1. Solve Heath Exc 1.4 (page 42). Show your calculations.
Consider the problem of evaluating the function sin(x), in particular, the propagated data error, i.e., the error in the function value due to a perturbation h in the argument x.
(a) Estimate the absolute error in evaluating sin(x).
(b) Estimate the relative error in evaluating sin(x).
(c) Estimate the condition number for this problem.
(d) For what values of the argument x is this problem highly sensitive?
2. Solve Heath Exc 1.6 (page 42). Show your calculations.
The sine function is given by the infinite series sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ... .
(a) What are the forward and backward errors if we approximate the sine function by using only the first term in the series, i.e., sin(x) \approx x, for x = 0.1, 0.5, and 1.0?
(b) What are the forward and backward errors if we approximate the sine function by using the first two terms in the series, i.e., sin(x) \approx x-x^3/6, for x = 0.1, 0.5, and 1.0?
3. Solve Heath Computer Problem 2.3 (page 100)
, and answer the following two questions:
(a). For a 5% relative error in the load vector b, what is the error in the output? Do a condition number analysis. (
The condition number analysis described in textbook can be found here )
(b). If this is a bridge, why do we anticipate more uncertainty in the load vector b than the matrix A?
4. While basking on Seba beach of Wabamun lake (west of Edmonton), in the distance I can just barely see the very top of the three power plant chimneys over at the eastern end of the lake. Determine the height of the chimney's with error margins. Assume the earth radius r=6366km, and make appropriate measurements with error margins in the map below. (You can assume the powerplant is in Kapasiwin).
5. Farmer McDonald runs an organic and ecological operation on his rural Alberta property. Instead of burning non-renewable fossil fuels, he takes advantage of the heat emitted by the animals. Each cow emits 300 W, and each pig 250 W. He is contemplating building a new three room barn (in the third room is Hay, which emits no heat), and has enlisted the help of the numerical methods students to help with the calculation of the temperature he can expect in each room. The architect has provided the following drawing of the barn plus inhabitants:
Let the outside temperature T_0 be a balmy -20C. The heat conductance between the various rooms are as follows: k12 = k13 = k23 = 45 W/C, k10 = k20 = 25 W/C, k30 = 105 W/C.
From the laws of physics we know that energy is conserved. Hence
for each room, the sum of the total heat energy emitted by the animals
and the heat energy leaking through the walls is zero.
The heat leakage through a wall depends on two things, the temperature
difference across the wall and the heat conductivity of the wall
and is given by P = k(To-Ti).
Hence for each room we can derive one heat balance equation in the unknown T1,T2,T3. Together we have three equations, which can be solved for the temperature in each room. Derive and solve these equations.
6. Solve Heath Exc 2.17 (page 97) Note: Show your solution steps in obtaining the LU factorization. Ie you cannot just type it into matlab, but have to hand-calculate this small LU factorization... (But you may use matlab to check your result.)
Write out the LU factorization of the following matrix (show both the L and U matrices explicitly):
[ 1 -1 0]
[-1 2 -1]
[ 0 -1 1]
7(Optional). Open ended question: When was this satellite picture of campus taken? Try to find cues in the image, calculate date and time. Describe your assumptions, conclusions and error analysis.
