Numerical Analysis

Numerical Analysis

A Second Course

1st Edition - January 1, 1972

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  • Author: James M. Ortega
  • eBook ISBN: 9781483268507

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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.

Table of Contents

  • Preface

    List of Commonly Used Symbols


    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



Product details

  • No. of pages: 214
  • Language: English
  • Copyright: © Academic Press 1972
  • Published: January 1, 1972
  • Imprint: Academic Press
  • eBook ISBN: 9781483268507

About the Author

James M. Ortega

About the Editor

Werner Rheinboldt

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