provides a foundational link to more specialized computational work in fluid dynamics, acoustics, and electromagnetism.

    PREFACE
    ACKNOWLEDGMENTS
    INTRODUCTION
    Discretization
    Conditioning
    Error
    On Methods of Computation
    INTERPOLATION OF FUNCTIONS. QUADRATURES
    ALGEBRAIC INTERPOLATION
    Existence and Uniqueness of Interpolating Polynomial
    Classical Piecewise Polynomial Interpolation
    Smooth Piecewise Polynomial Interpolation (Splines)
    Interpolation of Functions of Two Variables
    TRIGONOMETRIC INTERPOLATION
    Interpolation of Periodic Functions
    Interpolation of Functions on an Interval. Relation between Algebraic and Trigonometric Interpolation
    COMPUTATION OF DEFINITE INTEGRALS. QUADRATURES
    Trapezoidal Rule, Simpson’s Formula, and the Like
    Quadrature Formulae with No Saturation. Gaussian Quadratures
    Improper Integrals. Combination of Numerical and Analytical Methods
    Multiple Integrals
    SYSTEMS OF SCALAR EQUATIONS
    SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS: DIRECT METHODS
    Different Forms of Consistent Linear Systems
    Linear Spaces, Norms, and Operators
    Conditioning of Linear Systems
    Gaussian Elimination and Its Tri-Diagonal Version
    Minimization of Quadratic Functions and Its Relation to Linear Systems
    The Method of Conjugate Gradients
    Finite Fourier Series
    ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS
    Richardson Iterations and the Like
    Chebyshev Iterations and Conjugate Gradients
    Krylov Subspace Iterations
    Multigrid Iterations
    OVERDETERMINED LINEAR SYSTEMS. THE METHOD OF LEAST SQUARES
    Examples of Problems that Result in Overdetermined Systems
    Weak Solutions of Full Rank Systems. QR Factorization
    Rank Deficient Systems. Singular Value Decomposition
    NUMERICAL SOLUTION OF NONLINEAR EQUATIONS AND SYSTEMS
    Commonly Used Methods of Rootfinding
    Fixed Point Iterations
    Newton’s Method
    THE METHOD OF FINITE DIFFERENCES FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS
    NUMERCAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS
    Examples of Finite-Difference Schemes. Convergence
    Approximation of Continuous Problem by a Difference Scheme. Consistency
    Stability of Finite-Difference Schemes
    The Runge-Kutta Methods
    Solution of Boundary Value Problems
    Saturation of Finite-Difference Methods
    The Notion of Spectral Methods
    FINITE-DIFFERENCE SCHEMES FOR PARTIAL DIFFERENTIAL EQUATIONS
    Key Definitions and Illustrating Examples
    Construction of Consistent Difference Schemes
    Spectral Stability Criterion for Finite-Difference Cauchy Problems
    Stability for Problems with Variable Coefficients
    Stability for Initial Boundary Value Problems
    Explicit and Implicit Schemes for the Heat Equation
    DISCONTINUOUS SOLUTIONS AND METHODS OF THEIR COMPUTATION
    Differential Form of an Integral Conservation Law
    Construction of Difference Schemes
    DISCRETE METHODS FOR ELLIPTIC PROBLEMS
    A Simple Finite-Difference Scheme. The Maximum Principle
    The Notion of Finite Elements. Ritz and Galerkin Approximations
    THE METHODS OF BOUNDARY EQUATIONS FOR THE NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS
    BOUNDARY INTEGRAL EQUATIONS AND THE METHOD OF BOUNDARY ELEMENTS
    Reduction of Boundary Value Problems to Integral Equations
    Discretization of Integral Equations and Boundary Elements
    The Range of Applicability for Boundary Elements
    BOUNDARY EQUATIONS WITH PROJECTIONS AND THE METHOD OF DIFFERENCE POTENTIALS
    Formulation of Model Problems
    Difference Potentials
    Solution of Model Problems
    LIST OF FIGURES
    REFERENCED BOOKS
    REFERENCED JOURNAL ARTICLES
    INDEX

    Biography

    Victor S. Ryaben'kii, Semyon V Tsynkov

    “… presents the general methodology and principles of numerical analysis, illustrating the key concepts using numerical methods from real analysis, linear algebra, and differential equations. The book focuses on hoe to efficiently represent mathematical models for computer-based study. … this book provides a pedagogical account of the fundamentals of numerical analysis. … provides a foundation link to more specialized computational work in mathematics, science, and engineering. … Discusses three common numerical areas: interpolation and quadratures, linear and nonlinear solvers, and finite differences. Explains the most fundamental and universal concepts, including error, efficiency, complexity, stability, and convergence. Addresses advance topics, such as intrinsic accuracy limits, saturation of numerical methods by smoothness, and the method of difference potentials. Provides rigorous proofs for all important mathematical results. Includes numerous examples and exercises to illustrate key theoretical ideas and to enable independent study. ”
    — In Mathematical Reviews, Issue 2007g


    “It is an excellent book, having a wide spectrum of classical and advanced topics. The book has all the advantages of the Russian viewpoint as well as the Western one.”
    —David Gottlieb, Brown University, Providence, Rhode Island, USA