Warm-up

• Name, area, and a favorite textbook (take notes!)
  • If you are elsewhere, we’ll find someone for you to do this with!

Logistics

• Waitlist notes
• Time zone issues
• Julia and other languages
• Readings

Optimization background

• Variational notation
• Reminder that Lagrange multipliers exist!
• Direct solves, fixed point iterations
• Models of functions and models of solver dynamics

Least squares

• Normal equations and calculus
• Alternate inner products
• Cholesky, QR, and SVD
• Aside re min norm connection

Activity (submit a notebook or PDF for points)

• Show that if $\phi : \mathcal{V} \rightarrow \mathbb{R}$ is a quadratic form, then $a(v,w) = (\phi(v+w)-\phi(v)-\phi(w))/2$ is a bilinear form for which $\phi(v) = a(v,v)$.
• Show that adding rows to a matrix (e.g. more data in a least squares problem) can only increase the smallest singular value.
• Write a code to minimize $\sum_{j=0}^N |p(x_j)-\cos(x_j)|^2$ where $x_j$ are points in a uniform mesh on $[-\pi, \pi]$, with $p(x) = c_0 + c_1 x^2 + c_2 x^4 + c_3 x^6$.
• Argue that the coefficients converge to the coefficients that minimize $\int_{-\pi}^\pi |p(x)-\cos(x)|^2 \, dx$?
• Design (and answer) your own question that would be appropriate for this type of exercise!