Material for Exam 1:

• Pitfalls of numerical computing
  • finite precision arithmetic, roundoff error, machine precision ( eps), numerically stable vs. unstable calculations, relative error (how to decide you have a root, stopping criterion)
  • Example: computing with inside or outside polygons (Lectures + PS1), computing derivatives numerically (lectures + PS2)
• Root-finding
  • Bracketing, bisection, Newton's method, secant method = linear interpolation, Matlab's fzero (uses bracketing, bisection and/or secant).
    What is the idea behind Newton's method? behind the secant method?
    graphical interpretation of Newton and secant methods
    Why is Newton so fundamental? (easily generalized to complex numbers, multiple variables) How fast does Newton get to a root? How fast is bisection?

    How to choose a guess? (Lectures + PS1 + PS2)
• Plotting data
  • Lin-Lin, Log-Log, Lin-Log and Log-Lin Plots. why? when? how? Matlab's plot, loglog , semilogx , semilogy
  • Power law: y(x)= a x^b (cf. Fractals, self-similarity, PS2)
  • Exponential law: y(x)= a e^{b x} or y= a b ^x (cf. Radioactive decay, viscous decay, PS3)
• Curve-fitting and interpolation
  • Best fit in Least-Squares sense
    what does this mean? How to define the error function, where do equations come from?

    Polynomial fits, Matlab's polyfit (lectures + PS3)

    Polynomial fit of highest order vs. Piecewise polynomial interpolation: linear interpolation, Spline interpolation.

    What do those words mean? Matlab's spline (note also Matlab's interp1 and interp, etc...) (lectures + PS3)

    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 fits (lectures + PS3) x=A\b , curvefit (in optimization toolbox, see also "optdemo")
• Linear Systems of Equations
  • The basic idea: (Gaussian) elimination
  • Matrix-vector formulation A x = b
  • Solving linear systems using Matlab: x=A \ b

REVIEW: TUESDAY OCT 13th lecture will be devoted to a review of this material. However YOU HAVE TO ASK QUESTIONS. I will not cover new material nor will I prepare a review for sleeping students. I'll be there to answer your questions. We probably should do this in 1227 Engr Hall so we have more room to write.