Trial Exam

Time: 3 hrs
Tutor: Prof. Dr P.M.A. Sloot

Evaluation: All questions are worth 2 points. 


Question[1]
In the syllabus we mentioned the errors in the software of the'patriot missile defence' system used at the Gulf-war. At he original software for this system one used for a representation of 0.1 (decimal), a binary (base 2) representation of 23 bits after the binary dot.


1a:How many bits in base 2 are necessary for a representation of 100 hours in units of 0.1 (decimal) seconds?
1b:What is the binary representation of 0.1 (just like the expansion we showed with the number 5.2) ?
1c:Estimate the error caused by this chopped binary representation.


Question[2]
Question 1b and 1c shows  0.1 (decimal) has a not-finite periodical  binary expansion. One can see the bits in the error are equal to the bits in 0.1 moved over 20 digits: namely: -0.1 x 2-20.
The patriot defence system follows possible targets through dividing elapsed time by the speed of the target. Elapsed time was being calculated by: 
t1 = fl(0.1) x n1; 
t2 = fl(0.1) x n2; 
Δt = t2 -t1. 
fl(x) is the machine representation of the number x. fl(0.1) = 0.1 x (1-2-20) and n1, n2  are integers, obtained from the computer clock in units of 0.1 (decimal) seconds.

2a: What will be the relative error in fl(0.1)?
2b: Assume that after initialisation of the computer clock on 0 (exactly), the clock passed 8 hours after n1 was spotted and after two more seconds n2 was spotted. What will be the error in Δt because of the use of  fl(0.1) instead of 0,1 and what will be the relative error?


Question[3]
Some L-systems, like for instance the ones beneath, form a regular fractal structure.
ω: F - F - F - F
P: F= F - F + F + FF - F - F + F


3a: Explain how to determine the fractal dimension of a regular and an irregular fractal in the 2 dimensional plane.  Calculate the fractal dimension of the above-mentioned regular fractal.
3b: In the above-mentioned example the production rule generates per edge 8 new edges. Assume that the re-writing of one edge will cost 10 Flops. Name the axiom ω the 0-th rewriting. How much work will be the first re-writing of the axiom? And how many Flops will than be the n-th rewriting?
3c: If my computer implementation of this L-system has a performance of  40 Mflop/s, and I will do 20 rewritings, how long will it take my computer? How many hours will that be approximately?


Question[4]
4a: Why is being said that computer generated random numbers do not exist?
4b: How can I generate random numbers in the computer? Which are the parameters which will influence the quality of my random number generator?
4c: Explain how we can determine a integral with the use of random numbers.
4d: Explain  in which situation we better use numerical approximation methods of integration like "Simpson's rule"or the "Trapezium rule"  and when we are better of with stochastic approximation methods (like the Hit and Miss method and the Monte Carlo "Sample-mean" method). 


Question  [5] 
See two metal strips, one made of copper the other made of steel. Their thickness is similar. The strips are placed to each other and are being clenched  between two 'walls'. One side of the wall is being kept at 100 oC , the other at  0 oC (see figure). The metal strip is 10 cm tall, the copper strip 20 cm.

5a: Use the conduction law of Fourier to calculate the balance temperature on the crossing between steel and copper. Use the fact that you know the warmth flux ΔH in both strips has to be equal. The thermal conduction of steel is 
κsteel = 50.2 Wm-1 oC-1 
and of copper 
κcopper = 385 Wm-1 oC-1.
5b: Despite the fact the copper strip is far more taller, the temperature difference over the copper strip is smaller though than the temperature difference over the steel strip. Could you explain this?
5c: Given  a start-temperature for the entire strip (copper and steel) of Tstart. Describe the algorithm for the calculation of the temperature on a random location in the combined strip at a random moment of time. Note that the temperature on the crossing may not be assumed as a constant.