Elementary numerical analysis

Preface......Page out-0007.djvu
Contents......Page out-0011.djvu
1. Taylor Polynomials......Page out-0017.djvu
1.1 The Taylor Polynomial......Page out-0018.djvu
1.2 The Error in Taylor\'s Polynomial......Page out-0027.djvu
1.3 Polynomial Evaluation......Page out-0039.djvu
2. Error and Computer Arithmetic......Page out-0049.djvu
2.1 Floating-Point Numbers......Page out-0050.djvu
2.1.1 Accuracy of Floating-Point Representation......Page out-0054.djvu
2.1.2 Rounding and Chopping......Page out-0055.djvu
2.1.3 Consequences......Page out-0056.djvu
2.2 Errors: Definitions, Sources, and Examples......Page out-0059.djvu
2.2.1 Sources of Error......Page out-0061.djvu
2.2.2 Loss-of-Significance Errors......Page out-0063.djvu
2.2.3 Noise in Function Evaluation......Page out-0067.djvu
2.2.4 Underflow and Overflow Errors......Page out-0068.djvu
2.3 Propagation of Error......Page out-0073.djvu
2.4 Summation......Page out-0079.djvu
2.4.1 Rounding versus Chopping......Page out-0081.djvu
2.4.3 Calculation of Inner Products......Page out-0083.djvu
3. Rootfinding......Page out-0087.djvu
3.1 The Bisection Method......Page out-0088.djvu
3.1.1 Error Bounds......Page out-0090.djvu
3.2 Newton\'s Method......Page out-0095.djvu
3.2.1 Error Analysis......Page out-0099.djvu
3.2.2 Error Estimation......Page out-0101.djvu
3.3 Secant Method......Page out-0106.djvu
3.3.1 Error Analysis......Page out-0107.djvu
3.3.2 Comparison of Newton and Secant Methods......Page out-0110.djvu
3.3.3 The Matlab Function fzero......Page out-0111.djvu
3.4 Fixed Point Iteration......Page out-0113.djvu
3.4.1 Aitken Error Estimation and Extrapolation......Page out-0118.djvu
3.4.2 Higher-Order Iteration Formulas......Page out-0121.djvu
3.5 Ill-Behaving Rootfinding Problems......Page out-0125.djvu
3.5.1 Stability of Roots......Page out-0128.djvu
4. Interpolation and Approximation......Page out-0133.djvu
4.1 Polynomial Interpolation......Page out-0134.djvu
4.1.1 Linear Interpolation......Page out-0135.djvu
4.1.2 Quadratic Interpolation......Page out-0136.djvu
4.1.3 Higher-Degree Interpolation......Page out-0139.djvu
4.1.4 Divided Differences......Page out-0140.djvu
4.1.5 Properties of Divided Differences......Page out-0142.djvu
4.1.6 Newton\'s Divided Difference Interpolation Formula......Page out-0144.djvu
4.2 Error in Polynomials Interpolation......Page out-0154.djvu
4.2.1 Another Error Formula......Page out-0157.djvu
4.2.2 Behavior of the Error......Page out-0158.djvu
4.3 Interpolation Using Spline Functions......Page out-0163.djvu
4.3.2 Construction of the Interpolating Natural Cubic Spline......Page out-0165.djvu
4.3.3 Other Interpolation Spline Functions......Page out-0167.djvu
4.3.4 The Matlab Program spline......Page out-0170.djvu
4.4 The Best Approximation Problem......Page out-0175.djvu
4.4.1 Accuracy of the Minimax Approximation......Page out-0179.djvu
4.5 Chebyshev Polynomials......Page out-0181.djvu
4.5.1 The Triple Recursion Relation......Page out-0183.djvu
4.5.2 The Minimum Size Property......Page out-0184.djvu
4.6 A Near-Minimax Approximation Method......Page out-0187.djvu
4.6.1 Odd and Even Functions......Page out-0192.djvu
4.7 Least Squares Approximation......Page out-0194.djvu
4.7.1 Legendre Polynomials......Page out-0197.djvu
4.7.2 Solving for the Least Squares Approximation......Page out-0199.djvu
4.7.3 Generalizations of Least Squares Approximation......Page out-0201.djvu
5. Numerical Integration and Differentiation......Page out-0205.djvu
5.1 The Trapezoidal and Simpson Rules......Page out-0206.djvu
5.1.1 Simpson\'s Rule......Page out-0210.djvu
5.2 Error Formulas......Page out-0217.djvu
5.2.1 An Asymptotic Estimate of the Trapezoidal Error......Page out-0219.djvu
5.2.2 Error Formulas for Simpson\'s Rule......Page out-0221.djvu
5.2.3 Richardson Extrapolation......Page out-0224.djvu
5.2.4 Periodic Integrands......Page out-0225.djvu
5.3 Gaussian Numerical Integration......Page out-0233.djvu
5.4 Numerical Differentiation......Page out-0246.djvu
5.4.1 Differentiation Using Interpolation......Page out-0248.djvu
5.4.2 The Method of Undetermined Coefficients......Page out-0250.djvu
5.4.3 Effects of Error in Function Values......Page out-0252.djvu
6. Solution of Systems of Linear Equations......Page out-0257.djvu
6.1 Systems of Linear Equations......Page out-0258.djvu
