Preface Chapter 1 Mathematical Review and Error Analysis 1.1 Mathematical Review 1.2 Errors and Significant Digits 1.2.1 Truncation Error and Round-off Error 1.2.2 Absolute Error and Relative Error 1.2.3 Significant Digits 1.3 Avoid the Loss of Accuracy 1.3.1 Avoid the Subtraction of Nearly Equal Numbers. 1.3.2 Avoid Big Numbers 「Swallowing」 Small Numbers 1.3.3 Reduce Computations 1.3.4 Avoid Dividing by a Number with Small Absolute Value 1.3.5 Use Stable Algorithms 1.4 Exercises Chapter 2 Solutions of Equations in One Variable 2.1 The Bisection Method 2.2 Fixed-Point Iteration 2.2.1 Basic Concepts 2.2.2 Convergence and Error Estimation 2.2.3 Local Convergence and Order of Convergence 2.3 Newton』s Method and Secant Method 2.3.1 Newton』s Method 2.3.2 Secant Method 2.3.3 Newton』s Method for Finding Multiple Roots 2.4 Acceleration Techniques 2.5 Programs 2.6 Exercises Chapter 3 Interpolation 3.1 Lagrange Interpolation 3.2 Newton Interpolation 3.3 Aitken』s Method 3.4 Hermite Interpolation 3.5 Piecewise Polynomial Interpolation 3.6 Cubic Spline Interpolation 3.7 Programs 3.8 Exercises. Chapter 4 Curve Fitting and Orthogonal Polynomials 4.1 Least Square Method 4.2 Least Square Approximation 4.3 Orthogonal Polynomials. 4.4 Programs 4.4.1 Least Square Method 4.4.2 Least Square Approximation 4.5 Exercises Chapter 5 Direct Methods for Linear Systems 5.1 Gaussian Elimination Method 5.1.1 Linear Systems of Equation 5.1.2 Gaussian Elimination with Backward-Substitution 5.2 Gaussian Elimination with Partial Pivoting 5.3 Matrix Factorization
5.4 Two Special Types of Matrices 5.5 Gaussian Elimination on Tridiagonal Linear Systems 5.6 Norms of Vectors and Matrices 5.6.1 Norms of Vectors 5.6.2 Norms of Matrices 5.7 Ill-Conditioned Linear System and Condition Number 5.8 Programs 5.9 Exercises Chapter 6 Iterative Methods for Linear Systems 6.1 Iterative Methods 6.1.1 Jacobi Iterative Method. 6.1.2 Gauss-Seidel Iterative Method 6.1.3 SOR Method. 6.2 Convergence Analysis for Iterative Methods 6.3 Programs 6.4 Exercises Chapter 7 Numerical Differentiation and Integration 7.1 Numerical Differentiation 7.1.1 Three-Point Formulas and Five-Point Formulas 7.1.2 The Method by Using Cubic Spline Interpolating Function 7.1.3 Varying Step Size Midpoint Method 7.1.4 Richardson Extrapolation 7.2 Elements of Numerical Integration 7.3 Newton-Cotes Quadrature Formulas. 7.3.1 Basic Concepts of Newton-Cotes Quadrature Formulas 7.3.2 Some Common Newton-Cotes Formulas 7.4 Composite Numerical Integration 7.5 Romberg Integration 7.5.1 Recursive Trapezoidal Rule 7.5.2 Romberg Integration 7.6 Gaussian Quadrature 7.6.1 Basic Concepts 7.6.2 Two Common Gaussian Quadrature Formulas 7.6.3 Stability and Convergence 7.7 Programs 7.8 Exercises Chapter 8 Numerical Solutions of Ordinary Differential Equations 8.1 Elements of Initial Problems 8.2 Euler Method and Modified Euler Method 8.2.1 Euler Method and Trapezoidal Method. 8.2.2 Modified Euler Method 8.2.3 Local Truncation Error 8.3 Runge-Kutta Methods 8.3.1 Second-Order Runge-Kutta Methods 8.3.2 Some Common Third-and Fourth-Order Runge-Kutta Methods 8.4 Stability and Convergence 8.5 Multistep Methods 8.6 Programs 8.7 Exercises Chapter 9 Approximating Eigenvalues and Eigenvectors
9.1 Fundamental Theorems 9.2 The Power Method 9.3 Accelerating Convergence 9.4 Inverse Power Method 9.5 Householder』s Method 9.6 The QR method 9.7 Programs 9.8 Exercises References Appendix A English-Chinese Math Key Words Appendix B Some Math Expressions and Pronunciations
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