DATA301-26S1 Big Data Computing and Systems Project | UCNZ

DATA301 Big Data Computing and Systems

DATA 303 Tutorial 1

1. Calculate the 1, 2, and ∞ norms of the following vector
8
v =
1
1

2. Let x = (x1 x2) be a vector in R2. Assume for simplicity that |x1| |2. Show that
(a) ||||||||2;
(b) ||||2||||1; and
(c) ||x||12||||∞.
NB: since norms are always non-negative, these inequalities are equivalent to the inequalities you get by squaring both sides, where the direction of the inequality stays the same.

3. Find two different unit vectors u and w in R2 such that
||u + w|| ∞ = ||u||∞ + ||W||∞

4. Find two different unit vectors u and w in R2 such that

5. Using the triangle law, show that
||u + w||1= ||u||1 + ||||1
||a|| – ||b|| ≤ ||a – b|| ≤ ||a|| + ||b||

6. The surface of a lake has ice on it. The water temperature is 0°C. The air temperature above the lake is below zero. Ice formation requires the heat released by the freezing process must travel from the water/ice boundary through the ice layer to escape in the sub-zero temperature air. This heat flow is inversely proportional to the ice thickness y(t), where t is time. Hence y(t) satisfies the differential equation
dy dt y
(1)
where k is an unknown positive constant dependent on air temperature amongst other things. Ice starts forming on the lake at time t = 0 giving rise to the initial condition y(0) = 0.
(a) Find the value of A≥ 0 in terms of k such that the function y = A√t is a solution to (1). Note that this solution satisfies the initial condition y(0) = 0.
(b) Measurements of the ice thickness at various times yield the data
(ti, yi) = (1, 11); (4, 16); and (9,27)
Formulate a linear program which finds the least max norm fit of the model y =
Avt
to this data.

(c) Solve your LP graphically. What estimate of the value of k does your LP solution give?

7. A person wishes to model rainfall data of the form (si, ti, yi) where y; is the measured rainfall at the point (s, t) and s; and t, are the distance East and distance North from a reference point. The person wishes to fit a function of the form
y = F(s, t, x) = x1 + x28 + x3 + x48t
where 1,..., 4 are unknown parameters.
(a) Find the residual r at the ith data point;
(b) State the problem of finding the best 1-norm fit to the data;
(c) Express the problem of finding the best 1-norm fit to the data as a linear programming problem.