6.2 Matrix Arithmetic......Page out-0262.djvu
6.2.1 Arithmetic Operations......Page out-0263.djvu
6.2.2 Elementary Row Operations......Page out-0267.djvu
6.2.3 The Matrix Inverse......Page out-0268.djvu
6.2.4 Matrix Algebra Rules......Page out-0270.djvu
6.2.5 Solvability Theory of Linear Systems......Page out-0272.djvu
6.3 Gaussian Elimination......Page out-0278.djvu
6.3.1 Partial Pivoting......Page out-0284.djvu
6.3.2 Calculation of Inverse Matrices......Page out-0287.djvu
6.3.3 Operations Count......Page out-0290.djvu
6.4 The LU Factorization......Page out-0297.djvu
6.4.1 Compact Variants of Gaussian Elimination......Page out-0299.djvu
6.4.2 Tridiagonal Systems......Page out-0301.djvu
6.4.3 Matlab Built-in Functions for Solving Linear Systems......Page out-0305.djvu
6.5 Error in Solving Linear Systems......Page out-0308.djvu
6.5.1 The Residual Correction Method......Page out-0310.djvu
6.5.2 Stability in Solving Linear Systems......Page out-0311.djvu
6.6.1 Jacobi Method and Gauss-Seidel Method......Page out-0317.djvu
6.6.2 General Schema......Page out-0320.djvu
6.6.3 The Residual Correction Method......Page out-0324.djvu
7.1 Least Squares Data Fitting......Page out-0333.djvu
7.1.1 The Linear Least Squares Approximation......Page out-0336.djvu
7.1.2 Polynomial Least Squares Approximation......Page out-0338.djvu
7.2 The Eigenvalue Problem......Page out-0347.djvu
7.2.1 The Characteristic Polynomial......Page out-0349.djvu
7.2.2 Eigenvalues for Symmetric Matrices......Page out-0351.djvu
7.2.3 The Nonsymmetric Eigenvalue Problem......Page out-0353.djvu
7.2.4 The Power Method......Page out-0354.djvu
7.2.5 Convergence of the Power Method......Page out-0356.djvu
7.2.6 Matlab  Eigenvalue Calculations......Page out-0359.djvu
7.3.1 Newton\'s Method......Page out-0366.djvu
7.3.2 The General Newton Method......Page out-0370.djvu
7.3.3 A Modified Newton\'s Method......Page out-0375.djvu
8. Numerical Solutions of Ordinary Differential Equations......Page out-0381.djvu
8.1 Theory of Differential Equations: An Introduction......Page out-0382.djvu
8.1.1 General Solvability Theory......Page out-0386.djvu
8.1.2 Stability of the Initial Value Problem......Page out-0387.djvu
8.1.3 Direction Fields......Page out-0390.djvu
8.2 Euler\'s Method......Page out-0393.djvu
8.3 Convergence Analysis of Euler\'s Method......Page out-0400.djvu
8.3.1 Asymptotic Error Analysis......Page out-0404.djvu
8.3.2 Richardson Extrapolation......Page out-0405.djvu
8.4 Numerical Stability, Implicit Methods......Page out-0408.djvu
8.4.1 The Backward Euler Method......Page out-0410.djvu
8.4.2 The Trapezoidal Method......Page out-0414.djvu
8.5 Taylor and Runge-Kutta Methods......Page out-0422.djvu
8.5.1 Runge-Kutta Methods......Page out-0425.djvu
8.5.2 Error Prediction and Control......Page out-0429.djvu
8.5.3 Matlab Built-in Functions......Page out-0432.djvu
8.6 Multistep Methods......Page out-0437.djvu
8.7 Systems of Differential Equations......Page out-0446.djvu
8.7.1 Higher-Order Differential Equations......Page out-0448.djvu
8.7.2 Numerical Methods for Systems......Page out-0451.djvu
8.8 Finite Difference Method for Two-Point Boundary Value Problems......Page out-0456.djvu
9. Finite Difference Method for Partial Differential Equations......Page out-0465.djvu
9.1 The Poisson Equation......Page out-0467.djvu
9.2 One-Dimensional Heat Equation......Page out-0480.djvu
9.2.1 Semidiscretization......Page out-0481.djvu
9.2.2 Explicit Full Discretization......Page out-0482.djvu
9.3 One-Dimensional Wave Equation......Page out-0495.djvu
A. Mean Values Theorems......Page out-0505.djvu
B.1 Algebra......Page out-0515.djvu
B.2 Geometry......Page out-0516.djvu
B.3 Trigonometry......Page out-0519.djvu
B.4 Calculus......Page out-0521.djvu
C. Numerical Analysis Software Packages......Page out-0525.djvu
C.2 Public Domain Packages......Page out-0526.djvu
C.4 Symbolic Computation Environments......Page out-0529.djvu
C.5 Literature on Mathematical Software......Page out-0530.djvu
D. Matlab: An Introduction......Page out-0531.djvu
E. The Binary Number System......Page out-0539.djvu
E.1 Conversion from Decimal to Binary......Page out-0542.djvu
E.2 Hexadecimal Numbers......Page out-0544.djvu
Answers to Selected Problems......Page out-0547.djvu
Bibliography......Page out-0569.djvu
Index......Page out-0571.djvu