# 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 #