1 Mathematical Preliminaries 1.1 Review of Calculus 1.2 Roundoff Errors and Computer Arithmetic 1.3 Algorithms and Convergence 1.4 Numerical Software 2 Solutions of Equations in One Variable 2.1 The Bisection Method 2.2 Fixed-Point Iteration 2.3 Newtons Method 2.4 Error Analysis for Iterative Methods 2.5 Accelerating Convergence 2.6 Zeros of Polynomials Müllers Method 2.7 Survey of Methods and Software 3 Interpolation and Polynomial Approximation 3.1 Interpolation and the Lagrange Polynomial 3.2 Divided Differences 3.3 Hermite Interpolation 3.4 Cubic Spline Interpolation 3.5 Parametric Curves 3.6 Survey of Methods and Software 4. Numerical Differentiation and Integration 4.1 Numerical Differentiation 4.2 Richardsons Extrapolation 4.3 Elements of Numerical Integration 4.4 Composite numerical Integration 4.5 Romberg Integration 4.6 Adaptive Quadrature Methods 4.7 Gaussian Quadrature 4.8 Multiple Integrals 4.9 Improper Integrals 4.10 Survey of Methods and Software 5 Initial-Value Problems for ordinary Differential Equations 5.1 The Elementary Theory of Initial-Value Problems 5.2 Eulers Method 5.3 Higher-Order Taylor Methods 5.4 Runge-Kutta Methods 5.5 Error Control and the Runge-Kutta-Fehlberg Method 5.6 Multistep methods 5.7 Variable Step-Size Multistep Methods 5.8 Extrapolation Methods 5.9 Higher-Order Equations and Systems of Differential Equations 5.10 Stability 5.11 Stiff Differential Equations 5.12 Survey of Methods and Software 6 Direct Methods for Solving Linear Systems 6.1 Linear Systems of Equations 6.2 Pivoting Strategies 6.3 Linear Algebra and Matrix Inversion 6.4 The Determinant of a Matrix 6.5 matrix Factorization 6.6 Special Types of Matrices 6.7 Survey of Methods and Software 7 Iterative Techniques in Matrix algebra 7.1 Norms of Vectors and Matrices 7.2 Eigenvalues and Eigenvectors 7.3 Iterative Techniques for Solving Linear Systems 7.4 Error Bounds and Iterative Refinement 7.5 The Conjugate Gradient Method 7.6 Survey of Methods and Software 8 Approximation Theory 8.1 Discrete Least Squares Approximation 8.2 Orthogonal Polynomials and Least Squares Approximation 8.3 Chebyshev Polynomials and Econcomization of Power Series 8.4 Rational Function Approximation 8.5 Trigonometric Polynomial approximation 8.6 Fast Fourier Transforms 8.7 Survey of Methods and software 9 Approximating Eigenvalues 9.1 Linear algebra and Eigenvalues 9.2 The Power Method 9.3 Householders Method 9.4 The QR Algorithm 9.5 Survey of Methods and software 10 Numerical Solutions of Nonlinear Systems of Equations 10.1 Fixed Points for Functions of Several Variables 10.2 Newtons Method 10.3 Quasi-Newton Methods 10.4 Steepest Descent Techniques 10.5 Homotopy and Continuation Methods 10.6 Survey of Methods and Software 11 Boundary-Value Problems for Ordinary Differential Equations 11.1 The Linear Shooting Method 11.2 The Shooting Method for Nonlinear Problems 11.3 Finite-Difference Methods for Linear Problems 11.4 Finite-Difference Methods for Nonlinear Problems 11.5 The Rayleigh-Ritz Method 11.6 Survey of Methods and Software 12 Numerical solutions to Partial Differential Equations 12.1 Elliptic Partial Differential Equations 12.2 Parabolic Partial Differential Equations 12.3 Hyperbolic Partial Differential Equations 12.4 An Introduction to the Finite-Element Method 12.5 Survey of Methods and software Bibliography Answers to Selected Exercises Index
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