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2012WT2midterm1.solutions - 1 University of British...

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1University of British ColumbiaMath 307, Section 201Midterm 1Date:March 6, 2012Time:12:00 - 12:50pmName (print):Student ID Number:Signature:Instructor:Richard FroeseInstructions:1. No notes, books or calculators are allowed. A MATlAB/Octave formula sheet is provided onthe last page.2. Read the questions carefully and make sure you provide all the information that is asked forin the question.3. Show all your work. Answers without any explanation or without the correct accompanyingwork could receive no credit, even if they are correct.4. Answer the questions in the space provided. Continue on the back of the page if necessary.QuestionMarkMaximum1112838413Total40
2[3]1.(a) SupposeAis an invertible 2×2 matrix with matrix normkAk= 3 What can you sayabout the possible values ofbifb=A"10#? Isb="04#possible?"1#"1#
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(b) Suppose thatAx=bandA(x+Δx) =b+Δb. Write down the definitions of the relativeerror inxthe relative error forband the condition number cond(A). What is the basicinequality relating these quantities?
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[4](c) Suppose thatA-1= 1/3. How could you chooseband Δbsothat the relative error inxis large compared to the relative error inb? What can yousay about cond(A)?"1#"0#
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3[4]2.(a) Explain how you could decide whether there is a choice of coefficient vectora= [a1, a2, a3]Tsuch that the quadratic polynomiala1x2+a2x+a3passes through (1,0), (2,1), (3,1)and (4,2). Write down the MATLAB/Octave commands you would use to do this. (Youdo not have to carry out the computation)
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