# 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 ().
Two good samples of student project reports:
Due date: November 16!
Good samples of project reports:
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:
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
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