ECE 490, week of Apr 22

We will be covering some lower bounds for first order methods, and matching them algorithms of convex minimization, – based on Nesterov, ch. 2.

Class notes (pretty incomplete, to be used just as a study guide), here and here.

Homework, due by Midnight May 1st:

  1. Consider the 4-dimensional space spanned by the polynomials
    $$
    p(x)=ax^6+bx^4+cx^2+d.
    $$Find, using the Gram-Schmidt orthogonalization procedure,
    the orthonormal basis, if the scalar product is given by
    $$
    (p_1,p_2)_Q=\int_{0}^\infty e^{-x}p_1(x)p_2(x) dx.
    $$
  2. Consider quadratic form in $\Real^{100}$ given by
    \[
    Q=\left(
    \begin{array}{ccccc}
    2&1&1&\cdots&1\\
    1&2&1&\cdots&1\\
    1&1&2&\cdots&1\\
    \vdots&\vdots&\vdots&&\vdots\\
    1&1&1&\cdots&2\\
    \end{array}
    \right)
    \]
    (i.e. \(2\) on the diagonal,\(1\) elsewhere).

    1. Find the spectrum of \(Q\).
    2. Solve
      \[
      Qx=b,
      \]
      where \(b=(1,0,\ldots,0)\).
    3. Run two iterations of the conjugate gradient method for this system.
  3. Consider \(F(z)=z^3-9z+8\).
    1. Find roots of \(F\).
    2. Find (numerically) the for which starting points on the real line, the Newton method will converge to either of the roots.

Solutions

13 Responses to ECE 490, week of Apr 22

  1. Ruby Zhuang April 24, 2019 at 4:00 pm #

    For question 3 b), what are we finding?

    • yuliy April 24, 2019 at 4:48 pm #

      All the starting points (on the real line) from which Newton algorithm converges to each of the roots. It will give you a partition of the line into several intervals; you need to find them.

  2. Zhijian Yang April 25, 2019 at 2:26 pm #

    For the project presentation, how long shall each group present? Approximately how many slides for each group? Thanks.

    • yuliy April 29, 2019 at 9:43 am #

      5 minutes

  3. Guoming Yang April 26, 2019 at 3:38 pm #

    For Q2 is Q 100×100 or 10×10?

    • yuliy April 29, 2019 at 9:43 am #

      100×100

  4. Gwanhee Lee April 29, 2019 at 10:33 am #

    What does “numerically” mean in Question 3(b)?

  5. Ruby Zhuang April 30, 2019 at 1:48 pm #

    I can’t seem to find the submission portal on Gradescope.

    • Khaled Alshehri April 30, 2019 at 6:19 pm #

      You should be able to find it by now.

  6. Yiqing Xie May 3, 2019 at 10:13 pm #

    I don’t understand what a “comprehensive exam” is. Will the final exam contain materials before midterm? Thanks!

    • yuliy May 5, 2019 at 9:52 pm #

      Comprehensive means that the exam will contain material from all of the course.

  7. Ruby Zhuang May 4, 2019 at 11:45 pm #

    Could you post the solutions to HW 5? And I think the class notes are not the ones for this week. Thanks!

    • yuliy May 5, 2019 at 9:55 pm #

      Solutions come soon. Notes uploaded.

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