1st Edition
Introduction to Numerical Analysis and Scientific Computing
Designed for a one-semester course, Introduction to Numerical Analysis and Scientific Computing presents fundamental concepts of numerical mathematics and explains how to implement and program numerical methods. The classroom-tested text helps students understand floating point number representations, particularly those pertaining to IEEE simple and double-precision standards as used in scientific computer environments such as MATLAB® version 7.
Drawing on their years of teaching students in mathematics, engineering, and the sciences, the authors discuss computer arithmetic as a source for generating round-off errors and how to avoid the use of algebraic expression that may lead to loss of significant figures. They cover nonlinear equations, linear algebra concepts, the Lagrange interpolation theorem, numerical differentiation and integration, and ODEs. They also focus on the implementation of the algorithms using MATLAB®.
Each chapter ends with a large number of exercises, with answers to odd-numbered exercises provided at the end of the book. Throughout the seven chapters, several computer projects are proposed. These test the students' understanding of both the mathematics of numerical methods and the art of computer programming.
Computer Number Systems and Floating Point Arithmetic
Introduction
Conversion from Base 10 to Base 2
Conversion from Base 2 to Base 10
Normalized Floating Point Systems
Floating Point Operations
Computing in a Floating Point System
Finding Roots of Real Single-Valued Functions
Introduction
How to Locate the Roots of a Function
The Bisection Method
Newton's Method
The Secant Method
Solving Systems of Linear Equations by Gaussian Elimination
Mathematical Preliminaries
Computer Storage for Matrices. Data Structures
Back Substitution for Upper Triangular Systems
Gauss Reduction
LU Decomposition
Polynomial Interpolation and Splines Fitting
Definition of Interpolation
General Lagrange Polynomial Interpolation
Recurrence Formulae
Equally Spaced Data: Difference Operators
Errors in Polynomial Interpolation
Local Interpolation: Spline Functions
Concluding Remarks
Numerical Differentiation and Integration
Introduction
Mathematical Prerequisites
Numerical Differentiation
Richardson extrapolation
Richardson Extrapolation in Numerical Differentiation
Numerical Integration
Romberg Integration
Appendix
Advanced Numerical Integration
Numerical Integration for Nonuniform Partitions
Numerical Integration of Functions of Two Variables
Monte Carlo Simulations for Numerical Quadrature
Numerical Solutions of Ordinary Differential Equations (ODEs)
Introduction
Analytic Solutions to ODE
Mathematical Settings for Numerical Solutions to ODEs
Explicit Runge-Kutta Schemes
Adams Multistep Methods
Multistep Backward Difference Formulae
Finite-Difference Approximation to a Two-Points Boundary Value Problem
Bibliography
Index
Exercises and Computer Projects appear at the end of each chapter.
Biography
<p>Nabil Nassif received a Diplôme-Ingénieur from the Ecole Centrale de Paris and earned a master's degree in applied mathematics from Harvard University, followed by a PhD under the supervision of Professor Garrett Birkhoff. Since his graduation, Dr. Nassif has been affiliated with the Mathematics Department at the American University of Beirut, where he teaches and conducts research in the areas of mathematical modeling, numerical analysis and scientific computing. Professor Nassif has authored or co-authored about 50 publications in refereed journals and directed 12 PhD theses with an equal number of master's theses. During his career, Professor Nassif has also held several regular and visiting teaching positions in France, Switzerland, U.S.A. and Sweden.</p><p>Dolly Khoueiri Fayyad received her BSc and master's degrees from the American University of Beirut and her PhD degree from the University of Reims in France under the supervision of Professor Nabil Nassif. After earning her doctorate degree and before becoming a faculty member in the Mathematics Department of the American University of Beirut, she taught at the University of Louvain-la-Neuve in Belgium and then in the Sciences Faculty of Lebanon National University. Simultaneously, Dr. Fayyad has conducted research on the numerical solution of time-dependent partial differential equations and more particularly on semi-linear parabolic equations. She has also supervised several master's theses in her research areas.</p>
"... an introduction to basic topics of numerical analysis which can be covered in a one-semester course for students of Mathematics, Natural Sciences or Engineering. The topics covered include finding roots of nonlinear equations using the bisection method, Newton's method and the secant method; the Gaussian elimination method for solving linear systems; function interpolation and fitting; numerical differentiation and integration; and numerical methods for ordinary differential equations. The methods are introduced and their convergence and stability are discussed in some details. It also includes a chapter on computer number systems and floating point arithmetic. Computer codes written in MATLAB are also included. This book is suitable for undergraduate students and people who begin to learn about numerical analysis. Exercises and computer projects provided at the end of each chapter can help students to practice computational and programming skills."
—Trung Thanh Nguyen, in Zentralblatt MATH 1281