Linear Algebra

Topic: Accuracy of Computations 71In addition to returning a result that is likely to be reliable, most well-donecode will return a number, called the conditioning number that describes thefactor by which uncertainties in the input numbers could be magnified to becomeinaccuracies in the results returned (see [Rice]).The lesson of this discussion is that just because Gauss’ method always worksin theory, and just because computer code correctly implements that method,and just because the answer appears on green-bar paper, doesn’t mean that theanswer is reliable. In practice, always use a package where experts have workedhard 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]).0.000 3x + 1.556y = 1.5690.345 4x − 2.346y = 1.018(a) Solve it accurately, by hand. (b) Solve it by rounding at each step tofour significant digits.4 Rounding inside the computer often has an effect on the result. Assume thatyour machine 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 thereis an exact cancelation of the integer parts on the right side as well as on theleft.(b) Solve the system by hand, rounding to two decimal places, and with ε =0.001.