1st Edition
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 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
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.
Biography
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
