Classical and Modern Numerical Analysis Theory, Methods and Practice

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 theauthors' website to illustrate various concepts.

Mathematical Review and Computer Arithmetic Mathematical Review Computer Arithmetic Interval Computations Numerical Solution of Nonlinear Equations of One Variable Introduction 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 Introduction 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 Deflation 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 Introduction 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 Optimization 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 References Index 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.