Introduction to Numerical Analysis and Scientific Computing

By Nabil Nassif, Dolly Khuwayri Fayyad

© 2013 – Chapman and Hall/CRC

329 pages | 26 B/W Illus.

Purchasing Options:
Hardback: 9781466589483
pub: 2013-08-04
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Comp Exam Copy

About the Book

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.

Reviews

"… 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

Table of Contents

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.

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT004000
MATHEMATICS / Arithmetic
MAT021000
MATHEMATICS / Number Systems