Numerical Analysis 10th Edition by Richard Burden, Douglas Faires, Annette Burden 1305253663 9781305253667

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ISBN 10: 1305253663

ISBN 13: 9781305253667

Author: Richard L. Burden; J. Douglas Faires; Annette M. Burden

This well-respected text introduces the theory and application of modern numerical approximation techniques to students taking a one- or two-semester course in numerical analysis. Providing an accessible treatment that only requires a calculus prerequisite, the authors explain how, why, and when approximation techniques can be expected to work-and why, in some situations, they fail. A wealth of examples and exercises develop students’ intuition, and demonstrate the subject’s practical applications to important everyday problems in math, computing, engineering, and physical science disciplines. The first book of its kind when crafted more than 30 years ago to serve a diverse undergraduate audience, Burden, Faires, and Burden’s NUMERICAL ANALYSIS remains the definitive introduction to a vital and practical subject.

Numerical Analysis 10th Table of contents:

Chapter 1. Mathematical Preliminaries and Error Analysis

1.1. Review of Calculus

Limits and Continuity

Differentiability

Integration

Taylor Polynomials and Series

1.1. Exercise Set

1.2. Round-Off Errors and Computer Arithmetic

Binary Machine Numbers

Decimal Machine Numbers

Finite-Digit Arithmetic

Nested Arithmetic

1.2. Exercise Set

1.3. Algorithms and Convergence

Characterizing Algorithms

Rates of Convergence

1.3. Exercise Set

1.4. Numerical Software

General-Purpose Algorithms

Discussion Question

Key Concepts

Chapter Review

Chapter 2. Solutions of Equations in One Variable

2.1. The Bisection Method

Bisection Technique

2.1. Exercise Set

2.2. Fixed-Point Iteration

Fixed-Point Iteration

2.2. Exercise Set

2.3. Newton’s Method and Its Extensions

Newton’s Method

Convergence Using Newton’s Method

The Secant Method

The Method of False Position

2.3. Exercise Set

2.4. Error Analysis for Iterative Methods

Order of Convergence

Multiple Roots

2.4. Exercise Set

2.5. Accelerating Convergence

Aitken’s Δ 2 Method

Steffensen’s Method

2.5. Exercise Set

2.6. Zeros of Polynomials and Müller’s Method

Algebraic Polynomials

Horner’s Method

Complex Zeros: Müller’s Method

2.6. Exercise Set

2.7. Numerical Software and Chapter Review

Discussion Questions

Key Concepts

Chapter Review

Chapter 3. Interpolation and Polynomial Approximation

3.1. Interpolation and the Lagrange Polynomial

Lagrange Interpolating Polynomials

3.1. Exercise Set

3.2. Data Approximation and Neville’s Method

Neville’s Method

3.2. Exercise Set

3.3. Divided Differences

Divided Differences

Forward Differences

Backward Differences

Centered Differences

3.3. Exercise Set

3.4. Hermite Interpolation

Hermite Polynomials

Hermite Polynomials Using Divided Differences

3.4. Exercise Set

3.5. Cubic Spline Interpolation

Piecewise-Polynomial Approximation

Cubic Splines

Construction of a Cubic Spline

Natural Splines

Clamped Splines

3.5. Exercise Set

3.6. Parametric Curves

3.6. Exercise Set

3.7. Numerical Software and Chapter Review

Discussion Questions

Key Concepts

Chapter Review

Chapter 4. Numerical Differentiation and Integration

4.1. Numerical Differentiation

Three-Point Formulas

Three-Point Endpoint Formula

Three-Point Midpoint Formula

Five-Point Formulas

Five-Point Midpoint Formula

Five-Point Endpoint Formula

Second Derivative Midpoint Formula

Round-Off Error Instability

4.1. Exercise Set

4.2. Richardson’s Extrapolation

