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
Victor S. Ryaben'kii, Semyon V Tsynkov
Reviews for A Theoretical Introduction to Numerical Analysis
... 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