Numerical Analysis

Computer Science and Applied Mathematics: Numerical Analysis: A Second Course presents some of the basic theoretical results pertaining to the three major problem areas of numerical analysis—rounding error, discretization error, and convergence error. This book is organized into four main topics: mathematical stability and ill conditioning, discretization error, convergence of iterative methods, and rounding error. In these topics, this text specifically discusses the systems of linear algebraic equations, eigenvalues and eigenvectors, and differential and difference equations. The discretization error for initial and boundary value problems, systems of linear and nonlinear equations, and rounding error for Gaussian elimination are also elaborated. This publication is recommended for undergraduate level students and students taking a one-semester first-year graduate course for computer science and mathematics majors.

Preface

List of Commonly Used Symbols

Introduction

Chapter 1 Linear Algebra

1.1 Eigenvalues and Canonical Forms

1.2 Vector Norms

1.3 Matrix Norms

Part I Mathematically and Ill Conditioning

Chapter 2 Systems of Linear Algebraic Equations

2.1 Basic Error Estimates and Condition Numbers

2.2 A Posteriori Bounds and Eigenvector Computations

Chapter 3 Eigenvalues and Eigenvectors

3.1 Continuity Results

3.2 The Gerschgorin and Bauer-Fike Theorems

3.3 Special Results for Symmetric Matrices

Chapter 4 Differential and Difference Equations

4.1 Differential Equations

4.2 Difference Equations

Part II Discretization Error

Chapter 5 Discretization Error for Initial Value Problems

5.1 Consistency and Stability

5.2 Convergence and Order

Chapter 6 Discretization Error for Boundary Value Problems

6.1 The Maximum Principle

6.2 Matrix Methods

Part III Convergence of Iterative Methods

Chapter 7 Systems of Linear Equations

7.1 Convergence

7.2 Rate of Convergence

7.3 Applications to Differential Equations

Chapter 8 Systems of Nonlinear Equations

8.1 Local Convergence and Rate of Convergence

8.2 Error Estimates

8.3 Global Convergence

Part IV Rounding Error

Chapter 9 Rounding Error for Gaussian Elimination

9.1 Review of the Method

9.2 Rounding Error and Interchange Strategies

9.3 Backward Error Analysis

9.4 Iterative Refinement

Bibliography

Index