Material for Exam 2:

NOTE: since early October, I have been posting Lecture recaps after each lecture as well as handouts on the Lectures page . There will be some questions on the exams that will test your knowledge and understanding of basic Matlab commands and objects. The problem sets should have prepared you sufficiently for this.

• Curve-fitting
  • Best fit in Least-Squares sense
  • Polynomial fits (Matlab's polyfit)
  • Linear curve fitting vs. Nonlinear curve fitting
    • Linear : y(t)= a t² + b t + c or y(t)= a sin(t)+ b cos (2 t) + c sin(2 t) , ...
    • Nonlinear: y(t)= a sin(b t)+ c cos (d t) or y(t)= a e ^{b t}+ c e ^{d t} , ...
  • How to use Matlab for non-polynomial but linear fits ( a=A\y , What are a, A and y?)
• Linear Systems of Equations
  • Matrix-vector formulation A x = b
  • Solving linear systems using Matlab: x=A \ b
  • Defining a matrix in Matlab
• Data Analysis, Random Variables
  • Uniform distribution and Gaussian distribution (Matlab's rand, randn, erf)
  • Mean, variance, standard deviation, min, max (Matlab's mean, std, min, max)
  • Histograms (Matlab's hist)
• Numerical quadrature
  • Riemann sum, Trapezoidal and Midpoint rule (Matlab's trapz )
  • Simpson's rule and relation to trapezoidal (Matlab's quad )
  • Monte-Carlo integration
  • Integration given only discrete data points (Matlab's trapz + splineq )
• Numerical solution of Initial Value Problems
  • Initial value vs. boundary value problems
  • Euler's method: convergence, accuracy, stability
  • Runge-Kutta methods of 2nd and 4th order (Matlab's ode23, ode45 )
  • How to rewrite differential equations as a 1st order system.
  
  

  ---

  Fabian Waleffe
  1998-11-17