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.
Table of Contents
Computer Number Systems and Floating Point Arithmetic
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
How to Locate the Roots of a Function
The Bisection Method
The Secant Method
Solving Systems of Linear Equations by Gaussian Elimination
Computer Storage for Matrices. Data Structures
Back Substitution for Upper Triangular Systems
Polynomial Interpolation and Splines Fitting
Definition of Interpolation
General Lagrange Polynomial Interpolation
Equally Spaced Data: Difference Operators
Errors in Polynomial Interpolation
Local Interpolation: Spline Functions
Numerical Differentiation and Integration
Richardson Extrapolation in Numerical Differentiation
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)
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
Exercises and Computer Projects appear at the end of each chapter.
"... 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