### A MATLAB Approach, Second Edition

### A MATLAB Approach, Second Edition

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Numerical methods are a mainstay of researchers and professionals across the many mathematics, scientific, and engineering disciplines. The importance of these methods combined with the power and availability of today's computers virtually demand that students in these fields be well versed not only in the numerical techniques, but also in the use of a modern computational software package.

Updated to reflect the latest version of MATLAB, the second edition of An Introduction to Numerical Methods continues to fulfill both these needs. It introduces the theory and applications of the most commonly used techniques for solving numerical problems on a computer. It covers a wide range of useful algorithms, each presented with full details so that readers can visualize and interpret each step.

Highlights of the second edition:

Emphasis on understanding how the methods work, a simple, direct style, and thorough coverage make this book an outstanding initiation that allows students to see almost immediate results. It will boost their confidence in their ability to master the subject and give them valuable experience in the use of MATLAB.

INTRODUCTION

About the software MATLAB

An Introduction to MATLAB

Taylor Series

NUMBER SYSTEM AND ERRORS

Floating-Point Arithmetic

Round-Off Errors

Truncation Error

Interval Arithmetic

ROOTS OF EQUATIONS

The Bisection Method

The Method of False Position

Fixed-Point Iteration

The Secant method

Newton's Method

Convergence of the Newton and Secant Methods

Multiple Roots and the Modified Newton Method

Newton's Method for Nonlinear Systems

SYSTEM OF LINEAR EQUATIONS

Matrices and Matrix Operations

Naïve Gaussian Elimination

Gaussian Elimination with Scaled Partial Pivoting

LU Decomposition

Iterative Methods

INTERPOLATION

Polynomial Interpolation Theory

Newton's Divided Difference Interpolating Polynomial

The Error of the Interpolating Polynomial

Lagrange Interpolating Polynomial

INTERPOLATION WITH SPLINE FUNCTIONS

Piecewise Linear Interpolation

Quadratic Spline

Natural Cubic Splines

THE METHOD OF LEAST SQUARES

Linear Least Squares

Least Squares Polynomial

Nonlinear Least Squares

Trigonometric Least Squares Polynomial

NUMERICAL OPTIMIZATION

Analysis of Single-Variable Functions

Line Search Methods

Minimization Using Derivatives

NUMERICAL DIFFERENTIATION

Numerical Differentiation

Richardson's Formula

NUMERICAL INTEGRATION

Trapezoidal Rule

Simpson's Rule

Romberg Algorithm

Gaussian Quadrature

NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS

Euler's Method

Error Analysis

Higher Order Taylor Series Methods

Runge-Kutta Methods

Multistep Methods

Adams-Bashforth Method

Predictor-Corrector Methods

Adams-Moulton Method

Numerical Stability

Higher Order Equations and Systems of Differential Equations

Implicit Methods and Stiff Systems

Phase Plane Analysis: Chaotic Differential Equations

BOUNDARY-VALUE PROBLEMS

Finite-Difference Methods

Shooting Methods

EIGENVALUES AND EIGENVECTORS

Basic Theory

The Power Method

The Quadratic Method

Eigenvalues for Boundary-Value Problems

Bifurcations in Differential Equations

PARTIAL DIFFERENTIAL EQUATIONS

Parabolic Equations

Hyperbolic Equations

Elliptic Equations

Introduction to Finite Element Method

Bibliography and References

Appendices

Calculus Review

MATLAB Built-in Functions

Text MATLAB Functions

Answers to Selected Exercises

Index

Each chapter also contains a section of Applied Problems.

Numerical methods are a mainstay of researchers and professionals across the many mathematics, scientific, and engineering disciplines. The importance of these methods combined with the power and availability of today's computers virtually demand that students in these fields be well versed not only in the numerical techniques, but also in the use of a modern computational software package.

Updated to reflect the latest version of MATLAB, the second edition of An Introduction to Numerical Methods continues to fulfill both these needs. It introduces the theory and applications of the most commonly used techniques for solving numerical problems on a computer. It covers a wide range of useful algorithms, each presented with full details so that readers can visualize and interpret each step.

Highlights of the second edition:

Emphasis on understanding how the methods work, a simple, direct style, and thorough coverage make this book an outstanding initiation that allows students to see almost immediate results. It will boost their confidence in their ability to master the subject and give them valuable experience in the use of MATLAB.

