STM3CS ASSIGNMENT 1, 2020
Upload your PDF to the LMS by 4.00 pm on Tuesday the 17th of March. You will acknowledge authenticity electronically when submitting (no written statement or form needed).
Your answers should be self contained and comprehensible to a reader who does not have access to the questions. The context should be presented discursively, as in a professional paper and lead the reader to your presented solution. Where questions ask you to use software, you should include a printout or screenshot as part of your answer or a properly documented appendix. You will be assessed on the correctness (75%), and presentation and communication (25%) of your work in the assignment. There are some examples of excellent and reasonable presentation in the assignments section.
1. Use Gauss-Jordan elimination in R or MATLAB to find all solutions to the following systems of equations. Interpret the resultant matrix by writing down the solution set and give geometric interpretation of the solution set. (For example, if a solution set contains two parameters then it corresponds to a plane in 4-dimensional space). Be sure to include MATLAB/R input and output. (Note that the systems have been created so you can recycle most of your input.)
For the system (a) give the answer in a fraction form. In (b-d) give the approximate answer rounded to 8 decimal places.
In submitting your work, you are consenting that it may be copied and transmitted by the University for the detection of plagiarism.
(a)
(c)
−2a+5b+3c− d=10 2a−3b+4c+5d=3 −8a−4b−2c+3d=−2 7a+2b− c−4d=−3
−2a+5b+3c− d=10 −8a−4b−2c+3d=−2 18a+3b+ c−5d=−6 6a+9b+5c−4d=12
−2a+ 5b+3c− d=10 2a− 3b+4c+5d=3 8a− b −3d=−2 6a−15b−9c+3d=30
−2a+5b+3c− d=10 2a−3b+4c+5d=3 −8a−4b−2c+3d=−2 18a+3b+ c−5d=−6
2. Consider the iterative scheme Sn = 2Sn−1 + nSn−2 − 5Sn−3 , n 4.
(a) Manually, apply the relation to find the first 7 terms of the sequence if the initial conditions
areS1 =1,S2 =2andS3 =3(noteyoualreadyhavethefirst3sojust4moretodo).
(b) Write a documented function script that finds the first n terms of the above sequence for any user defined initial conditions and use this to verify your answers to (a). (You could modify and fully document a copy of your script from Lab 2.)
(c) Use your function to find the first 12 terms of the sequences with initial conditions; (i) S1 =4,S2 =2andS3 =−3 (ii) S1 =2,S2 =−6andS3 =5
(iii) S1 =−3,S2 =1andS3 =0.
3. Consider the function f (x) = 4√x − 3
(a) Find the fixed points of f exactly algebraically
(b) Use the derivative test to determine the nature (repelling or attracting) of the fixed points.
(c) Starting at x0 = 8,
(i) give detailed calculations of the first four steps, x1, x2, x3 and x4 of the fixed point iteration, working correct to 4 decimal.
(b)
(d)
(ii) Modify (and add detailed comments to) your fixed point function from Lab 2 for this case to give the first 12 iterates of the fixed point iteration (correct to 14 decimal places) and use this to give an approximation to the fixed point. Compare the final iterate with the appropriate fixed point from part (a) and comment on the accuracy of the final iterate . (You should include the modified function as part of your answer or as a properly refer- enced appendix.)
(iii) Using the function from (ii) give an approximation to the fixed point after 50 iterations (correct to 14 decimal places). Comment on the accuracy of the obtained result in comparison with the result from part (ii).
(d) Now apply the function from (c(ii)) for the starting point x0 = 1.1 to find an approximation to the fixed point after 30 iterations (correct to 14 decimal places). Using the obtained result discuss the properties of the attracting and repealing fixed points of a map.