# programming exercise 1.

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

Use any language or library. The assignments are due by midnight of 3.14 (Monday).

#### 3 credits students

Consider the piecewise linear function $$f$$ on $$[-4,4]$$
with nodes at the integer points $$-4,-3,\ldots, 3, 4$$ (it means that the function is continuous and linear on each interval $$[k,k+1]$$, such that $$f(k)$$ is the $$(k+5)$$-th digit of your UIN.

Piecewise linear function (red) is approximated (not especially well) by a polynomial (blue).

Find the polynomial
$p=a_o+a_1x+\ldots+a_4x^4$
of degree $$4$$ minimizing the mean square error
$\int_{-4}^4 |p(x)-f(x)|^2 dx.$
Submit the solution in the following format:

• text file with the first line giving the coefficients of $$f$$,

[a_o a_1 a_2 a_3 a_4]

followed by

• the code and
• the execution output.
• Also attach a pdf  where $$f$$ and $$p$$ are plotted on $$[-4,4]$$ simultaneously.

#### 4 credits students

Consider the spring networks: a network of points, $$p_{ij}, i=0,\ldots,10, j=0,\ldots,10,$$, where two points $$p_{ij}$$ and $$p_{kl}$$ are connected by a spring if either $$i=k, |j-l|=1$$ (horizontal neighbors) or $$|i-k|=1, j=l$$ (vertical neighbors).

Rectangular network to embed.

A configuration is a 2D placement of the network nodes, that is an assignment of two coordinates,
$(x_{ij}, y_{ij})$
to each node $$p_{ij}$$.

The task is to minimize
$f=\frac12\left(\sum_{(i,j)\sim (k,l)}(x_{ij}-x_{kl})^2+(y_{ij}-y_{kl})^2\right),$
(the summation is over all neighboring pairs of nodes), subject to constraints
$x_{0,0}=0,y_{0,0}=0,x_{0,10}=0,y_{0,10}=10,x_{10,0}=10,y_{10,0}=0,x_{10,10}=10,y_{10,10}=10,$
and
$x_{5,5}=A_e, y_{5,5}=A_o,$
where $$A_e$$ is the sum of digits of your UIN at even places, and $$A_o$$ is the sum of digits of your UIN at odd places.

Submit the solution in the following format:

• text file with the coefficients $$x_{ij}$$:

x=[x(0,0) x(0,1) ... x(0,10); x(1,0) ... x(10,10)]

followed by coefficients $$y_{ij}$$:

y=[y(0,0) y(0,1) ... y(0,10); y(1,0) ... y(10,10)];

• the code and
• the execution output.

### 16 Responses to programming exercise 1.

1. John February 27, 2016 at 10:46 am #

When is this due? Thanks.

• yuliy February 29, 2016 at 10:00 pm #

Due date posted.

2. George February 27, 2016 at 6:00 pm #

Hello professor.

One question about the 4 credit problem. What is the set in which the assigned coordinate numbers should belong? Integer pairs in [0,10]x[0,10]?

• yuliy February 29, 2016 at 10:01 pm #

x’s and y’s are real numbers.

3. Jiading February 29, 2016 at 2:07 pm #

Hello, professor! I wonder in the 4-credit problem, should I create the configuration table myself? If so, is there any constraints to follow? Or should it be completely random?

• yuliy February 29, 2016 at 10:02 pm #

You have to find the optimal x’s and y’s, not random.

4. Ravi March 1, 2016 at 4:46 pm #

Hello Professor Yuliy,

I have a doubt regarding the Programming Exercise-01 -4 credits problem. Actually, I could not understand how each x and y have i,j co-ordinates. Are there 2 different planes, one plane called as X plane and the other Y-plane with A_e and A_o separately for each of them respectively. Then the springs would be along X-plane and Y-plane separately without interfering each other. It looks like a finite element topology optimization problem. Even the stiffness of the spring is not given., does that mean that I have to assume a stiffness with the co-ordinates at the 4 corners and the center fixed as per the problem constraints.

If possible can you elaborate on the problem?

• yuliy March 6, 2016 at 11:56 pm #

x(i,j) and y(i,j) are variables indexed by i,j.

Stiffnesses of the springs are equal.

5. Anon March 5, 2016 at 4:58 pm #

I have a question about the submission format.

It says to submit it by:
x=[x(0,0) x(0,1) … x(10,0); x(1,0) … x(10,10)]

but should this be
x=[x(0,0) x(0,1) … x(0,10); x(1,0), x(1,1), …, x(1,10); x(2,0), x(2,1), …, x(2,10); x(3, 0) … x(10,10)]

• yuliy March 6, 2016 at 11:55 pm #

Thank you! corrected.

6. Chen March 5, 2016 at 7:07 pm #

For credit 4 problem, we can solve x and y separately as they are independent. Am I right?

• yuliy March 6, 2016 at 11:53 pm #

Yes.

7. Ann March 8, 2016 at 4:53 pm #

Hi Professor, how should we submit our assignment? There seems to be no place to submit it on compass2g.

• yuliy March 14, 2016 at 12:08 am #

Compass has the assignment now.

8. Quan March 11, 2016 at 9:53 pm #

Hi Professor,

Is iterative method for the 3 credits problem acceptable? Am I supposed to find a better algorithm in order to get full credits?

• yuliy March 14, 2016 at 12:08 am #

Yes – but you will have to indicate why you believe your solution is a good one (criteria of convergence).