INTRODUCTION

About the software MATLAB

An Introduction to MATLAB

Taylor Series

NUMBER SYSTEM AND ERRORS

Floating-Point Arithmetic

Round-Off Errors

Truncation Error

Interval Arithmetic

ROOTS OF EQUATIONS

The Bisection Method

The Method of False Position

Fixed-Point Iteration

The Secant method

Newton's Method

Convergence of the Newton and Secant Methods

Multiple Roots and the Modified Newton Method

Newton's Method for Nonlinear Systems

SYSTEM OF LINEAR EQUATIONS

Matrices and Matrix Operations

Naïve Gaussian Elimination

Gaussian Elimination with Scaled Partial Pivoting

LU Decomposition

Iterative Methods

INTERPOLATION

Polynomial Interpolation Theory

Newton's Divided Difference Interpolating Polynomial

The Error of the Interpolating Polynomial

Lagrange Interpolating Polynomial

INTERPOLATION WITH SPLINE FUNCTIONS

Piecewise Linear Interpolation

Quadratic Spline

Natural Cubic Splines

THE METHOD OF LEAST SQUARES

Linear Least Squares

Least Squares Polynomial

Nonlinear Least Squares

Trigonometric Least Squares Polynomial

NUMERICAL OPTIMIZATION

Analysis of Single-Variable Functions

Line Search Methods

Minimization Using Derivatives

NUMERICAL DIFFERENTIATION

Numerical Differentiation

Richardson's Formula

NUMERICAL INTEGRATION

Trapezoidal Rule

Simpson's Rule

Romberg Algorithm

Gaussian Quadrature

NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS

Euler's Method

Error Analysis

Higher Order Taylor Series Methods

Runge-Kutta Methods

Multistep Methods

Adams-Bashforth Method

Predictor-Corrector Methods

Adams-Moulton Method

Numerical Stability

Higher Order Equations and Systems of Differential Equations

Implicit Methods and Stiff Systems

Phase Plane Analysis: Chaotic Differential Equations

BOUNDARY-VALUE PROBLEMS

Finite-Difference Methods

Shooting Methods

EIGENVALUES AND EIGENVECTORS

Basic Theory

The Power Method

The Quadratic Method

Eigenvalues for Boundary-Value Problems

Bifurcations in Differential Equations

PARTIAL DIFFERENTIAL EQUATIONS

Parabolic Equations

Hyperbolic Equations

Elliptic Equations

Introduction to Finite Element Method

Bibliography and References

Appendices

Calculus Review

MATLAB Built-in Functions

Text MATLAB Functions

Answers to Selected Exercises

Index

Each chapter also contains a section of Applied Problems.

Numerical methods are a mainstay of researchers and professionals across the many mathematics, scientific, and engineering disciplines. The importance of these methods combined with the power and availability of today's computers virtually demand that students in these fields be well versed not only in the numerical techniques, but also in the use of a modern computational software package.

Updated to reflect the latest version of MATLAB, the second edition of An Introduction to Numerical Methods continues to fulfill both these needs. It introduces the theory and applications of the most commonly used techniques for solving numerical problems on a computer. It covers a wide range of useful algorithms, each presented with full details so that readers can visualize and interpret each step.

Highlights of the second edition:

Emphasis on understanding how the methods work, a simple, direct style, and thorough coverage make this book an outstanding initiation that allows students to see almost immediate results. It will boost their confidence in their ability to master the subject and give them valuable experience in the use of MATLAB.

