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Introduction to Numerical Analysis and Scientific Computing

By Nabil Nassif, Dolly Khuwayri Fayyad

Chapman and Hall/CRC – 2013 – 329 pages

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    978-1-46-658948-3
    August 4th 2013

Description

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.

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.

Name: Introduction to Numerical Analysis and Scientific Computing (Hardback)Chapman and Hall/CRC 
Description: By Nabil Nassif, Dolly Khuwayri Fayyad. 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...
Categories: Computational Numerical Analysis, Mathematics & Statistics for Engineers, Mathematical Physics