Linear Algebra

68 Chapter One. Linear Systemsworked hard to counter what can go wrong.Exercises1 Using two decimal places, add 253 and 2/3.2 This intersect-the-lines problem contrasts with the example discussed above.(1, 1)x + 2y = 33x − 2y = 1Illustrate that in this system some small change in the numbers will produce onlya small change in the solution by changing the constant in the bottom equation to1.008 and solving. Compare it to the solution of the unchanged system.3 Solve this system by hand ([Rice]).(a) Solve it accurately, by hand.four significant digits.0.000 3x + 1.556y = 1.5690.345 4x − 2.346y = 1.018(b) Solve it by rounding at each step to4 Rounding inside the computer often has an effect on the result. Assume that yourmachine has eight significant digits.(a) Show that the machine will compute (2/3) + ((2/3) − (1/3)) as unequal to((2/3) + (2/3)) − (1/3). Thus, computer arithmetic is not associative.(b) Compare the computer’s version of (1/3)x + y = 0 and (2/3)x + 2y = 0. Istwice the first equation the same as the second?5 Ill-conditioning is not only dependent on the matrix of coefficients. This example[Hamming] shows that it can arise from an interaction between the left and rightsides of the system. Let ε be a small real.3x + 2y + z = 62x + 2εy + 2εz = 2 + 4εx + 2εy − εz = 1 + ε(a) Solve the system by hand. Notice that the ε’s divide out only because there isan exact cancellation of the integer parts on the right side as well as on the left.(b) Solve the system by hand, rounding to two decimal places, and with ε = 0.001.