INTRODUCTION

About the software MATLAB

An Introduction to MATLAB

Taylor Series

NUMBER SYSTEM AND ERRORS

Floating-Point Arithmetic

Round-Off Errors

Truncation Error

Interval Arithmetic

ROOTS OF EQUATIONS

The Bisection Method

The Method of False Position

Fixed-Point Iteration

The Secant method

Newton's Method

Convergence of the Newton and Secant Methods

Multiple Roots and the Modified Newton Method

Newton's Method for Nonlinear Systems

SYSTEM OF LINEAR EQUATIONS

Matrices and Matrix Operations

Naïve Gaussian Elimination

Gaussian Elimination with Scaled Partial Pivoting

LU Decomposition

Iterative Methods

INTERPOLATION

Polynomial Interpolation Theory

Newton's Divided Difference Interpolating Polynomial

The Error of the Interpolating Polynomial

Lagrange Interpolating Polynomial

INTERPOLATION WITH SPLINE FUNCTIONS

Piecewise Linear Interpolation

Quadratic Spline

Natural Cubic Splines

THE METHOD OF LEAST SQUARES

Linear Least Squares

Least Squares Polynomial

Nonlinear Least Squares

Trigonometric Least Squares Polynomial

NUMERICAL OPTIMIZATION

Analysis of Single-Variable Functions

Line Search Methods

Minimization Using Derivatives

NUMERICAL DIFFERENTIATION

Numerical Differentiation

Richardson's Formula

NUMERICAL INTEGRATION

Trapezoidal Rule

Simpson's Rule

Romberg Algorithm

Gaussian Quadrature

NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS

Euler's Method

Error Analysis

Higher Order Taylor Series Methods

Runge-Kutta Methods

Multistep Methods

Adams-Bashforth Method

Predictor-Corrector Methods

Adams-Moulton Method

Numerical Stability

Higher Order Equations and Systems of Differential Equations

Implicit Methods and Stiff Systems

Phase Plane Analysis: Chaotic Differential Equations

BOUNDARY-VALUE PROBLEMS

Finite-Difference Methods

Shooting Methods

EIGENVALUES AND EIGENVECTORS

Basic Theory

The Power Method

The Quadratic Method

Eigenvalues for Boundary-Value Problems

Bifurcations in Differential Equations

PARTIAL DIFFERENTIAL EQUATIONS

Parabolic Equations

Hyperbolic Equations

Elliptic Equations

Introduction to Finite Element Method

Bibliography and References

Appendices

Calculus Review

MATLAB Built-in Functions

Text MATLAB Functions

Answers to Selected Exercises

Index

Each chapter also contains a section of Applied Problems.

Highlights of the second edition:

INTRODUCTION

About the software MATLAB

An Introduction to MATLAB

Taylor Series

NUMBER SYSTEM AND ERRORS

Floating-Point Arithmetic

Round-Off Errors

Truncation Error

Interval Arithmetic

ROOTS OF EQUATIONS

The Bisection Method

The Method of False Position

Fixed-Point Iteration

The Secant method

Newton's Method

Convergence of the Newton and Secant Methods

Multiple Roots and the Modified Newton Method

Newton's Method for Nonlinear Systems

SYSTEM OF LINEAR EQUATIONS

Matrices and Matrix Operations

Naïve Gaussian Elimination

Gaussian Elimination with Scaled Partial Pivoting

LU Decomposition

Iterative Methods

INTERPOLATION

Polynomial Interpolation Theory

Newton's Divided Difference Interpolating Polynomial

The Error of the Interpolating Polynomial

Lagrange Interpolating Polynomial

INTERPOLATION WITH SPLINE FUNCTIONS

Piecewise Linear Interpolation

Quadratic Spline

Natural Cubic Splines

THE METHOD OF LEAST SQUARES

Linear Least Squares

Least Squares Polynomial

Nonlinear Least Squares

Trigonometric Least Squares Polynomial

NUMERICAL OPTIMIZATION

Analysis of Single-Variable Functions

Line Search Methods

Minimization Using Derivatives

NUMERICAL DIFFERENTIATION

Numerical Differentiation

Richardson's Formula

NUMERICAL INTEGRATION

Trapezoidal Rule

Simpson's Rule

Romberg Algorithm

Gaussian Quadrature

NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS

Euler's Method

Error Analysis

Higher Order Taylor Series Methods

Runge-Kutta Methods

Multistep Methods

Adams-Bashforth Method

Predictor-Corrector Methods

Adams-Moulton Method

Numerical Stability

Higher Order Equations and Systems of Differential Equations

Implicit Methods and Stiff Systems

Phase Plane Analysis: Chaotic Differential Equations

BOUNDARY-VALUE PROBLEMS

Finite-Difference Methods

Shooting Methods

EIGENVALUES AND EIGENVECTORS

Basic Theory

The Power Method

The Quadratic Method

Eigenvalues for Boundary-Value Problems

Bifurcations in Differential Equations

PARTIAL DIFFERENTIAL EQUATIONS

Parabolic Equations

Hyperbolic Equations

Elliptic Equations

Introduction to Finite Element Method

Bibliography and References

Appendices

Calculus Review

MATLAB Built-in Functions

Text MATLAB Functions

Answers to Selected Exercises

Index

Each chapter also contains a section of Applied Problems.

Highlights of the second edition:

