数值分析-第2版

PrologueP1 OverviewP2 Numerical Analysis SoftwareP3 Textbooks and MonographsP3．1 Selected Textbooks on Numerical AnalysisP3．2 Monographs and Books on Specialized TopicsP4 Journals1 Machine Arithmetic and Related Matters1．1 Real Numbers， Machine Numbers， and Rounding1．1．1 Real Numbers1．1．2 Machine Numbers1．1．3 Rounding1．2 Machine Arithmetic1．2．1 A Model of Machine Arithmetic1．2．2 Error Propagation in Arithmetic Operations： Cancellation Error1．3 The Condition of a Problem1．3．1 Condition Numbers1．3．2 Examples1．4 The Condition of an Algorithm1．5 Computer Solution of a Problem; Overall Error1．6 Notes to Chapter 1Exercises and Machine Assignments to Chapter 1ExercisesMachine AssignmentsSelected Solutions to ExercisesSelected Solutions to Machine Assignments2 Approximation and Interpolation2．1 Least Squares Approximation2．1．1 Inner Products2．1．2 The Normal Equations2．1．3 Least Squares Error; Convergence2．1．4 Examples of Orthogonal Systems2．2 Polynomial Interpolation2．2．1 Lagrange Interpolation Formula： Interpolation Operator．2．2．2 Interpolation Error2．2．3 Convergence．．2．2．4 Chebyshev Polynomials and Nodes2．2．5 Barycentric Formula2．2．6 Newton's Formula2．2．7 Hermite Interpolation2．2．8 Inverse Interpolation2．3 Approximation and Interpolation by Spline Functions2．3．1 Interpolation by Piecewise Linear Functions2．3．2 A Basis for St（A）2．3．3 Least Squares Approximation2．3．4 Interpolation by Cubic Splines2．3．5 Minimality Properties of Cubic Spline Interpolants2．4 Notes to Chapter 2Exercises and Machine Assignments to Chapter 2ExercisesMachine AssignmentsSelected Solutions to ExercisesSelected Solutions to Machine Assignments3 Numerical Differentiation and Integration3．1 Numerical Differentiation3．1．1 A General Differentiation Formula for Unequally Spaced Points3．1．2 Examples3．1．3 Numerical Differentiation with Perturbed Data3．2 Numerical Integration3．2．1 The Composite Trapezoidal and Simpson's Rules3．2．2 （Weighted） Newton-Cotes and Gauss Formulae3．2．3 Properties of Gaussian Quadrature Rules3．2．4 Some Applications of the Gauss Quadrature Rule3．2．5 Approximation of Linear Functionals： Method f Interpolation vs． Method of Undetermined Coefficients3．2．6 Peano Representation of Linear Functionals3．2．7 Extrapolation Methods3．3 Notes to Chapter 3Exercises and Machine Assignments to Chapter 3ExercisesMachine AssignmentsSelected Solutions to ExercisesSelected Solutions to Machine Assignments4 Nonlinear Equations4．1 Examples4．1．1 A Transcendental Equation4．1．2 A Two-Point Boundary Value Problem4．1．3 A Nonlinear Integral Equation4．1．4 s-Orthogonal Polynomials4．2 Iteration， Convergence， and Efficiency4．3 The Methods of Bisection and Sturm Sequences4．3．1 Bisection Method4．3．2 Method of Sturm Sequences4．4 Method of False Position4．5 Secant Method4．6 Newton's Method4．7 Fixed Point Iteration4．8 Algebraic Equations4．8．1 Newton's Method Applied to an Algebraic Equation4．8．2 An Accelerated Newton Method for Equations with Real Roots4．9 Systems of Nonlinear Equations4．9．1 Contraction Mapping Principle4．9．2 Newton's Method for Systems of Equations4．10 Notes to Chapter 4Exercises and Machine Assignments to Chapter 4ExercisesMachine AssignmentsSelected Solutions to ExercisesSelected Solutions to Machine Assignments5 Initial Value Problems for ODEs： One-Step Methods5．1 Examples5．2 Types of Differential Equations5．3 Existence and Uniqueness5．4 Numerical Methods5．5 Local Description of One-Step Methods5．6 Examples of One-Step Methods5．6．1 Euler's Method5．6．2 Method of Taylor Expansion5．6．3 Improved Euler Methods5．6．4 Second-Order Two-Stage Methods5．6．5 Runge-Kutta Methods5．7 Global Description of One-Step Methods5．7．1 Stability5．7．2 Convergence5．7．3 Asymptotics of Global Error5．8 Error Monitoring and Step Control5，8．1 Estimation of Global Error5，8．2 Truncation Error Estimates5，8．3 Step Control5．9 Stiff Problems5，9．1 A-Stability5．9．2 Pad6 Approximation5．9．3 Examples of A-Stable One-Step Methods5．9．4 Regions of Absolute Stability5．10 Notes to Chapter 5Exercises and Machine Assignments to Chapter 5ExercisesMachine AssignmentsSelected Solutions to ExercisesSelected Solutions to Machine Assignments6 Initial Value Problems for ODEs： Multistep Methods6．1 Local Description of Multistep Methods6．1．1 Explicit and Implicit Methods6．1．2 Local Accuracy6．1．3 Polynomial Degree vs． Order6．2 Examples of Multistep Methods6．2．1 Adams-Bashforth Method6．2．2 Adams-Moulton Method6．2．3 Predictor-Corrector Methods6．3 Global Description of Multistep Methods6．3．1 Linear Difference Equations6．3．2 Stability and Root Condition6．3．3 Convergence6．3．4 Asymptotics of Global Error6．3．5 Estimation of Global Error6．4 Analytic Theory of Order and Stability6．4．1 Analytic Characterization of Order6．4．2 Stable Methods of Maximum Order6．4．3 Applications6．5 Stiff Problems6．5．1 A-Stability6．5．2 A（c0-Stability6．6 Notes to Chapter 6Exercises and Machine Assignments to Chapter 6ExercisesMachine AssignmentsSelected Solutions to ExercisesSelected Solutions to Machine Assignments7 Two-Point Boundary Value Problems for ODEs7．1 Existence and Uniqueness7．1．1 Examples7．1．2 A Scalar Boundary Value Problem7．1．3 General Linear and Nonlinear Systems7．2 Initial Value Techniques7．2．1 Shooting Method for a Scalar Boundary Value Problem7．2．2 Linear and Nonlinear Systems7．2．3 Parallel Shooting7．3 Finite Difference Methods7．3．1 Linear Second-Order Equations7．3．2 Nonlinear Second-Order Equations7．4 Variational Methods7．4．1 Variational Formulation7．4．2 The Extremal Problem7．4．3 Approximate Solution of the Extremal Problem7．5 Notes to Chapter 7Exercises and Machine Assignments to Chapter 7ExercisesMachine AssignmentsSelected Solutions to ExercisesSelected Solutions to Machine AssignmentsReferencesIndex