G071503 Numerical Analysis.doc

The original Chinese preliminary version of this syllabus can be accessed from/new/student/tplan/G071503-3.htmEdited by LiPing HE and Baorui SONG in May, 2003Modified by Jin ZENG and Guobiao ZHOU in July, 2004Reviewed by Jianguo HUANGRearranged and translated into English by Weiming SHI (GTID: 902416684)on January 6, 2008Numerical AnalysisCourse Number: G071503Credits/Lecture Hours: 3.0/54Semester: FallCourse Title: Numerical Analysis Department: Department of MathematicsInstructor: guobiao zhouProspected Students: Majors in Mechanical Engineering, Electrical Engineering, Material Science and Engineering, Economics and Management, Life Science and Biotechnology, Physics, Mechanics or majors in Humanities that are required to take this course.Prerequisites: Linear Algebra, Calculus, Programming LanguageMain Topics: Numerical Algebra, Numerical Approximation, Numerical solution of nonlinear equations, Numerical integral, Numerical solution of ordinary differential equationsTextbook and Reference：1 Qinyang LI Numerical Analysis (4th Edition), Tsinghua University Press, 20052 Zhizhong SUN, Weiping YUAN, Zhenchu WEN, Numerical Analysis, Southeast University Press, 20023 J.Stoer and R. Bulirsch, Introduction to Numerical Analysis (2nd Edition), Springer-Verlag, Berlin-New York, 19934 Atkinson K E, An Introduction to Numerical Analysis, John Wiley & Sons, 1989Course Content:Part IIntroductionContent: Motivation, properties and basic requirement of computational methods; fundamental concepts of error and error analysisRequirement: Understand the role of computational methods in solving practical problems and the content, research methods, study methods and history of this course.Understand the error, the operation and analysis of the error in computational methods, the key issues in approximation calculation and the fundamental concepts like the stability of numerical algorithm, convergence and convergence rate.PART II Interpolation and Approximation2.1 Polynomial Interpolation2.1.1 Lagrange interpolation2.1.2 Newton interpolation2.1.3 Hermite interpolation2.2 Piecewise Interpolation2.2.1 Problems in polynomial interpolations2.2.2 Piecewise linear interpolation2.2.3 Piecewise cubic Hermite interpolation2.3 Cubic Spline Interpolation2.4 The Least-squares Fitting of Curves2.5 The Least-squares Approximation and Orthogonal Polynomial2.6 Best Uniform Approximation (optional)PART IIINumerical Integration3.1 Basis Ideas in Numerical Integration3.2 Newton-Cotes Formula3.2.1 Newton-Cotes Formula3.2.2 Compound Newton-Cotes Formula3.3 Variable Step Size and Richardson Acceleration Method3.4 Gaussian Quadrature Method3.4.1 Algebra Accuracy3.4.2 Gauss Integral Formula3.4.3 Gauss Point3.4.4 Properties of Gauss Integral FormulaPART IVThe Numerical Solution of Ordinary Differential Equations4.1 Eular Method and Its Variant4.2 Runge-Kutta Method4.2.1 Taylor Series Method4.2.2 Basic Idea of Runge-Kutta Method4.2.3 Second-order Runge-Kutta Method4.2.4 Fourth-order Runge-Kutta Method4.3 Convergence and Stability of Single-step Method4.4 Linear Multi-step MethodPART VNumerical Methods for Nonlinear Equations5.1 Search Method5.1.1 Step-by-step Method and Its Properties and Problems in Applications5.1.2 Bisection Method and Its Properties and Problems in Applications5.2 Iterative Method5.2.1 Fundamental Principle of Iterative Method5.2.2 Convergence and Convergence Rate of Iterative Method5.3 Newton Method and Secant MethodPART VIDirect Methods Solving Linear Equation System6.1 Gauss Elimination Method6.2 LU Decomposition Method6.3 The Norm of Vector and Matrix6.4 Error AnalysisPART VIIIterative Methods Solving Linear Equation System7.1 Basic Iterative Method7.1.1 Jacobi Iteration Method7.1.2 Gauss-Seidel Iteration Method7.2 Convergence of Iterative Method7.3 SOR MethodPART VIIINumerica