INTRODUCTION

About the software MATLAB

An Introduction to MATLAB

Taylor Series

NUMBER SYSTEM AND ERRORS

Floating-Point Arithmetic

Round-Off Errors

Truncation Error

Interval Arithmetic

ROOTS OF EQUATIONS

The Bisection Method

The Method of False Position

Fixed-Point Iteration

The Secant method

Newton's Method

Convergence of the Newton and Secant Methods

Multiple Roots and the Modified Newton Method

Newton's Method for Nonlinear Systems

SYSTEM OF LINEAR EQUATIONS

Matrices and Matrix Operations

Naïve Gaussian Elimination

Gaussian Elimination with Scaled Partial Pivoting

LU Decomposition

Iterative Methods

INTERPOLATION

Polynomial Interpolation Theory

Newton's Divided Difference Interpolating Polynomial

The Error of the Interpolating Polynomial

Lagrange Interpolating Polynomial

INTERPOLATION WITH SPLINE FUNCTIONS

Piecewise Linear Interpolation

Quadratic Spline

Natural Cubic Splines

THE METHOD OF LEAST SQUARES

Linear Least Squares

Least Squares Polynomial

Nonlinear Least Squares

Trigonometric Least Squares Polynomial

NUMERICAL OPTIMIZATION

Analysis of Single-Variable Functions

Line Search Methods

Minimization Using Derivatives

NUMERICAL DIFFERENTIATION

Numerical Differentiation

Richardson's Formula

NUMERICAL INTEGRATION

Trapezoidal Rule

Simpson's Rule

Romberg Algorithm

Gaussian Quadrature

NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS

Euler's Method

Error Analysis

Higher Order Taylor Series Methods

Runge-Kutta Methods

Multistep Methods

Adams-Bashforth Method

Predictor-Corrector Methods

Adams-Moulton Method

Numerical Stability

Higher Order Equations and Systems of Differential Equations

Implicit Methods and Stiff Systems

Phase Plane Analysis: Chaotic Differential Equations

BOUNDARY-VALUE PROBLEMS

Finite-Difference Methods

Shooting Methods

EIGENVALUES AND EIGENVECTORS

Basic Theory

The Power Method

The Quadratic Method

Eigenvalues for Boundary-Value Problems

Bifurcations in Differential Equations

PARTIAL DIFFERENTIAL EQUATIONS

Parabolic Equations

Hyperbolic Equations

Elliptic Equations

Introduction to Finite Element Method

Bibliography and References

Appendices

Calculus Review

MATLAB Built-in Functions

Text MATLAB Functions

Answers to Selected Exercises

Index

Each chapter also contains a section of Applied Problems.

Highlights of the second edition:

INTRODUCTION

About the software MATLAB

An Introduction to MATLAB

Taylor Series

NUMBER SYSTEM AND ERRORS

Floating-Point Arithmetic

Round-Off Errors

Truncation Error

Interval Arithmetic

ROOTS OF EQUATIONS

The Bisection Method

The Method of False Position

Fixed-Point Iteration

The Secant method

Newton's Method

Convergence of the Newton and Secant Methods

Multiple Roots and the Modified Newton Method

Newton's Method for Nonlinear Systems

SYSTEM OF LINEAR EQUATIONS

Matrices and Matrix Operations

Naïve Gaussian Elimination

Gaussian Elimination with Scaled Partial Pivoting

LU Decomposition

Iterative Methods

INTERPOLATION

Polynomial Interpolation Theory

Newton's Divided Difference Interpolating Polynomial

The Error of the Interpolating Polynomial

Lagrange Interpolating Polynomial

INTERPOLATION WITH SPLINE FUNCTIONS

Piecewise Linear Interpolation

Quadratic Spline

Natural Cubic Splines

THE METHOD OF LEAST SQUARES

Linear Least Squares

Least Squares Polynomial

Nonlinear Least Squares

Trigonometric Least Squares Polynomial

NUMERICAL OPTIMIZATION

Analysis of Single-Variable Functions

Line Search Methods

Minimization Using Derivatives

NUMERICAL DIFFERENTIATION

Numerical Differentiation

Richardson's Formula

NUMERICAL INTEGRATION

Trapezoidal Rule

Simpson's Rule

Romberg Algorithm

Gaussian Quadrature

NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS

Euler's Method

Error Analysis

Higher Order Taylor Series Methods

Runge-Kutta Methods

Multistep Methods

Adams-Bashforth Method

Predictor-Corrector Methods

Adams-Moulton Method

Numerical Stability

Higher Order Equations and Systems of Differential Equations

Implicit Methods and Stiff Systems

Phase Plane Analysis: Chaotic Differential Equations

BOUNDARY-VALUE PROBLEMS

Finite-Difference Methods

Shooting Methods

EIGENVALUES AND EIGENVECTORS

Basic Theory

The Power Method

The Quadratic Method

Eigenvalues for Boundary-Value Problems

Bifurcations in Differential Equations

PARTIAL DIFFERENTIAL EQUATIONS

Parabolic Equations

Hyperbolic Equations

Elliptic Equations

Introduction to Finite Element Method

Bibliography and References

Appendices

Calculus Review

MATLAB Built-in Functions

Text MATLAB Functions

Answers to Selected Exercises

Index

Each chapter also contains a section of Applied Problems.