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).

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 grid to embed.

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.