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Summary
This reader-friendly introduction to the fundamental concepts and techniques of numerical analysis/numerical methods develops concepts and techniques in a clear, concise, easy-to- read manner, followed by fully-worked examples. Application problems drawn from the literature of many different fields prepares readers to use the techniques covered to solve a wide variety of practical problems.Rootfinding. Systems of Equations. Eigenvalues and Eigenvectors. Interpolation and Curve Fitting. Numerical Differentiation and Integration. Numerical Methods for Initial Value Problems of Ordinary Differential Equations. Second-Order One-Dimensional Two-Point Boundary Value Problems. Finite Difference Method for Elliptic Partial Differential Equations. Finite Difference Method for Parabolic Partial Differential Equations. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation.For anyone interested in numerical analysis/methods and their applications in many fields
Table of Contents
(NOTE: Each chapter begins with An Overview.) 1. Getting Started.
Algorithms. Convergence. Floating Point Numbers. Floating Point Arithmetic.
2. Rootfinding.
Bisection Method. Method of False Position. Fixed Point Iteration. Newton's Method. The Secant Method and Muller's Method. Accelerating Convergence. Roots of Polynomials.
3. Systems of Equations.
Gaussian Elimination. Pivoting Strategies. Norms. Error Estimates. LU Decomposition. Direct Factorization. Special Matrices. Iterative Techniques for Linear Systems: Basic Concepts and Methods. Iterative Techniques for Linear Systems: Conjugate-Gradient Method. Nonlinear Systems.
4. Eigenvalues and Eigenvectors.
The Power Method. The Inverse Power Method. Deflation. Reduction to Tridiagonal Form. Eigenvalues of Tridiagonal and Hessenberg Matrices.
5. Interpolation and Curve Fitting.
Lagrange Form of the Interpolating Polynomial. Neville's Algorithm. The Newton Form of the Interpolating Polynomial and Divided Differences. Optimal Interpolating Points. Piecewise Linear Interpolation. Hermite and Hermite Cubic Interpolation. Regression.
6. Numerical Differentiation and Integration.
Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.
7. Numerical Methods for Initial Value Problems of Ordinary Differential Equations.
Introduction. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.
8. Second-Order One-Dimensional Two-Point Boundary Value Problems.
Finite Difference Method, Part I: The Linear Problem with Dirichlet Boundary Conditions. Finite Difference Method, Part II: The Linear Problem with Non-Dirichlet Boundary Conditions. Finite Difference Method, Part III: Nonlinear Problems. The Shooting Method, Part I: Linear Boundary Value Problems. The Shooting Method, Part II: Nonlinear Boundary Value Problems.
9. Finite Difference Method for Elliptic Partial Differential Equations.
The Poisson Equation on a Rectangular Domain, I: Dirichlet Boundary Conditions. The Poisson Equation on a Rectangular Domain, II: Non-Dirichlet Boundary Conditions. Solving the Discrete Equations: Relaxation Schemes. Local Mode Analysis of Relaxation and the Multigrid Method. Irregular Domains.
10. Finite Difference Method for Parabolic Partial Differential Equations.
The Heat Equation with Dirichlet Boundary Conditions. Stability. More General Parabolic Equations. Non-Dirichlet Boundary Conditions. Polar Coordinates. Problems in Two Space Dimensions.
11. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation.
Appendix A. Important Theorems from Calculus. Appendix B. Algorithm for Solving a Tridiagonal System of Linear Equations.
References. Index.
Excerpts
This is an introduction to the fundamental concepts and techniques of numerical analysis and numerical methods for undergraduates, as well as for graduate engineers arid applied scientists receiving their first exposure to numerical analysis. Applications drawn from the literature of many different fields will prepare students to use the techniques covered to solve a wide variety of practical problems. There is also sufficient mathematical detail to prepare students to embark upon an investigation of more advanced topics, especially in PDEs. The presentation style is what I like to call tell and show. This means that the concepts and techniques are first developed in a clear, concise, and easy-to-read manner, and then illustrated with at least one fully worked example. In total, nearly 250 fully worked examples are presented to help the students grasp the sequence of calculations associated with a particular method and gain better insight into algorithm operation.The text is organized around mathematical problems, with each chapter devoted to a single type of problem (e.g., rootfinding, numerical calculus: differentiation and integration, the matrix eigenvalue problem, and elliptic partial differential equations). Within each chapter the presentation begins with the simplest and most basic methods, progressing gradually to more advanced topics. Early chapters generally contain easier material, while later chapters proceed at increasing levels of difficulty and complexity. Throughout, emphasis is placed on understanding and being able to work with the key concepts of rate/order of convergence and stability, and assessing the accuracy of numerical results. This emphasis helps students develop skill in numerically verifying theoretical convergence speed. More importantly, the text emphasizes that it is not sufficient to obtain the correct answer from a numerical algorithm; one must also check that convergence toward the correct answer is happening at the correct speed. I have always felt very strongly that a textbook must provide students with some means of checking their understanding and honing their skills, some means of making the knowledge their own. This is invariably accomplished through the exercises. This text features more than 1200 numbered exercises (many with multiple parts) organized into exercise sets at the end of each section. Each exercise set contains problems designed to provide students with the opportunity to practice (with paper, pencil, and calculator) the sequence of calculations associated with a particular method. The exercises usually also require the verification of theoretical error bounds and/or theoretical rates of convergence. Additional exercises may require the derivation of a method, an examination of conditions under which methods perform better or worse than predicted by theory, or extension of material presented in the section. Many exercises require students to code a numerical method on the computer and then use that computer code, and many exercises are application problems that require interpretation of results. Distinctive FeaturesA quick scan of the table of contents will reveal that certain topics typically found in a book of this nature, such as approximation (orthogonal least-squares, FFT, rational function approximation) and optimization, have been omitted. In place of these topics is an extensive coverage of material not usually found, or only briefly discussed, in other texts. This extensive coverage includes treatment of non-Dirichlet boundary conditions, and artificial singularities for one-dimensional boundary value problems; treatment of non-Dirichlet boundary conditions, the multigrid method and irregular domains for elliptic partial differential equations; treatment of source and decay terms, non-Dirichlet boundary conditions, polar coordinates and problems in two space dimensions for parabolic partial differential equations; and treatment of the ad