We have A 1 0 0 106 1 0 108 0 1 What is the condition numbe

We have A = (1 0 0 -10^-6 1 0 10^8 0 1). What is the condition number kappa_1 (A)? Suppose we solved the system Ax = b using two different algorithms and obtained two computed solutions x and x. To check the accuracy of the computed solutions, we computed the residuals r = b - Ax and r = b - Ax and found out that ||r||_1 lessthanorequalto ||r||_1. Can you conclude that the solution x is more accurate than x, i.e., closer to the actual solution? If your answer is yes, prove it. If your answer is no, present an example (i.e., present a specific vector b) and show that the conclusion is not true with an explanation on why.

Solution

There are many different types of linear systems or equations each having their own special characteristics. Thus we can hardly expect any particular direct method, like Gauss elimination, to be the best possible method to use in all circumstances. Moreover, ir we do use a direct method, our computed solution will almost certainly be incorrect due to round-off error. Therefore we need some way to determine the size or the error or any computed solution and also some way to improve this computed solution. In this section, we will briefly develop a little of the theoretical background that is needed both for the analysis of errcrs and for the analysis of the \"iterative\" algorithms of Section 2.4 which generate a seq uence of approximate solutions to Ax = b. The subject of error analysis must be approached somewhat carefully, since a particular computed solution (say xJ to Ax = b may be considered badly in error or quite acceptable depending on how we intend to use the vector XC\" For example, let x, denote the \"true\" solution of Ax = b and let r = AXe - b be the residual vector. Then, r = AXe - b = AXe - Ax, or

Xc - x, = A - 1r. (2.34) If A -1 has some very large entries, then Eq. (2.34) demonstrates the fact that the residual vector, r, might be small, and yet Xe might be quite far from x,, Depending on the context of the problem that gave rise to the equation Ax = b, we might be happy with having AXe - b small or we might need Xe to be near x,,