Instructions: a ) Show your work neatly and concisely and put a box around each nal solution, if applicable. b ) No credit will be given for a nal solution if there is no indication of how it was derived. c ) Partial credit will be given even if your nal solution is incorrect. 1. [10 points] True or False: All positive integers are represented exactly by IEEE double precision oating point numbers. If true, provide proof. If false, provide the smallest counter-example. 2. Given x,y R and x = (x), y = (y) we know (from class) that the relative errors induced by the roundo are bounded by the machine precision = 253 so that |xx| |x| |yy| |y| a ) [15 points] Based on this, show that for some xed constant K > 0, the relative error for the product xy = (xy) may be bounded as |xyxy| |xy| K Hint: You may nd it convenient to let x = x+x and y = y+y where x and y are the (absolute) roundos. b ) [10 points] Similarly investigate the relative error for the sum x + y = (x + y). Discuss why a similar bound does not exist for this case. 3. Consider the quadratic equation ax2 + bx + c = 0 with a = 1, b = 100 + 1014 and c = 1012 1 a ) [5 points] Find the exact roots of the quadratic equation. b ) [5 points] Using the formula x1,2 = bb2 4ac 2a , compute the roots on a computer. c ) [5 points] What is the relative error? d ) [5 points] Show that by rationalizing the numerator of the formula in part b we get to x1,2 = 2c bb2 4ac e ) [5 points] Using the rationalized formula repeat parts b and c. What do you observe? 4. Consider the linear system 1.130x6.990y = 14.20 1.013x6.099y = 14.22 a ) [5 points] Use Matlab Linsolve or rref command to nd the approximate solution of the system. b ) [5 points] Find the norm residual |Axb|. c ) [6 points] Using four-digit rounding arithmetic, solve the linear system. d ) [6 points] Using four-digit chopping arithmetic, solve the linear system.