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by Brian Bradie

Cover type: HardbackEdition: 06

Copyright: 2006

Publisher: Prentice Hall, Inc.

Published: 2006

International: No

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This student-friendly text develops concepts and techniques in a clear, concise, easy-to-read manner, followed by fully-worked examples. Application problems drawn from the literature of many different fields prepares students to use the techniques covered to solve a wide variety of practical problems.

**1. Getting Started. **

Algorithms. Convergence. Floating Point Numbers. Floating Point Arithmetic.

**2. Rootfinding. **

Bisection Method. Method of False Position. Fixed Point Iteration. Newton's Method. The Secant Method and Muller's Method. Accelerating Convergence. Roots of Polynomials.

**3. Systems of Equations. **

Gaussian Elimination. Pivoting Strategies. Norms. Error Estimates. LU Decomposition. Direct Factorization. Special Matrices. Iterative Techniques for Linear Systems: Basic Concepts and Methods. Iterative Techniques for Linear Systems: Conjugate-Gradient Method. Nonlinear Systems.

**4. Eigenvalues and Eigenvectors. **

The Power Method. The Inverse Power Method. Deflation. Reduction to Tridiagonal Form. Eigenvalues of Tridiagonal and Hessenberg Matrices.

**5. Interpolation and Curve Fitting. **

Lagrange Form of the Interpolating Polynomial. Neville's Algorithm. The Newton Form of the Interpolating Polynomial and Divided Differences. Optimal Interpolating Points. Piecewise Linear Interpolation. Hermite and Hermite Cubic Interpolation. Regression.

**6. Numerical Differentiation and Integration. **

Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.

**7. Numerical Methods for Initial Value Problems of Ordinary Differential Equations. **

Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.

**8. Second-Order One-Dimensional Two-Point Boundary Value Problems. **

Finite Difference Method, Part I: The Linear Problem with Dirichlet Boundary Conditions. Finite Difference Method, Part II: The Linear Problem with Non-Dirichlet Boundary Conditions. Finite Difference Method, Part III: Nonlinear Problems. The Shooting Method, Part I: Linear Boundary Value Problems. The Shooting Method, Part II: Nonlinear Boundary Value Problems.

**9. Finite Difference Method for Elliptic Partial Differential Equations. **

The Poisson Equation on a Rectangular Domain, I: Dirichlet Boundary Conditions. The Poisson Equation on a Rectangular Domain, II: Non-Dirichlet Boundary Conditions. Solving the Discrete Equations: Relaxation Schemes. Local Mode Analysis of Relaxation and the Multigrid Method. Irregular Domains.

**10. Finite Difference Method for Parabolic Partial Differential Equations. **

The Heat Equation with Dirichlet Boundary Conditions. Stability. More General Parabolic Equations. Non-Dirichlet Boundary Conditions. Polar Coordinates. Problems in Two Space Dimensions.

**11. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation. **

Advection Equation, I: Upwind Differencing. Advection Equation, II: MacCormack Method. Convection-Diffusion Equation. The Wave Equation.

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Summary

This student-friendly text develops concepts and techniques in a clear, concise, easy-to-read manner, followed by fully-worked examples. Application problems drawn from the literature of many different fields prepares students to use the techniques covered to solve a wide variety of practical problems.

Table of Contents
Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.

**1. Getting Started. **

Algorithms. Convergence. Floating Point Numbers. Floating Point Arithmetic.

**2. Rootfinding. **

Bisection Method. Method of False Position. Fixed Point Iteration. Newton's Method. The Secant Method and Muller's Method. Accelerating Convergence. Roots of Polynomials.

**3. Systems of Equations. **

Gaussian Elimination. Pivoting Strategies. Norms. Error Estimates. LU Decomposition. Direct Factorization. Special Matrices. Iterative Techniques for Linear Systems: Basic Concepts and Methods. Iterative Techniques for Linear Systems: Conjugate-Gradient Method. Nonlinear Systems.

**4. Eigenvalues and Eigenvectors. **

The Power Method. The Inverse Power Method. Deflation. Reduction to Tridiagonal Form. Eigenvalues of Tridiagonal and Hessenberg Matrices.

**5. Interpolation and Curve Fitting. **

Lagrange Form of the Interpolating Polynomial. Neville's Algorithm. The Newton Form of the Interpolating Polynomial and Divided Differences. Optimal Interpolating Points. Piecewise Linear Interpolation. Hermite and Hermite Cubic Interpolation. Regression.

