应用数学导论 2010-2011秋季学期 2009级数理科学

任课老师

Projects
Matlab tutorial
Project 1. Interpolation and approximation (Description, lagrange.m, leastsquare.m). 
       Two good samples of student project reports: Wu Zhuojie, and Zhang Yichi
Project 2. Numerical quadratures (Description)
        Due date: November 16!
        Good samples of project reports: project2_mayao.pdf, project2_suchunzhi.pdf, project2_tanglingpeng.pdf, project2_wuzhuojie.pdf
        Euler-Maclaurin like formula for composite Simpson's rule (Most of students found the superconvergence phenomenon of the T2 numerical quadrature  in Project 2. Click the link for an interesting analysis for this phenomenon.)
Project 3. Numerical linear algebra (Description)
        Due date: December 2.
        Good samples: project3_HanLi.pdf, project3_MaoQitao.pdf, project3_YuYiwei.pdf
Project 4. Random number generator (Description)
        Due Date: December 16.
Outline of Contents:
Chapter 0.              Introduction and overview (Notes)
Chapter 1.              Interpolation and approximation(Notes)
              Polynomial interpolation (Lagrange, Newton, Remainder theorem)
         Runge phenomenon,
         piecewise polynomial approximation, splines (curve fitting),
         orthogonal polynomials, L^2 approximation
Chapter 2.              Numerical integration and differentiation (Notes)
         central, forward, and backward difference, accuracy, extrapolation
         Trap rule, Euler-Maclaurin formula, Simpson's rule 
         Gaussin quadrature
Chapter 3.              Iteration and convergence (Notes)
          solving linear and nonlinear equations,
          Jacobi, GS, steepest decent,
          Newton's iteration, etc.
Chapter 4.              linear algebra and MATLAB (Notes)
         tridiagonal matrix, LU and QR decomposition
         power method, QR iteration for eigenvalues
Chapter 5.              elementary probablility and sampling
          elementary prob, conditional prob,
          expectations, moments, variance,
          random number generators,
          illustration of law of large numbers and CLT
          simple Monte Carlo, variance, and variance reduction
          statistics
          random matrices
Chapter 6.              asymptotic analysis
          dominated balance, Laplace integrals, oscillatory integrals,
          Stirling formula, Gamma function,
          power series solutions of ODEs,
          matched asymptotics, removing secular terms
References:
Zhang and Li, Numerical Analysis
 
Conte and de Boor, Elementary Numerical Analysis
 
Quarteroni and Saleri, Scientific Computing with MATLAB
 
Numerical Recipes
 
Bender and Orszag, Advanced mathematical methods for scientists and engineers.
 
Nayfeh's book, and other books on asymptotics, Cole's book
 
MATLAB User's Guide