4.2. Exercise Set

4.3. Elements of Numerical Integration

The Trapezoidal Rule

Simpson’s Rule

Measuring Precision

Closed Newton-Cotes Formulas

Open Newton-Cotes Formulas

4.3. Exercise Set

4.4. Composite Numerical Integration

Round-Off Error Stability

4.4. Exercise Set

4.5. Romberg Integration

4.5. Exercise Set

4.6. Adaptive Quadrature Methods

4.6. Exercise Set

4.7. Gaussian Quadrature

Legendre Polynomials

Gaussian Quadrature on Arbitrary Intervals

4.7. Exercise Set

4.8. Multiple Integrals

Gaussian Quadrature for Double Integral Approximation

Nonrectangular Regions

Triple Integral Approximation

4.8. Exercise Set

4.9. Improper Integrals

Left Endpoint Singularity

Right-Endpoint Singularity

Infinite Singularity

4.9. Exercise Set

4.10. Numerical Software and Chapter Review

Discussion Questions

Key Concepts

Chapter Review

Chapter 5. Initial-Value Problems for Ordinary Differential Equations

5.1. The Elementary Theory of Initial-Value Problems

Well-Posed Problems

5.1. Exercise Set

5.2. Euler’s Method

Error Bounds for Euler’s Method

5.2. Exercise Set

5.3. Higher-Order Taylor Methods

Taylor Method of Order n

5.3. Exercise Set

5.4. Runge-Kutta Methods

Runge-Kutta Methods of Order Two

Midpoint Method

Higher-Order Runge-Kutta Methods

Computational Comparisons

5.4. Exercise Set

5.5. Error Control and the Runge-Kutta-Fehlberg Method

Runge-Kutta-Fehlberg Method

5.5. Exercise Set

5.6. Multistep Methods

Adams-Bashforth Explicit Methods

Adams-Moulton Implicit Methods

Predictor-Corrector Methods

5.6. Exercise Set

5.7. Variable Step-Size Multistep Methods

5.7. Exercise Set

5.8. Extrapolation Methods

5.8. Exercise Set

5.9. Higher-Order Equations and Systems of Differential Equations

Higher-Order Differential Equations

5.9. Exercise Set

5.10. Stability

One-Step Methods

Multistep Methods

5.10. Exercise Set

5.11. Stiff Differential Equations

5.11. Exercise Set

5.12. Numerical Software

Discussion Questions

Key Concepts

Chapter Review

Chapter 6. Direct Methods for Solving Linear Systems

6.1. Linear Systems of Equations

Matrices and Vectors

Operation Counts

6.1. Exercise Set

6.2. Pivoting Strategies

Partial Pivoting

Scaled Partial Pivoting

Complete Pivoting

6.2. Exercise Set

6.3. Linear Algebra and Matrix Inversion

Matrix Arithmetic

Matrix-Vector Products

Matrix-Matrix Products

Square Matrices

Inverse Matrices

Transpose of a Matrix

6.3. Exercise Set

6.4. The Determinant of a Matrix

6.4. Exercise Set

6.5. Matrix Factorization

Permutation Matrices

6.5. Exercise Set

6.6. Special Types of Matrices

Diagonally Dominant Matrices

Positive Definite Matrices

Band Matrices

Tridiagonal Matrices

6.6. Exercise Set

6.7. Numerical Software

Discussion Questions

Key Concepts

Chapter Review

Chapter 7. Iterative Techniques in Matrix Algebra

7.1. Norms of Vectors and Matrices

Vector Norms

Distance Between Vectors in ℝ n

Matrix Norms and Distances

7.1. Exercise Set

7.2. Eigenvalues and Eigenvectors

Spectral Radius

Convergent Matrices

7.2. Exercise Set

7.3. The Jacobi and Gauss-Siedel Iterative Techniques

Jacobi’s Method

The Gauss-Seidel Method

General Iteration Methods

7.3. Exercise Set

7.4. Relaxation Techniques for Solving Linear Systems

7.4. Exercise Set

7.5. Error Bounds and Iterative Refinement

Condition Numbers

Iterative Refinement

7.5. Exercise Set

7.6. The Conjugate Gradient Method

Preconditioning

7.6. Exercise Set

7.7. Numerical Software

Discussion Questions

Key Concepts

Chapter Review

Chapter 8. Approximation Theory

8.1. Discrete Least Squares Approximation

Linear Least Squares