**6. Numerical Differentiation and Integration. **

Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations.

**7. Numerical Methods for Initial Value Problems of Ordinary Differential Equations. **

**8. Second-Order One-Dimensional Two-Point Boundary Value Problems. **

Finite Difference Method, Part I: The Linear Problem with Dirichlet Boundary Conditions. Finite Difference Method, Part II: The Linear Problem with Non-Dirichlet Boundary Conditions. Finite Difference Method, Part III: Nonlinear Problems. The Shooting Method, Part I: Linear Boundary Value Problems. The Shooting Method, Part II: Nonlinear Boundary Value Problems.

**9. Finite Difference Method for Elliptic Partial Differential Equations. **

The Poisson Equation on a Rectangular Domain, I: Dirichlet Boundary Conditions. The Poisson Equation on a Rectangular Domain, II: Non-Dirichlet Boundary Conditions. Solving the Discrete Equations: Relaxation Schemes. Local Mode Analysis of Relaxation and the Multigrid Method. Irregular Domains.

**10. Finite Difference Method for Parabolic Partial Differential Equations. **

The Heat Equation with Dirichlet Boundary Conditions. Stability. More General Parabolic Equations. Non-Dirichlet Boundary Conditions. Polar Coordinates. Problems in Two Space Dimensions.

**11. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation. **

Advection Equation, I: Upwind Differencing. Advection Equation, II: MacCormack Method. Convection-Diffusion Equation. The Wave Equation.

Publisher Info

Publisher: Prentice Hall, Inc.

Published: 2006

International: No

Published: 2006

International: No

**1. Getting Started. **

Algorithms. Convergence. Floating Point Numbers. Floating Point Arithmetic.

**2. Rootfinding. **

Bisection Method. Method of False Position. Fixed Point Iteration. Newton's Method. The Secant Method and Muller's Method. Accelerating Convergence. Roots of Polynomials.

**3. Systems of Equations. **

Gaussian Elimination. Pivoting Strategies. Norms. Error Estimates. LU Decomposition. Direct Factorization. Special Matrices. Iterative Techniques for Linear Systems: Basic Concepts and Methods. Iterative Techniques for Linear Systems: Conjugate-Gradient Method. Nonlinear Systems.

**4. Eigenvalues and Eigenvectors. **

The Power Method. The Inverse Power Method. Deflation. Reduction to Tridiagonal Form. Eigenvalues of Tridiagonal and Hessenberg Matrices.

**5. Interpolation and Curve Fitting. **

Lagrange Form of the Interpolating Polynomial. Neville's Algorithm. The Newton Form of the Interpolating Polynomial and Divided Differences. Optimal Interpolating Points. Piecewise Linear Interpolation. Hermite and Hermite Cubic Interpolation. Regression.

**6. Numerical Differentiation and Integration. **

**7. Numerical Methods for Initial Value Problems of Ordinary Differential Equations. **

**8. Second-Order One-Dimensional Two-Point Boundary Value Problems. **

Finite Difference Method, Part I: The Linear Problem with Dirichlet Boundary Conditions. Finite Difference Method, Part II: The Linear Problem with Non-Dirichlet Boundary Conditions. Finite Difference Method, Part III: Nonlinear Problems. The Shooting Method, Part I: Linear Boundary Value Problems. The Shooting Method, Part II: Nonlinear Boundary Value Problems.

**9. Finite Difference Method for Elliptic Partial Differential Equations. **

The Poisson Equation on a Rectangular Domain, I: Dirichlet Boundary Conditions. The Poisson Equation on a Rectangular Domain, II: Non-Dirichlet Boundary Conditions. Solving the Discrete Equations: Relaxation Schemes. Local Mode Analysis of Relaxation and the Multigrid Method. Irregular Domains.

**10. Finite Difference Method for Parabolic Partial Differential Equations. **

The Heat Equation with Dirichlet Boundary Conditions. Stability. More General Parabolic Equations. Non-Dirichlet Boundary Conditions. Polar Coordinates. Problems in Two Space Dimensions.

**11. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation. **

Advection Equation, I: Upwind Differencing. Advection Equation, II: MacCormack Method. Convection-Diffusion Equation. The Wave Equation.