Polynomial Least Squares

8.1. Exercise Set

8.2. Orthogonal Polynomials and Least Squares Approximation

Linearly Independent Functions

Orthogonal Functions

8.2. Exercise Set

8.3. Chebyshev Polynomials and Economization of Power Series

Minimizing Lagrange Interpolation Error

Minimizing Approximation Error on Arbitrary Intervals

Reducing the Degree of Approximating Polynomials

8.3. Exercise Set

8.4. Rational Function Approximation

Padé Approximation

Continued-Fraction Approximation

Chebyshev Rational Function Approximation

8.4. Exercise Set

8.5. Trigonometric Polynomial Approximation

Orthogonal Trigonometric Polynomials

Discrete Trigonometric Approximation

8.5. Exercise Set

8.6. Fast Fourier Transforms

8.6. Exercise Set

8.7. Numerical Software

Discussion Questions

Key Concepts

Chapter Review

Chapter 9. Approximating Eigenvalues

9.1. Linear Algebra and Eigenvalues

9.1. Exercise Set

9.2. Orthogonal Matrices and Similarity Transformations

9.2. Exercise Set

9.3. The Power Method

Accelerating Convergence

Symmetric Matrices

Inverse Power Method

Deflation Methods

9.3. Exercise Set

9.4. Householder’s Method

Householder Transformations

9.4. Exercise Set

9.5. The QR Algorithm

Rotation Matrices

Accelerating Convergence

9.5. Exercise Set

9.6. Singular Value Decomposition

Constructing a Singular Value Decomposition for an m × n Matrix A

Least Squares Approximation

Other Applications

9.6. Exercise Set

9.7. Numerical Software

Discussion Questions

Key Concepts

Chapter Review

Chapter 10. Numerical Solutions of Nonlinear Systems of Equations

10.1. Fixed Points for Functions of Several Variables

Fixed Points in ℝ n

Accelerating Convergence

10.1. Exercise Set

10.2. Newton’s Method

The Jacobian Matrix

10.2. Exercise Set

10.3. Quasi-Newton Methods

Sherman-Morrison Formula

10.3. Exercise Set

10.4. Steepest Descent Techniques

The Gradient of a Function

10.4. Exercise Set

10.5. Homotopy and Continuation Methods

Continuation Method

10.5. Exercise Set

10.6. Numerical Software

Discussion Questions

Key Concepts

Chapter Review

Chapter 11. Boundary-Value Problems for Ordinary Differential Equations

11.1. The Linear Shooting Method

Linear Boundary-Value Problems

Linear Shooting

Reducing Round-Off Error

11.1. Exercise Set

11.2. The Shooting Method for Nonlinear Problems

Newton Iteration

11.2. Exercise Set

11.3. Finite-Difference Methods for Linear Problems

Discrete Approximation

Employing Richardson’s Extrapolation

11.3. Exercise Set

11.4. Finite-Difference Methods for Nonlinear Problems

Newton’s Method for Iterations

Employing Richardson’s Extrapolation

11.4. Exercise Set

11.5. The Rayleigh-Ritz Method

Variational Problems

Piecewise-Linear Basis

B -Spline Basis

11.5. Exercise Set

11.6. Numerical Software

Discussion Questions

Key Concepts

Chapter Review

Chapter 12. Numerical Solutions to Partial Differential Equations

12.1. Elliptic Partial Differential Equations

Selecting a Grid

Finite-Difference Method

Choice of Iterative Method

12.1. Exercise Set

12.2. Parabolic Partial Differential Equations

Forward-Difference Method

Stability Considerations

Backward-Difference Method

Crank-Nicolson Method

12.2. Exercise Set

12.3. Hyperbolic Partial Differential Equations

Improving the Initial Approximation

12.3. Exercise Set

12.4. An Introduction to the Finite-Element Method

Defining the Elements

Triangulating the Region

12.4. Exercise Set

12.5. Numerical Software

Discussion Questions

Key Concepts

Chapter Review

Bibliography

Formula Sheets

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